ALMOST RING THEORY

OFER GABBER AND LORENZO RAMERO

sixth (and �nal) release

Ofer Gabber I.H.E.S. Le Bois-Marie 35, route de Chartres F-91440 Bures-sur-Yvette

e-mail address: gabber@ihes.fr

Lorenzo Ramero Université de Bordeaux I Laboratoire de Mathématiques 351, cours de la Liberation F-33405 Talence Cedex

e-mail address: ramero@math.u-bordeaux.fr
web page: http://www.math.u-bordeaux.fr/∼ramero

Contents

1. Introduction 3 2. Homological theory 7 2.1. Some ring-theoretic preliminaries 7 2.2. Categories of almost modules and algebras 11 2.3. Uniform spaces of almost modules 16 2.4. Almost homological algebra 22 2.5. Almost homotopical algebra 29 3. Almost ring theory 37 3.1. Flat, unramified and étale morphisms 38 3.2. Nilpotent deformations of almost algebras and modules 40 3.3. Nilpotent deformations of torsors 46 3.4. Descent 54 3.5. Behaviour of étale morphisms under Frobenius 63 4. Fine study of almost projective modules 70 4.1. Almost traces 70 4.2. Endomorphisms of 𝔾̂_{m}. 76 4.3. Modules of almost finite rank 81 4.4. Localisation in the flat site 86 4.5. Construction of quotients by flat equivalence relations 94 5. Henselization and completion of almost algebras 99 5.1. Henselian pairs 100 5.2. Criteria for unramified morphisms 103 5.3. Topological algebras and modules 108 5.4. Henselian approximation of structures over adically complete rings 113 5.5. Lifting theorems for henselian pairs 123 5.6. Smooth locus of an affine almost scheme 130 5.7. Quasi-projective almost schemes 137 5.8. Lifting and descent of torsors 142 6. Valuation theory 147 6.1. Ordered groups and valuations 147 6.2. Basic ramification theory 155 6.3. Algebraic extensions 158 6.4. Logarithmic differentials 165 6.5. Transcendental extensions 169 6.6. Deeply ramified extensions 176 7. Analytic geometry 181 7.1. Derived completion functor 182 7.2. Cotangent complex for formal schemes and adic spaces 187 7.3. Deformations of formal schemes and adic spaces 195 7.4. Analytic geometry over a deeply ramified base 203 7.5. Semicontinuity of the discriminant 207 8. Appendix 214 8.1. Simplicial almost algebras 214 8.2. Fundamental group of an almost algebra 218 References 225 Index 227

Le bruit des vagues était encore plus paresseux, plus étale qu’a midi. C’était le même soleil, la même lumière sur le même sable qui se prolongeait ici.

A. Camus - L’étranger

1. Introduction

From a pragmatic standpoint, one can describe the theory of almost rings as a useful tool for performing calculations of Galois cohomology groups. Indeed, this is the main application of Faltings’ “almost purity theorem”, which is the technical heart of [34].

Though almost ring theory is developed here as an independent branch of mathematics, stretching somewhere in between commutative algebra and category theory, the original applications to Galois cohomology still provide the main motivation and influence largely the evolution of the subject.

It is therefore fitting to introduce the present work by reviewing briefly the main ideas behind these calculations. Let us consider first a complete discretely valued field K of zero characteristic, with perfect residue field of characteristic p > 0, and uniformizer π; we denote by K^{+} the ring of integers of K. The valuation v of K extends uniquely to any algebraic extension, and we want to normalize the value group in such a way that v(p) = 1 in every such extension.

Let E be a finite Galois extension of K, with Galois group G. Typically, one is given a discrete E^{+}[G]-module M (such that the G-action on M is semilinear, that is, compatible with the G-action on E^{+}), and is interested in studying the (modified) Tate cohomology Ĥ^{i} := Ĥ^{i}(G, M) (for i ∈ ℤ). (Recall that Ĥ^{i}(M) agrees with Galois cohomology R^{i}Γ^{G}M for i > 0, with Galois homology for i < −1, and for i = 0 it equals M^{G}/Tr_{E/K}(M), the G-invariants divided by the image of the trace map).

In such a situation, the scalar multiplication map E^{+}⊗_{ℤ}M → M induces natural cup product pairings Ĥ^{i}(G, E^{+}) ⊗_{ℤ} Ĥ^{j} → Ĥ^{i+j}. Especially, the action of (E^{+})^{G} = K^{+} on Ĥ^{i} factors through K^{+}/Tr_{E/K}(E^{+}); in other words, the image of E^{+} under the trace map annihilates the modified Tate cohomology.

If now the extension E is tamely ramified over K, then Tr_{E/K}(E^{+}) = K^{+}, so the Ĥ^{i} vanish for all i ∈ ℤ. Even sharper results can be achieved when the extension is unramified. Indeed, in such case E^{+} is a G-torsor for the étale topology of K^{+}, hence, some basic descent theory tells us that the natural map

E^{+} ⊗_{K^{+}} RΓ^{G}M → M[0]

is an isomorphism in the derived category of the category of E^{+}[G]-modules (where we have denoted by M[0] the complex consisting of M placed in degree zero).

In Tate’s paper [64] there occurs a variant of the above situation : instead of the finite extension E one considers the algebraic closure K^{a} of K, so that G is the absolute Galois group of K, and the discrete G-module M is replaced by the topological module C(χ), where C is the p-adic completion of K^{a}, whose natural G-action we “twist” by a continuous character χ : G → K^{×}. Then the relevant H^{•} is the continuous Galois cohomology H_{cont}^{•}(G, C(χ)), which is defined in general as the homology of a complex of continuous cochains. Under the present assumptions, H^{i} can be computed by the formula:

H_{cont}^{i}(G, C(χ)) := (lim_{\overleftarrow{n}} H^{i}(G, K^{a+}(χ) ⊗_{ℤ} ℤ/p^{n}ℤ)) ⊗_{ℤ} ℚ.

Let now K_{∞} be a totally ramified Galois extension with Galois group H isomorphic to ℤ_{p}. Tate realized that, for cohomological purposes, the extension K_{∞} plays the role of a maximal totally ramified Galois extension of K. More precisely, let L be any finite extension of K, and set L_{n} := L · K_{n}, where K_{n} is the subfield of K_{∞} fixed by H^{p^{n}} ≃ p^{n} · ℤ_{p}. The extension K_{n} ⊂ L_{n} is unramified if and only if the different ideal 𝒟_{n} := 𝒟_{L_{n}^{+}/K_{n}^{+}} equals L_{n}^{+}. In case this fails, the valuation v(δ_{n}) of a generator δ_{n} of 𝒟_{n} will be a strictly positive rational number, giving a quantitative measure for the ramification of the extension. With this notation, [64, §3.2, Prop.9] reads

(1.0.1)  lim_{n→∞} v(δ_{n}) = 0

(indeed, v(δ_{n}) approaches zero about as fast as p^{−n}). In this sense, one can say that the extension K_{∞} ⊂ L_{∞} := L · K_{∞} is almost unramified. One immediate consequence is that the maximal ideal 𝔪 of K_{∞}^{+} is contained in Tr_{L_{∞}/K_{∞}}(L_{∞}^{+}). If, additionally, L is a Galois extension of K, we can consider the subgroup

G_{∞} := Gal(L_{∞}/K_{∞}) ⊂ Gal(L/K)

and the foregoing implies that 𝔪 annihilates H_{cont}^{i}(G_{∞}, M), for every i > 0, and every topological L_{∞}^{+}[G_{∞}]-module M. More precisely, the homology of the cone of the natural morphism

(1.0.2)  L_{∞}^{+} ⊗_{K_{∞}^{+}} RΓ^{G_{∞}}M → M[0]

is annihilated by 𝔪 in all degrees, i.e. it is almost zero. Equivalently, one says that the maps on homology induced by (1.0.2)are almost isomorphisms in all degrees.

Tate goes on to apply these cohomological vanishings to the study of p-divisible groups; in turns, this study enables him to establish a comparison between the étale and the Hodge cohomology of an abelian scheme over K^{+}, which has become the prototype for all subsequent investigation of p-adic Hodge theory.

A first generalization of (1.0.1) can be found in the work [37] by Fresnel and Matignon; one interesting aspect of this work is that it does away with any consideration of local class field theory (which was used to get the main estimates in [64]); instead, Fresnel and Matignon write a general extension L as a tower of monogenic subextensions, whose structure is sufficiently well understood to allow a direct and very explicit analysis. The main tool in [37] is a notion of different ideal 𝒟_{E^{+}/K^{+}} for a possibly infinite algebraic field extension K ⊂ E; then the extension K_{∞} considered in [64] is replaced by any extension E of K such that 𝒟_{E^{+}/K^{+}} = (0), and (1.0.1)is generalized by the claim that 𝒟_{F^{+}/E^{+}} = F^{+}, for every finite extension E ⊂ F.

In some sense, the arguments of [37] anticipate those used by Faltings in the first few paragraphs of his fundamental article [33]. There we find, first of all, a further extension of (1.0.1): the residue field of K is now not necessarily perfect, instead one assumes only that it admits a finite p-basis; then the relevant K_{∞} is an extension whose residue field is perfect, and whose value group is p-divisible. This generalization paves the way to the almost purity theorem, of which it represents the one-dimensional case. In order to state and prove the higher dimensional case, Faltings invents the method of “almost étale extensions”, and indeed sketches in a few pages a whole program of “almost commutative algebra”, with the aim of transposing to the almost context as much as possible of the classical theory. So, for instance, if A is a given K_{∞}^{+}-algebra, and M is an A-module, one says that M is almost flat if, for every A-module N, the natural map of complexes

M ⊗^{L}_{A} N → M ⊗_{A} N[0]

induces almost isomorphisms on homology in all degrees. Similarly, M is almost projective if the same holds for the map of complexes Hom_{A}(M, N)[0] → RHom_{A}(M, N). Then, according to [33], a map A → B of K_{∞}^{+}-algebras is called almost étale if B is almost projective as an A-module and as a B ⊗_{A} B-module (moreover, B is required to be almost finitely generated : the discussion of finiteness conditions in almost ring theory is a rather subtle business, and we dedicate the better part of section 2.3 to its clarification).

With this new language, the almost purity theorem should be better described as an almost version of Abhyankar’s lemma, valid for morphisms A → B of K^{+}-algebras that are étale in characteristic zero and possibly wildly ramified on the locus of positive characteristic. The actual statement goes as follows. Suppose that A admits global étale coordinates, that is, there exists an étale map K^{+}[T_{1}^{±1}, ..., T_{d}^{±1}] → A; whereas in the tamely ramified case a finite ramified base change K^{+} → K^{+}[π^{1/n}] (with (p, n) = 1) suffices to kill all ramification, the infinite extension A → A_{∞} := A[T_{1}^{±1/p^{∞}}, ..., T_{d}^{±1/p^{∞}}] ⊗_{K^{+}} K_{∞}^{+} is required in the wildly ramified case, to kill almost all ramification, which means that the normalization B_{∞} of A_{∞} ⊗_{A} B is almost étale over A_{∞}.

Faltings has proposed two distinct strategies for the proof of his theorem : the first one, presented in [33], consists in adapting Grothendieck’s proof of Zariski-Nagata’s purity^{1}; a more recent one ([34]) uses the action of Frobenius on some local cohomology modules, and is actually valid under more general assumptions (one does not require the existence of étale coordinates, but only a weaker semi-stable reduction hypothesis on the special fibre).

As a corollary, one deduces cohomological vanishings generalizing the foregoing : indeed, suppose that the extension of fraction fields Frac(A) ⊂ Frac(B) is Galois with group G; then, granting almost purity, B_{∞} is an “almost G-torsor” over A_{∞}, therefore, for any B_{∞}[G]-module M, the natural map of complexes B_{∞} ⊗_{A_{∞}} RΓ^{G}M → M[0] induces almost isomorphisms on homology. Finally, these results can be used (together with a lot of hard work) to deduce comparison theorems between p-adic étale cohomology and deRham (or other kinds of) cohomology, for arbitrary smooth projective varieties over K. This method can even be extended to treat cohomology with not necessarily constant coefficients (see [34]), thereby providing the most comprehensive approach to p-adic Hodge theory found so far.

The purpose of our text is to fully work out the foundations of “almost commutative algebra” outlined by Faltings; in the process we generalize and simplify considerably the theory, and also extend it in directions that were not explored in [33], [34].

It turns out that most of almost ring theory can be built up satisfactorily from a very slim and general set of assumptions: our basic setup, introduced in section 2.1, consists of a ring V and an ideal 𝔪 ⊂ V such that 𝔪 = 𝔪^{2}; starting from (2.5.14) we also assume that 𝔪 ⊗_{V} 𝔪 is a flat V -module : simple considerations show this to be a natural hypothesis, often verified in practice.

The V -modules killed by 𝔪 are the objects of a (full) Serre subcategory Σ of the category V -Mod of all V -modules, and the quotient V^{a}-Mod := V -Mod/Σ is an abelian category which we call the category of almost V -modules. It is easy to check that the usual tensor product of V -modules descends to a bifunctor ⊗ on almost V -modules, so that V^{a}-Mod is a monoidal abelian category in a natural way. Then an almost ring is just an almost V -module A endowed with a “multiplication” morphism A ⊗ A → A satisfying certain natural axioms. Together with the obvious morphisms, these gadgets form a category V^{a}-Alg. Given any almost V -algebra A, one can then define the notion of A-module and A-algebra, just like for usual rings. The purpose of the game is to reconstruct in this new framework as much as possible (and useful) of classical linear and commutative algebra. Essentially, this is the same as the ideology informing Deligne’s paper [23], which sets out to develop algebraic geometry in the context of abstract tannakian categories. We could also claim an even earlier ancestry, in that some of the leading motifs resonating throughout our text, can be traced as far back as Gabriel’s memoir [38] “Des catégories abéliennes”.

In evoking Deligne’s and Gabriel’s works, we have unveiled another source of motivation whose influence has steadily grown throughout the long gestation of our paper. Namely, we have come to view almost ring theory as a contribution to that expanding body of research of still uncertain range and shifting boundaries, that we could call “abstract algebraic geometry”. We would like to encompass under this label several heterogeneous developments: notably, it should include various versions of non-commutative geometry that have been proposed in the last twenty years, but also the relative schemes of [43], as well as Deligne’s ideas for algebraic geometry over symmetric monoidal categories.

1At the time of writing, there are still some obscure points in this proof

The common thread loosely unifying these works is the realization that ”geometric spaces” do not necessarily consist of set-theoretical points, and - perhaps more importantly - functions on such ”spaces” do not necessarily form (sheaves of) commutative rings. Much effort has been devoted to extending the reach of geometric intuition to non-commutative algebras; alternatively, one can retain commutativity, but allow “structure sheaves” which take values in tensor categories other than the category of rings. As a case in point, to any given almost ring A one can attach its spectrum Spec A, which is just A viewed as an object of the opposite of the category V^{a}-Alg. Spec A has even a natural flat topology, which allows to define more general almost schemes by gluing (i.e. taking colimits of) diagrams of affine spectra; all this is explained in section 5.7, where we also introduce quasi-projective almost schemes and investigate some basic properties of the smooth locus of a quasi-projective almost scheme. By way of illustration, these generalities are applied in section 5.8 in order to solve a deformation problem for torsors over affine almost group schemes; let us stress that the problem in question is stated purely in terms of affine objects (i.e. almost rings and “almost Hopf algebras”), but the solution requires the introduction of certain auxiliary almost schemes that are not affine.

Having soared into the thin air of abstract geometry, we come back to earth in the last two chapters, which deal with applications to valuation theory and to p-adic analytic geometry : especially the reader will find there our own contributions to almost purity. This general presentation has thus come full circle, and we defer to the introductory remarks at the beginning of the respective chapters for a more detailed description of our results.

Acknowledgements The second author is very much indebted to Gerd Faltings for many patient explanations on the method of almost étale extensions. Next he would like to acknowledge several interesting discussions with Ioannis Emmanouil. He is also much obliged to Pierre Deligne, for a useful list of critical remarks. Finally, he owes a special thank to Roberto Ferretti, who has read the first tentative versions of this work, has corrected many slips and has made several valuable suggestions.

This project began in 1997 while the second author was supported the IHES, and has been brought to completion while the second author was a guest of the Laboratoire de Théorie des Nombres of the University of Paris VI.

2. Homological theory

As explained in the introduction, in order to define a category of almost modules one requires a pair (V, 𝔪) consisting of a ring V and an ideal 𝔪 ⊂ V such that 𝔪 = 𝔪^{2}. In section 2.1 we collect a few useful ring-theoretic preliminaries concerning such pairs. In section 2.2 we introduce the category V^{a}-Mod of almost modules : it is a quotient V -Mod/Σ of the category of V -modules, where Σ is the thick subcategory of the V -modules killed by 𝔪. V^{a}-Mod is an abelian tensor category and its commutative unitary monoids, called almost algebras, are the chief objects of study in this work. The first useful observation is that the localization functor V -Mod → V^{a}-Mod admits both left and right adjoints. Taken together, these functors exhibit the kind of exactness properties that one associates to open embeddings of topoi, perhaps a hint of some deeper geometrical structure, still to be unearthed.

After these generalities, we treat in section 2.3 the question of finiteness conditions for almost modules. Let A denote an almost algebra, fixed for the rest of this introduction. It is certainly possible to define as usual a notion of finitely generated A-module, however this turns out to be too restrictive a class for applications. The main idea here is to define a uniform structure on the set of equivalence classes of A-modules ; then we will say that an A-module is almost finitely generated if its isomorphism class lies in the topological closure of the subspace of finitely generated A-modules. Similarly we define almost finitely presented A-modules. The uniform structure also comes handy when we want to construct operators on almost modules : if one can show that the operator in question is uniformly continuous on a class 𝒞 of almost modules, then its definition extends right away by continuity to the topological closure \overline{𝒞} of 𝒞. This is exemplified by the construction of the (almost) Fitting ideals for A-modules, at the end of section 2.3.

In section 2.4 we introduce the basic toolkit of homological algebra, beginning with the notion of flat almost module, which poses no problem, since we do have a tensor product in our category. The notion of projectivity is more subtle : it turns out that the category of A-modules usually does not have enough projectives. The useful notion is almost projectivity: simply one uses the standard definition, except that the role of the Hom functor is played by the internal alHom functor. The scarcity of projectives should not be regarded as surprising or pathological: it is quite analogous to the lack of enough projective objects in the category of quasi-coherent 𝒪_{X}-modules on a non-affine scheme X.

Section 2.5 introduces the cotangent complex of a morphism of almost rings, and establishes its usual properties, such as transitivity and Tor-independent base change theorems. These foundations will be put to use in chapter 3, to study infinitesimal deformations of almost algebras.

2.1. Some ring-theoretic preliminaries. Unless otherwise stated, every ring is commutative with unit. This section collects some results of general nature that will be used throughout this work.

2.1.1. Our basic setup consists of a fixed base ring V containing an ideal 𝔪 such that 𝔪^{2} = 𝔪. Starting from (2.5.14), we will also assume that 𝔪̃ := 𝔪 ⊗_{V} 𝔪 is a flat V -module. Example 2.1.2. (i) The main example is given by a non-discrete valuation ring (V, | · |) of rank one; in this case 𝔪 will be the maximal ideal.

(ii) Take 𝔪 := V. This is the “classical limit”. In this case almost ring theory reduces to usual ring theory. Thus, all the discussion that follows specialises to, and sometimes gives alternative proofs for, statements about rings and their modules.

2.1.3. Let M be a given V -module. We say that M is almost zero if 𝔪M = 0. A map ϕ of V -modules is an almost isomorphism if both Ker ϕ and Coker ϕ are almost zero V -modules.

Remark 2.1.4. (i) It is easy to check that a V -module M is almost zero if and only if 𝔪⊗_{V} M = 0. Similarly, a map M → N of V -modules is an almost isomorphism if and only if the induced map 𝔪̃ ⊗_{V} M → 𝔪̃ ⊗_{V} N is an isomorphism. Notice also that, if 𝔪 is flat, then 𝔪 ≃ 𝔪̃.

(ii) Let V → W be a ring homomorphism. For a V -module M set M_{W} := W ⊗_{V} M. We have an exact sequence

(2.1.5)  0 → K → 𝔪_{W} → 𝔪W → 0

where K := Tor_{1}^{V} (V/𝔪, W) is an almost zero W -module. By (i) it follows that 𝔪 ⊗_{V} K ≃ (𝔪W) ⊗_{W} K ≃ 0. Then, applying 𝔪_{W} ⊗_{W} − and − ⊗_{W} (𝔪W) to (2.1.5)we derive

𝔪_{W} ⊗_{W} 𝔪_{W} ≃ 𝔪_{W} ⊗_{W} (𝔪W) ≃ (𝔪W) ⊗_{W} (𝔪W)

i.e. 𝔪̃_{W} ≃ (𝔪W)^{∼}. In particular, if 𝔪̃ is a flat V -module, then 𝔪̃_{W} is a flat W -module. This means that our basic assumptions on the pair (V, 𝔪) are stable under arbitrary base extension. Notice that the flatness of 𝔪 does not imply the flatness of 𝔪W. This partly explains why we insist that 𝔪̃, rather than 𝔪, be flat.

2.1.6. Before moving on, we want to analyze in some detail how our basic assumptions relate to certain other natural conditions that can be postulated on the pair (V, 𝔪). Indeed, let us consider the following two hypotheses : (A) 𝔪 = 𝔪^{2} and 𝔪 is a filtered union of principal ideals. (B) 𝔪 = 𝔪^{2} and, for all integers k > 1, the k-th powers of elements of 𝔪 generate 𝔪.

Clearly (A) implies (B). Less obvious is the following result. Proposition 2.1.7. (i) (A) implies that 𝔪̃ is flat.

(ii) If 𝔪̃ is flat then (B) holds.

Proof. Suppose that (A) holds, so that 𝔪 = colim_{α∈I} V x_{α}, where I is a directed set parametrizing elements x_{α} ∈ 𝔪 (and α ≤ β ⇔ V x_{α} ⊂ V x_{β}). For any α ∈ I we have natural isomorphisms

(2.1.8)  V x_{α} ≃ V/Ann_{V} (x_{α}) ≃ (V x_{α}) ⊗_{V} (V x_{α}).

For α ≤ β, let j_{αβ} : V x_{α} ↪ V x_{β} be the imbedding; we have a commutative diagram

\xymatrix{V \ar[r]^{μ_{z^{2}}} \ar[d]_{π_{α}} & V \ar[d]^{π_{β}} \\ (V x_{α}) ⊗_{V} (V x_{α}) \ar[r]^{j_{αβ}⊗j_{αβ}} & (V x_{β}) ⊗_{V} (V x_{β})}

where z ∈ V is such that x_{α} = z·x_{β}, μ_{z^{2}} is multiplication by z^{2} and π_{α} is the projection induced by (2.1.8) (and similarly for π_{β}). Since 𝔪 = 𝔪^{2}, for all α ∈ I we can find β such that x_{α} is a multiple of x_{β}^{2}. Say x_{α} = y ·x_{β}^{2}; then we can take z := y ·x_{β}, so z^{2} is a multiple of x_{α} and in the above diagram Ker π_{α} ⊂ Ker μ_{z^{2}}. Hence one can define a map λ_{αβ} : (V x_{α}) ⊗_{V} (V x_{α}) → V such that π_{β} ∘ λ_{αβ} = j_{αβ} ⊗ j_{αβ} and λ_{αβ} ∘ π_{α} = μ_{z^{2}}. It now follows that for every V -module N, the induced morphism Tor_{1}^{V} (N, (V x_{α}) ⊗_{V} (V x_{α})) → Tor_{1}^{V} (N, (V x_{β}) ⊗_{V} (V x_{β})) is the zero map. Taking the colimit we derive that 𝔪̃ is flat. This shows (i). In order to show (ii) we consider, for any prime number p, the following condition

(∗_{p}) 𝔪/p · 𝔪 is generated (as a V -module) by the p-th powers of its elements.

Clearly (B) implies (∗_{p}) for all p. In fact we have :

Claim 2.1.9. (B) holds if and only if (∗_{p}) holds for every prime p.

Proof of the claim: Suppose that (∗_{p}) holds for every prime p. The polarization identity

k! · x_{1} · x_{2} · ... · x_{k} = ∑_{I⊂{1,2,...,k}}(−1)^{k−|I|} · (∑_{i∈I} x_{i})^{k}

shows that if N := ∑_{x∈𝔪} V x^{k} then k! · 𝔪 ⊂ N. To prove that N = 𝔪 it then suffices to show that for every prime p dividing k! we have 𝔪 = p · 𝔪 + N. Let ϕ : V/pV → V/pV be the Frobenius (x ↦ x^{p}); we can denote by (V/pV)^{ϕ} the ring V/pV seen as a V/pV -algebra via the homomorphism ϕ. Also set ϕ^{∗}M := M ⊗_{V/pV} (V/pV)^{ϕ} for a V/pV -module M. Then the map ϕ^{∗}(𝔪/p · 𝔪) → (𝔪/p · 𝔪) (defined by raising to p-th power) is surjective by (∗_{p}). Hence so is (ϕ^{r})^{∗}(𝔪/p · 𝔪) → (𝔪/p · 𝔪) for every r > 0, which says that 𝔪 = p · 𝔪 + N when k = p^{r}, hence for every k.

Next recall (see [6, Exp. XVII 5.5.2]) that, if M is a V -module, the module of symmetric tensors TS^{k}(M) is defined as (⊗_{V}^{k} M)^{S_{k}}, the invariants under the natural action of the symmetric group S_{k} on ⊗_{V}^{k} M. We have a natural map Γ^{k}(M) → TS^{k}(M) that is an isomorphism when M is flat (see loc. cit. 5.5.2.5; here Γ^{k} denotes the k-th graded piece of the divided power algebra). Claim 2.1.10. The group S_{k} acts trivially on ⊗_{V}^{k} 𝔪 and the map 𝔪̃⊗_{V} 𝔪 → 𝔪̃ (x⊗y⊗z ↦ x⊗yz) is an isomorphism. Proof of the claim: The first statement is reduced to the case of transpositions and to k = 2. There we can compute : x ⊗ yz = xy ⊗ z = y ⊗ xz = yz ⊗ x. For the second statement note that the imbedding 𝔪 ↪ V is an almost isomorphism, and apply remark 2.1.4(i).

Suppose now that 𝔪̃ is flat and pick a prime p. Then S_{p} acts trivially on ⊗_{V}^{p} 𝔪̃. Hence

(2.1.11)  Γ^{p}(𝔪̃) ≃ ⊗_{V}^{p} 𝔪̃ ≃ 𝔪̃.

But Γ^{p}(𝔪̃) is spanned as a V -module by the products γ_{i_{1}}(x_{1}) · ... · γ_{i_{k}}(x_{k}) (where x_{i} ∈ 𝔪̃ and ∑_{j} i_{j} = p). Under the isomorphism (2.1.11) these elements map to \binom{p}{i_{1},...,i_{k}} · x_{1}^{i_{1}} · ... · x_{k}^{i_{k}}; but such an element vanishes in 𝔪̃/p·𝔪̃ unless i_{k} = p for some k. Therefore 𝔪̃/p·𝔪̃ is generated by p-th powers, so the same is true for 𝔪/p · 𝔪, and by the above, (B) holds, which shows (ii). □

Theorem 2.1.12. Let (ε_{i} | i ∈ I) be a family of generators of 𝔪 and, for every subset S ⊂ I, denote by 𝔪_{S} ⊂ 𝔪 the subideal generated by (ε_{i} | i ∈ S). Then we have:

(i) Every countable subset S ⊂ I is contained in another countable S′ ⊂ I such that: (a) 𝔪_{S′}^{2} = 𝔪_{S′}.

(b) If 𝔪̃ is a flat V -module, the same holds for 𝔪_{S′} ⊗_{V} 𝔪_{S′}. (ii) Suppose that 𝔪 is countably generated as a V -module. Then :

(a) 𝔪̃ is countably presented as a V -module.

(b) If 𝔪̃ is a flat V -module, then it is of homological dimension ≤ 1.

Proof. For every i ∈ I, we can write ε_{i} = ∑_{j} x_{ij}ε_{j}, for certain x_{ij} ∈ 𝔪. For any i, j ∈ I such that x_{ij} ≠ 0, let us write x_{ij} = ∑_{k} x_{ijk}ε_{k} for some x_{ijk} ∈ V. We say that a subset S ⊂ I is saturated if the following holds: whenever i ∈ S and x_{ijk} ≠ 0, we have j, k ∈ S. Claim 2.1.13. Every countable subset S ⊂ I is contained in a countable saturated subset S_{∞} ⊂ I. Proof of the claim: We let S_{0} := S and define recursively S_{n} for every n > 0 as follows. Suppose S_{n−1} has already been given; then we set

S_{n} := S_{n−1} ∪ {i ∈ I | there exists a ∈ S_{n−1}, b ∈ I such that either x_{aib} ≠ 0 or x_{abi} ≠ 0}.

Notice that, for every i ∈ I we have x_{ij} = 0 for all but finitely many j ∈ I, hence x_{ijk} = 0 for all but finitely many j, k ∈ I. It follows easily that S_{∞} := ⋃_{n∈ℕ} S_{n} will do.

(i.a) is now straightforward, when one remarks that 𝔪_{S} = 𝔪_{S}^{2} for every saturated subset S.

Next, let S ⊂ I be any saturated subset. Clearly the family (ε_{i} ⊗ ε_{j} | i, j ∈ S) generates 𝔪̃_{S} := 𝔪_{S} ⊗_{V} 𝔪_{S} and ε_{i} · ε_{j} · (ε_{k} ⊗ ε_{l}) = ε_{k} · ε_{l} · (ε_{i} ⊗ ε_{j}) for all i, j, k, l ∈ S. Let F (S) be the V -module defined by generators (e_{ij})_{i,j∈S}, subject to the relations:

ε_{i} · ε_{j} · e_{kl} = ε_{k} · ε_{l} · e_{ij}  e_{ik} = ∑_{j} x_{ij}e_{jk}  for all i, j, k, l ∈ S.

We get an epimorphism π_{S} : F (S) → 𝔪̃_{S} by e_{ij} ↦ ε_{i} ⊗ ε_{j}. The relations imply that, if x := ∑_{k,l} y_{kl}e_{kl} ∈ Ker π_{S}, then ε_{i} · ε_{j} · x = 0, so 𝔪_{S} · Ker π_{S} = 0. Whence 𝔪_{S} ⊗_{V} Ker π_{S} = 0 and 1_{𝔪_{S}} ⊗_{V} π_{S} is an isomorphism. We consider the diagram

\xymatrix{𝔪_{S} ⊗_{V} F (S) \ar[r]^{∼} \ar[d]_{ϕ} & 𝔪_{S} ⊗_{V} 𝔪̃_{S} \ar[d]^{ψ} \\ F (S) \ar[r]^{π_{S}} & 𝔪̃_{S}}

where ϕ and ψ are induced by scalar multiplication. We already know that ψ is an isomorphism, and since F (S) = 𝔪_{S} · F (S), we see that ϕ is an epimorphism, so π_{S} is an isomorphism. If now I is countable, this shows that (ii.a) holds. Now (ii.b) follows from (ii.a) and the following lemma 2.1.16. In order to show (i.b) we will use the following well known criterion. Claim 2.1.14. ([51, Ch.I, Th.1.2]). Let R be a ring, M an R-module. Then M is R-flat if and only if, for every finitely presented R-module N, every morphism N → M factors as a composition N → L → M where L is a free R-module of finite rank.

In order to apply claim 2.1.14 we show: Claim 2.1.15. Let S ⊂ I be a countable saturated subset. Then there exists a countable saturated subset σ(S) ⊂ I containing S, with the following property. For every finitely presented V - module N and every morphism f : N → 𝔪̃_{S}, we can find a commutative diagram:

\xymatrix{N \ar[r]^{f} \ar[d] & 𝔪̃_{S} \ar[d]^{j} \\ L \ar[r] & 𝔪̃_{σ(S)}}

where L is free V -module of finite rank and j is the natural map. Proof of the claim: In view of (ii.a), we can write 𝔪̃_{S} ≃ colim_{α∈A} M_{α}, where A is some filtered countable set and every M_{α} is a finitely presented V -module. Given f as in the claim, we can find α ∈ A such that f factors thorugh the natural map ι_{α} : M_{α} → 𝔪̃_{S}, so we are reduced to prove the claim for N = M_{α} and f = ι_{α}. However, by assumption 𝔪̃ is flat, so by claim 2.1.14 the composition M_{α} →^{ι_{α}} 𝔪̃_{S} → 𝔪̃ factors through some map L_{α} → 𝔪̃, with L_{α} free of finite rank over V. Furthermore, thanks to claim 2.1.13 we have 𝔪̃ = colim_{J} 𝔪̃_{J}, where J runs over the family of all countable saturated subsets of I. It follows that, for some countable saturated S_{α} ⊃ S, the map L_{α} → 𝔪̃ factors through 𝔪̃_{S_{α}}. Clearly σ(S) := ⋃_{α∈A} S_{α} will do.

Finally, for any countable subset S ⊂ I, let us set S′ := ⋃_{n∈ℕ} σ^{n}(S_{∞}), where S_{∞} is the saturation of S as in claim 2.1.13. One verifies easily that (i.b) holds for this choice of S′. □

The proof of following well known lemma is due to D.Lazard ([51, Ch.I, Th.3.2]), up to some slight imprecisions which were corrected in [55, pp.49-50]. For the convenience of the reader we reproduce the argument. Lemma 2.1.16. Let R be any ring. A flat countably presented R-module has homological dimension ≤ 1.

Proof. Let M be flat and countable presented over R, and choose a presentation F_{1} →^{ϕ} F_{0} → M → 0 with F_{0} and F_{1} free R-modules of (infinite) countable rank. Let (e_{j} | j ∈ ℕ) be a basis of F_{0} and (f_{i} | i ∈ ℕ) a basis of F_{1}. Say that ϕ(f_{i}) = ∑_{i} x_{ij}e_{j} with x_{ij} ∈ R, for every i ∈ ℕ. For every i ∈ ℕ we define the finite set S_{i} := {j ∈ ℕ | x_{ij} ≠ 0}; also, let S′_{n} := ⋃_{i≤n}({i} ∪ S_{i}). Define R-modules G_{n} := ⊕_{i≤n}f_{i}R, H_{n} := ⊕_{j∈S′_{n}}e_{j}R; the restriction of ϕ induces maps ϕ_{n} : G_{n} → H_{n} and we have M ≃ colim_{n∈ℕ} Coker ϕ_{n}. By claim 2.1.14, the

natural map Coker ϕ_{n} → M factors as a composition Coker ϕ_{n} →^{α_{n}} L_{n} →^{ψ_{n}} M, where L_{n} is a free R-module of finite rank. Then ψ_{n} factors through a map ψ′_{n} : L_{n} → Coker ϕ_{t(n)} for some t(n) ∈ ℕ. We further define ψ′′_{n} : L_{n} → Coker ϕ_{k(n)} as the composition of ψ′_{n} with the natural transition map Coker ϕ_{t(n)} → Coker ϕ_{k(n)}, where k(n) is suitably chosen, so that k(n) > max(n, t(n)) and the composition ψ′′_{n} ∘ α_{n} : Coker ϕ_{n} → Coker ϕ_{k(n)} is the natural transition map. We define by induction on n ∈ ℕ a direct system of maps τ_{n} : L_{h_{n}} → L_{h_{n+1}}, as follows. Set h_{0} := 0; next, suppose that h_{n} ∈ ℕ has already been given up to some n ∈ ℕ; we let h_{n+1} := k(h_{n}) and τ_{n} := α_{h_{n+1}} ∘ ψ′′_{h_{n}}. Clearly M ≃ colim_{n∈ℕ}L_{h_{n}}; set L := ⊕_{n∈ℕ}L_{h_{n}} and let

τ : L → L be the map given by the rule τ(x_{n} | n ∈ ℕ) := (x_{n} − τ_{n}(x_{n−1}) | n ∈ ℕ). We derive a short exact sequence: 0 → L →^{τ} L → M → 0, whence the claim. □

2.2. Categories of almost modules and algebras. If 𝒞 is a category, and X, Y two objects of 𝒞, we will usually denote by Hom_{𝒞} (X, Y) the set of morphisms in 𝒞 from X to Y and by 1_{X} the identity morphism of X. Moreover we denote by 𝒞 ^{o} the opposite category of 𝒞 and by s.𝒞 the category of simplicial objects over 𝒞, that is, functors Δ^{o} → 𝒞, where Δ is the category whose objects are the ordered sets [n] := {0, ..., n} for each integer n ≥ 0 and where a morphism ϕ : [p] → [q] is a non-decreasing map. A morphism f : X → Y in s.𝒞 is a sequence of morphisms f_{[n]} : X[n] → Y [n], n ≥ 0 such that the obvious diagrams commute. We can imbed 𝒞 in s.𝒞 by sending each object X to the “constant” object s.X such that s.X[n] = X for all n ≥ 0 and s.X[ϕ] = 1_{X} for all morphisms ϕ in Δ.

2.2.1. If 𝒞 is an abelian category, D(𝒞) will denote the derived category of 𝒞. As usual we have also the full subcategories D^{+}(𝒞), D^{−}(𝒞) of complexes of objects of 𝒞 that are exact for sufficiently large negative (resp. positive) degree. If R is a ring, the category of R-modules (resp. R-algebras) will be denoted by R-Mod (resp. R-Alg). Most of the times we will write Hom_{R}(M, N) instead of Hom_{R-Mod}(M, N).

We denote by Set the category of sets. The symbol ℕ denotes the set of non-negative integers; in particular 0 ∈ ℕ.

2.2.2. The full subcategory Σ of V -Mod consisting of all V -modules that are almost isomorphic to 0 is clearly a Serre subcategory and hence we can form the quotient category V -Mod/Σ. There is a localization functor

V -Mod → V -Mod/Σ M ↦ M^{a}

that takes a V -module M to the same module, seen as an object of V -Mod/Σ. In particular, we have the object V^{a} associated to V; it seems therefore natural to use the notation V^{a}-Mod for the category V -Mod/Σ, and an object of V^{a}-Mod will be indifferently referred to as “a V^{a}-module” or “an almost V -module”. In case we need to stress the dependance on the ideal 𝔪, we can write (V, 𝔪)^{a}-Mod.

Since the almost isomorphisms form a multiplicative system (see e.g. [67, Exerc.10.3.2]), it is possible to describe the morphisms in V^{a}-Mod via a calculus of fractions, as follows. Let V -al.Iso be the category that has the same objects as V -Mod, but such that Hom_{V -al.Iso}(M, N) consists of all almost isomorphisms M → N. If M is any object of V -al.Iso we write (V -al.Iso/M) for the category of objects of V -al.Iso over M (i.e. morphisms ϕ : X → M).

If ϕ_{i} : X_{i} → M (i = 1, 2) are two objects of (V -al.Iso/M) then Hom_{(V -al.Iso/M)}(ϕ_{1}, ϕ_{2}) consists of all morphisms ψ : X_{1} → X_{2} in V -al.Iso such that ϕ_{1} = ϕ_{2} ∘ ψ. For any two V -modules M, N we define a functor ℱ_{N} : (V -al.Iso/M)^{o} → V -Mod by associating to an object ϕ : P → M the V -module Hom_{V} (P, N) and to a morphism α : P → Q the map Hom_{V} (Q, N) → Hom_{V} (P, N) : β ↦ β ∘ α. Then we have

(2.2.3)  Hom_{V^{a}-Mod}(M^{a}, N^{a}) = colim_{(V -al.Iso/M)^{o}}ℱ_{N}.

However, formula (2.2.3) can be simplified considerably by remarking that for any V -module M, the natural morphism 𝔪̃ ⊗_{V} M → M is an initial object of (V -al.Iso/M). Indeed, let ϕ : N → M be an almost isomorphism; the diagram

\xymatrix{𝔪̃ ⊗_{V} N \ar[r]^{∼} \ar[d] & 𝔪̃ ⊗_{V} M \ar[d] \\ N \ar[r]^{ϕ} & M}

(cp. remark 2.1.4(i)) allows one to define a morphism ψ : 𝔪̃ ⊗_{V} M → N over M. We need to show that ψ is unique. But if ψ_{1}, ψ_{2} : 𝔪̃ ⊗_{V} M → N are two maps over M, then Im(ψ_{1} −ψ_{2}) ⊂ Ker(ϕ) is almost zero, hence Im(ψ_{1} −ψ_{2}) = 0, since 𝔪̃⊗_{V} M = 𝔪·(𝔪̃⊗_{V} M). Consequently, (2.2.3)boils down to

(2.2.4)  Hom_{V^{a}-Mod}(M^{a}, N^{a}) = Hom_{V} (𝔪̃ ⊗_{V} M, N).

In particular Hom_{V^{a}-Mod}(M, N) has a natural structure of V -module for any two V^{a}-modules M, N, i.e. Hom_{V^{a}-Mod}(−, −) is a bifunctor that takes values in the category V -Mod.

2.2.5. One checks easily (for instance using (2.2.4)) that the usual tensor product induces a bifunctor − ⊗_{V} − on almost V -modules, which, in the jargon of [24] makes of V^{a}-Mod an abelian tensor category. Then an almost V -algebra is just a commutative unitary monoid in the tensor category V^{a}-Mod. Let us recall what this means. Quite generally, let (𝒞 , ⊗, U) be any abelian tensor category, so that ⊗ : 𝒞 × 𝒞 → 𝒞 is a biadditive functor, U is the identity object of 𝒞 (see [24, p.105]) and for any two objects M and N in 𝒞 we have a “commutativity constraint” (i.e. a functorial isomorphism θ_{M|N} : M ⊗ N → N ⊗ M that “switches the two factors”) and a functorial isomorphism ν_{M} : U ⊗ M → M. Then a 𝒞 -monoid A is an object of 𝒞 endowed with a morphism μ_{A} : A ⊗ A → A (the “multiplication” of A) satisfying the associativity condition

μ_{A} ∘ (1_{A} ⊗ μ_{A}) = μ_{A} ∘ (μ_{A} ⊗ 1_{A}).

We say that A is unitary if additionally A is endowed with a “unit morphism” \underline{1}_{A} : U → A satisfying the (left and right) unit property :

μ_{A} ∘ (\underline{1}_{A} ⊗ 1_{A}) = ν_{A} μ_{A} ∘ (\underline{1}_{A} ⊗ 1_{A}) ∘ θ_{A|U} = μ_{A} ∘ (1_{A} ⊗ \underline{1}_{A}).

Finally A is commutative if μ_{A} = μ_{A} ∘ θ_{A|A} (to be rigorous, in all of the above one should indicate the associativity constraints, which we have omitted : see [24]). A commutative unitary monoid will also be simply called an algebra. With the morphisms defined in the obvious way, the 𝒞 -monoids form a category; furthermore, given a 𝒞 -monoid A, a left A-module is an object M of 𝒞 endowed with a morphism σ_{M/A} : A ⊗ M → M such that σ_{M/A} ∘ (1_{A} ⊗ σ_{M/A}) = σ_{M/A} ∘ (μ_{A} ⊗ 1_{M}). Similarly one defines right A-modules and A-bimodules. In the case of bimodules we have left and right morphisms σ_{M,l} : A ⊗ M → M, σ_{M,r} : M ⊗ A → M and one imposes that they “commute”, i.e. that

σ_{M,r} ∘ (σ_{M,l} ⊗ 1_{A}) = σ_{M,l} ∘ (1_{A} ⊗ σ_{M,r}).

Clearly the (left resp. right) A-modules (and the A-bimodules) form an additive category with A-linear morphisms defined as one expects. One defines the notion of a submodule as an equivalence class of monomorphisms N → M such that the composition A ⊗ N → A ⊗ M → M factors through N. Especially, a two-sided ideal of A is an A-sub-bimodule I → A. For given submodules I, J of A one denotes IJ := Im(I ⊗J → A⊗A →^{μ_{A}} A). For an A-module M, the annihilator Ann_{A}(M) of M is the largest ideal j : I → A of A such that the composition I ⊗ M ⟶^{j⊗1_{M}} A ⊗ M ⟶^{σ_{M/A}} M is the zero morphism.

2.2.6. If f : M → N is a morphism of left A-modules, then Ker(f) exists in the underlying abelian category 𝒞 and one checks easily that it has a unique structure of left A-module which makes it a submodule of M. If moreover ⊗ is right exact when either argument is fixed, then also Coker f has a unique A-module structure for which N → Coker f is A-linear. In this case the category of left A-modules is abelian. Similarly, if A is a unitary 𝒞 -monoid, then one defines the notion of unitary left A-module by requiring that σ_{M/A} ∘ (\underline{1}_{A} ⊗ 1_{M}) = ν_{M} and these form an abelian category when ⊗ is right exact.

2.2.7. Specialising to our case we obtain the category V^{a}-Alg of almost V -algebras and, for every almost V -algebra A, the category A-Mod of unitary left A-modules. Clearly the localization functor restricts to a functor V -Alg → V^{a}-Alg and for any V -algebra R we have a localization functor R-Mod → R^{a}-Mod.

Next, if A is an almost V -algebra, we can define the category A-Alg of A-algebras. It consists of all the morphisms A → B of almost V -algebras.

2.2.8. Let again (𝒞 , ⊗, U) be any abelian tensor category. By [24, p.119], the endomorphism ring End_{𝒞} (U) of U is commutative. For any object M of 𝒞, denote M_{∗} = Hom_{𝒞} (U, M); then M ↦ M_{∗} defines a functor 𝒞 → End_{𝒞} (U)-Mod. Moreover, if A is a 𝒞 -monoid, A_{∗} is an associative End_{𝒞} (U)-algebra, with multiplication given as follows. For a, b ∈ A_{∗} let a · b := μ_{A} ∘ (a ⊗ b) ∘ ν_{U}^{−1}. Similarly, if M is an A-module, M_{∗} is an A_{∗}-module in a natural way, and in this way we obtain a functor from A-modules and A-linear morphisms to A_{∗}- modules and A_{∗}-linear maps. Using [24, Prop.1.3], one can also check that End_{𝒞} (U) = U_{∗} as End_{𝒞} (U)-algebras, where U is viewed as a 𝒞 -monoid using ν_{U}.

2.2.9. All this applies especially to our categories of almost modules and almost algebras : in this case we call M ↦ M_{∗} the functor of almost elements. So, if M is an almost module, an almost element of M is just an honest element of M_{∗}. Using (2.2.4) one can show easily that for every V -module M the natural map M → (M^{a})_{∗} is an almost isomorphism.

2.2.10. Let A be a V^{a}-algebra. For any two A-modules M, N, the set Hom_{A-Mod}(M, N) has a natural structure of A_{∗}-module and we obtain an internal Hom functor by letting

alHom_{A}(M, N) = Hom_{A-Mod}(M, N)^{a}.

This is the functor of almost homomorphisms from M to N.

2.2.11. For any A-module M we have also a functor of tensor product M ⊗_{A} − on A-modules which, in view of the following proposition 2.2.13 can be shown to be a left adjoint to the functor alHom_{A}(M, −). It can be defined as M ⊗_{A} N := (M_{∗} ⊗_{A_{∗}} N_{∗})^{a}.

With this tensor product, A-Mod is an abelian tensor category as well, and A-Alg could also be described as the category of (A-Mod)-algebras. Under this equivalence, a morphism ϕ : A → B of almost V -algebras becomes the unit morphism \underline{1}_{B} : A → B of the corresponding monoid. We will sometimes drop the subscript and write simply \underline{1}. Remark 2.2.12. Let V → W be a map of base rings, W taken with the extended ideal 𝔪W. Then W^{a} is an almost V -algebra so we have defined the category W^{a}-Mod using base ring V and the category (W, 𝔪W)^{a}-Mod using base W. One shows easily that they are equivalent: we have an obvious functor (W, 𝔪W)^{a}-Mod → W^{a}-Mod and a quasi-inverse is provided by M ↦ M_{∗}. Similar base comparison statements hold for the categories of almost algebras.

Proposition 2.2.13. Let A be a V^{a}-algebra, R a V -algebra.

(i) There is a natural isomorphism A ≃ A_{∗}^{a} of almost V -algebras.

(ii) The functor M ↦ M_{∗} from R^{a}-Mod to R-Mod (resp. from R^{a}-Alg to R-Alg) is right adjoint to the localization functor R-Mod → R^{a}-Mod (resp. R-Alg → R^{a}-Alg).

(iii) The counit of the adjunction M_{∗}^{a} → M is a natural isomorphism from the composition of the two functors to the identity functor 1_{R^{a}-Mod} (resp. 1_{R^{a}-Alg}).

Proof. (i) has already been remarked. We show (ii). In light of remark 2.2.12 (applied with W = R) we can assume that V = R. Let M be a V -module and N an almost V -module; we have natural bijections

Hom_{V^{a}-Mod}(M^{a}, N) ≃ Hom_{V^{a}-Mod}(M^{a}, (N_{∗})^{a}) ≃ Hom_{V} (𝔪̃ ⊗_{V} M, N_{∗})
≃ Hom_{V} (M, Hom_{V} (𝔪̃, N_{∗})) ≃ Hom_{V} (M, Hom_{V^{a}-Mod}(V, (N_{∗})^{a}))
≃ Hom_{V} (M, N_{∗})

which proves (ii). Now (iii) follows by inspecting the proof of (ii), or by [38, Ch.III Prop.3]. □

Remark 2.2.14. (i) Let M_{1}, M_{2} be two A-modules. By proposition 2.2.13(iii) it is clear that a morphism ϕ : M_{1} → M_{2} of A-modules is uniquely determined by the induced morphism M_{1∗} → M_{2∗}. On this basis, we will very often define morphisms of A-modules (or A-algebras) by saying how they act on almost elements.

(ii) It is a bit tricky to deal with preimages of almost elements under morphisms: for instance, if ϕ : M_{1} → M_{2} is an epimorphism (by which we mean that Coker ϕ ≃ 0) and m_{2} ∈ M_{2∗}, then it is not true in general that we can find an almost element m_{1} ∈ M_{1∗} such that ϕ_{∗}(m_{1}) = m_{2}. What remains true is that for arbitrary ε ∈ 𝔪 we can find m_{1} such that ϕ_{∗}(m_{1}) = ε · m_{2}.

(iii) The existence of the right adjoint M ↦ M_{∗} follows also directly from [38, Chap.III §3 Cor.1 or Chap.V §2].

Corollary 2.2.15. The categories A-Mod and A-Alg are both complete and cocomplete.

Proof. We recall that the categories A_{∗}-Mod and A_{∗}-Alg are both complete and cocomplete. Now let I be any small indexing category and M : I → A-Mod be any functor. Denote by M_{∗} : I → A_{∗}-Mod the composed functor i ↦ M(i)_{∗}. We claim that colim_{I} M = (colim_{I} M_{∗})^{a}. The proof is an easy application of proposition 2.2.13(iii). A similar argument also works for limits and for the category A-Alg. □

2.2.16. For any V^{a}-algebra A, The abelian category A-Mod satisfies axiom (AB5) (see e.g. [67, §A.4]) and it has a generator, namely the object A itself. It then follows by a general result that A-Mod has enough injectives.

Corollary 2.2.17. The functor M ↦ M_{∗} from R^{a}-Mod to R-Mod sends injectives to injectives and injective envelopes to injective envelopes.

Proof. The functor M ↦ M_{∗} is right adjoint to an exact functor, hence it preserves injectives. Now, let J be an injective envelope of M; to show that J_{∗} is an injective envelope of M_{∗}, it suffices to show that J_{∗} is an essential extension of M_{∗}. However, if N ⊂ J_{∗} and N ∩ M_{∗} = 0, then N^{a} ∩ M = 0, hence 𝔪N = 0, but J_{∗} does not contain 𝔪-torsion, thus N = 0. □

2.2.18. Note that the essential image of M ↦ M_{∗} is closed under limits. Next recall that the forgetful functor A_{∗}-Alg → Set (resp. A_{∗}-Mod → Set) has a left adjoint A_{∗}[−] : Set → A_{∗}-Alg (resp. A^{(−)} : Set → A_{∗}-Mod) that assigns to a set S the free A_{∗}-algebra A_{∗}[S] (resp. the free A_{∗}-module A_{∗}^{(S)}) generated by S. If S is any set, it is natural to write A[S] (resp. A^{(S)}) for the A-algebra (A_{∗}[S])^{a} (resp. for the A-module (A_{∗}^{(S)})^{a}. This yields a left adjoint, called the free A-algebra functor Set → A-Alg (resp. the free A-module functor Set → A-Mod) to the “forgetful” functor A-Alg → Set (resp. A-Mod → Set) B ↦ B_{∗}.

2.2.19. Now let R be any V -algebra; we want to construct a left adjoint to the localisation functor R-Mod → R^{a}-Mod. For a given R^{a}-module M, let

(2.2.20)  M_{!} := 𝔪̃ ⊗_{V} (M_{∗}).

We have the natural map (unit of adjunction) R → R_{∗}^{a}, so that we can view M_{!} as an R-module. Proposition 2.2.21. Let R be a V -algebra.

(i) The functor R^{a}-Mod → R-Mod defined by (2.2.20)is left adjoint to localisation. (ii) The unit of the adjunction M → M_{!}^{a} is a natural isomorphism from the identity functor

1_{R^{a}-Mod} to the composition of the two functors.

Proof. (i) follows easily from (2.2.4)and (ii) follows easily from (i). □

Corollary 2.2.22. Suppose that 𝔪̃ is a flat V -module. Then we have :

(i) the functor M ↦ M_{!} is exact;

(ii) the localisation functor R-Mod → R^{a}-Mod sends injectives to injectives.

Proof. By proposition 2.2.21 it follows that M ↦ M_{!} is right exact. To show that it is also left exact when 𝔪̃ is a flat V -module, it suffices to remark that M ↦ M_{∗} is left exact. Now, by (i), the functor M ↦ M^{a} is right adjoint to an exact functor, so (ii) is clear. □

2.2.23. Let B be any A-algebra. The multiplication on B_{∗} is inherited by B_{!}, which is therefore a non-unital ring in a natural way. We endow the V -module V ⊕ B_{!} with the ring structure determined by the rule: (v, b) · (v′, b′) := (v · v′, v · b′ + v′ · b + b · b′) for all v, v′ ∈ V and b, b′ ∈ B_{!}. Then V ⊕ B_{!} is a (unital) ring. We notice that the V -submodule generated by all the elements of the form (x · y, −x ⊗ y ⊗ \underline{1}) (for arbitrary x, y ∈ 𝔪) forms an ideal I of V ⊕ B_{!}. Set B_{!!} := (V ⊕ B_{!})/I. Thus we have a sequence of V -modules

(2.2.24)  0 → 𝔪̃ → V ⊕ B_{!} → B_{!!} → 0

which in general is only right exact. Definition 2.2.25. We say that B is an exact V^{a}-algebra if the sequence (2.2.24)is exact. Remark 2.2.26. Notice that if 𝔪̃ →^{∼} 𝔪 (e.g. when 𝔪 is flat), then all V^{a}-algebras are exact. In the general case, if B is any A-algebra, then V^{a} × B is always exact. Indeed, we have (V^{a} × B)_{∗} ≃ V_{∗}^{a} × B_{∗} and, by remark 2.1.4(i), 𝔪̃ ⊗_{V} V_{∗}^{a} ≃ 𝔪̃.

Clearly we have a natural isomorphism B ≃ B_{!!}^{a}. Proposition 2.2.27. The functor B ↦ B_{!!} is left adjoint to the localisation functor A_{!!}-Alg → A-Alg.

Proof. Let B be an A-algebra, C an A_{!!}-algebra and ϕ : B → C^{a} a morphism of A-algebras. By proposition 2.2.21 we obtain a natural A_{∗}-linear morphism B_{!} → C. Together with the structure morphism V → C this yields a map ϕ̃ : V ⊕ B_{!} → C which is easily seen to be a ring homomorphism. It is equally clear that the ideal I defined above is mapped to zero by ϕ̃, hence the latter factors through a map of A_{!!}-algebras B_{!!} → C. Conversely, such a map induces a morphism of A-algebras B → C^{a} just by taking localisation. It is easy to check that the two procedures are inverse to each other, which shows the assertion. □

Remark 2.2.28. (i) The functor of almost elements commutes with arbitrary limits, because all right adjoints do. It does not in general commute with colimits, not even with arbitrary infinite direct sums. Dually, the functors M ↦ M_{!} and B ↦ B_{!!} commute with all colimits. In particular, the latter commutes with tensor products.

(ii) Resume the notation of remark 2.2.12. The change of setup functor (W, 𝔪W)^{a}-Mod → W^{a}-Mod commutes with the operations M ↦ M_{∗} and M ↦ M_{!}. The corresponding functor F : (W, 𝔪W)^{a}-Alg → W^{a}-Alg satisfies the identity: F (B)_{!!} ⊗_{W_{!!}^{a}} W = B_{!!} for every (W, 𝔪W)^{a}-algebra B.

2.3. Uniform spaces of almost modules. Let A be a V^{a}-algebra. For any cardinal number c, we let ℳ_{c}(A) be the set of isomorphism classes of A-modules which admit a set of generators of cardinality ≤ c. In the following we fix some (very) large infinite cardinality ω, and suppose that the isomorphism classes of all our A-modules lie in ℳ_{ω}(A). The choice of ω is required to avoid set-theoretical inconsistencies, but it is immaterial for our purposes, so we will henceforth just write ℳ (A) instead of ℳ_{ω}(A). Definition 2.3.1. Let A be a V^{a}-algebra and M an A-module.

(i) We define a uniform structure on the set ℐ_{A}(M) of A-submodules of M, as follows. For every finitely generated ideal 𝔪_{0} ⊂ 𝔪, the subset of ℐ_{A}(M) × ℐ_{A}(M) given by E_{M}(𝔪_{0}) := {(M_{0}, M_{1}) | 𝔪_{0}M_{0} ⊂ M_{1} and 𝔪_{0}M_{1} ⊂ M_{0}} is an entourage for the uniform structure, and the subsets of this kind form a fundamental system of entourages. (ii) We define a uniform structure on ℳ (A) as follows. For every finitely generated ideal 𝔪_{0} ⊂ 𝔪 and every integer n ≥ 0 we define the entourage E_{ℳ} (𝔪_{0}) ⊂ ℳ (A) × ℳ (A), which consists of all pairs of A-modules (M, M′) such that there exist a third module N and morphisms ϕ : N → M, ψ : N → M′, such that 𝔪_{0} annihilates the kernel and cokernel of ϕ and ψ. We declare that the E_{ℳ} (𝔪_{0}) form a fundamental system of entourages for the uniform structure of ℳ (A).

Remark 2.3.2. Notice that the entourage E_{ℳ} (𝔪_{0}) can be defined equivalently by all the pairs of A-modules (M, M′) such that there exists a third module L and morphisms ϕ′ : M → L, ψ : M′ → L such that 𝔪_{0} annihilates the kernel and cokernel of ϕ and ϕ. Indeed, given a pair (M, M′) ∈ E_{ℳ} (𝔪_{0}), and a datum (N, ϕ, ψ) as in definition 2.3.1(ii), a datum (L, ϕ′, ψ′) satisfying the above condition is obtained from the push out diagram

(2.3.3)  \xymatrix{N \ar[r]^{ϕ} \ar[d]_{ψ} & M \ar[d]^{ϕ′} \\ M′ \ar[r]^{ψ′} & L.}

Conversely, given a datum (L, ϕ′, ψ′), one obtains another diagram as (2.3.3), by letting N be the fibred product of M and M′ over L.

2.3.4. We will also need occasionally a notion of “Cauchy product” : let ∏_{n=0}^{∞} I_{n} be a formal infinite product of ideals I_{n} ⊂ A. We say that the formal product satisfies the Cauchy condition (or briefly : is a Cauchy product) if, for every neighborhood 𝒰 of A in ℐ_{A}(A) there exists n_{0} ≥ 0 such that ∏_{m=n}^{n+p} I_{m} ∈ 𝒰 for all n ≥ n_{0} and all p ≥ 0. Lemma 2.3.5. Let M be an A-module.

(i) ℐ_{A}(M) with the uniform structure of definition 2.3.1 is complete and separated.

(ii) The following maps are uniformly continuous :

(a) ℐ_{A}(M) × ℐ_{A}(M) → ℐ_{A}(M) : (M′, M′′) ↦ M′ ∩ M′′.

(b) ℐ_{A}(M) × ℐ_{A}(M) → ℐ_{A}(M) : (M′, M′′) ↦ M′ + M′′.

(c) ℐ_{A}(A) × ℐ_{A}(A) → ℐ_{A}(A) : (I, J) ↦ IJ. (iii) For any A-linear morphism ϕ : M → N, the following maps are uniformly continuous:

(a) ℐ_{A}(M) → ℐ_{A}(N) : M′ ↦ ϕ(M′).
(b) ℐ_{A}(N) → ℐ_{A}(M) : N′ ↦ ϕ^{−1}(N′).

Proof. (i) : The separation property is easily verified. We show that ℐ_{A}(M) is complete. Therefore, suppose that ℱ is some Cauchy filter of ℐ_{A}(M). Concretely, this means that for every finitely generated 𝔪_{0} ⊂ 𝔪, there exists F (𝔪_{0}) ∈ ℱ such that 𝔪_{0}I ⊂ J for every I, J ∈ F (𝔪_{0}). Let L := ⋃_{F ∈ℱ} (⋂_{I∈F} I). We claim that L is the limit of our filter. Indeed, for a given finitely generated 𝔪_{0} ⊂ 𝔪, we have 𝔪_{0}I ⊂ ⋂_{J∈F (𝔪_{0})} J, for every I ∈ F (𝔪_{0}), whence 𝔪_{0}I ⊂ L. On the other hand, if I ∈ F ⊂ F (𝔪_{0}), we can write: 𝔪_{0}L = ⋃_{F′⊂F} 𝔪_{0}(⋂_{J∈F′} J) ⊂ ⋃_{F′⊂F} (⋂_{J∈F′} 𝔪_{0}J) ⊂ ⋃_{F′⊂F} I = I (where F′ runs over all the subsets F′ ∈ ℱ such that F′ ⊂ F). This shows that (L, I) ∈ E_{M}(𝔪_{0}) whenever I ∈ F (𝔪_{0}), which implies the claim. (ii) and (iii) are easy and will be left to the reader. □

Remark 2.3.6. In general, the uniform space ℳ (A) is not separated. In view of proposition 3.2.30, a counterexample is provided by remark 3.2.29. Lemma 2.3.7. Let ϕ : M → N be an A-linear morphism, B an A-algebra. The following maps are uniformly continuous :

(i) ℳ (A) → ℐ_{A}(A) : M ↦ Ann_{A}(M).

(ii) ℐ_{A}(M) × ℐ_{A}(N) → ℳ (A) : (M′, N′) ↦ (ϕ(M′) + N′)/ϕ(M′).

(iii) ℳ (A) × ℳ (A) → ℳ (A) : (M′, M′′) ↦ alHom_{A}(M′, M′′).

(iv) ℳ (A) × ℳ (A) → ℳ (A) : (M′, M′′) ↦ M′ ⊗_{A} M′′.

(v) ℳ (A) → ℳ (B) : M ↦ B ⊗_{A} M.

(vi) ℳ (A) → ℳ (A) : M ↦ Λ_{A}^{r}M for any r ≥ 0, provided (B) holds.

Proof. We show (iv) and leave the others to the reader. By symmetry, we reduce to verifying that, if (M′, M′′) ∈ E_{ℳ} (𝔪_{0}) and N is an arbitrary A-module, then (N ⊗_{A} M′, N ⊗_{A} M′′) ∈ E_{ℳ} (𝔪_{0}^{2}). Then we can further assume that there is a morphism ϕ : M′ → M′′ with 𝔪_{0} · Ker ϕ = 𝔪_{0} · Coker ϕ = 0. We factor ϕ_{!} as an epimorphism followed by a monomorphism M′_{!} →^{ϕ_{1}} Im ϕ_{!} →^{ϕ_{2}} M′′_{!}, and then we reduce to checking that the kernels and cokernels of both 1_{N} ⊗_{A} ϕ_{1} and 1_{N} ⊗_{A} ϕ_{2} are killed by 𝔪_{0}. This is clear for 1_{N} ⊗_{A} ϕ_{1}, and it follows easily for 1_{N} ⊗_{A} ϕ_{2} as well, by using the Tor sequences. □

Definition 2.3.8. For a subset S of a topological space T, let \overline{S} denote the adherence of S in T. Let M be an A-module.

(i) M is said to be finitely generated if its isomorphism class lies in ⋃_{n∈ℕ} ℳ_{n}(A).

(ii) M is said to be almost finitely generated if its isomorphism class lies in \overline{⋃_{n∈ℕ} ℳ_{n}(A)} . (iii) M is said to be uniformly almost finitely generated if there exists an integer n ≥ 0 such

that the isomorphism class of M lies in 𝒰_{n}(A) := \overline{ℳ_{n}(A)} . Then we will say that n is a uniform bound for M.

(iv) M is said to be finitely presented if it is isomorphic to the cokernel of a morphism of free finitely generated A-modules. We denote by ℱ 𝒫(A) ⊂ ℳ (A) the subset of the isomorphism classes of finitely presented A-modules.

(v) M is almost finitely presented if its isomorphism class lies in \overline{ℱ 𝒫(A)} . Remark 2.3.9. Under condition (A), an A-module M lies in 𝒰_{n}(A) if and only if, for every ε ∈ 𝔪 there exists an A-linear morphism A^{n} → M whose cokernel is killed by ε. Proposition 2.3.10. Let M be an A-module.

(i) M is almost finitely generated if and only if for every finitely generated ideal 𝔪_{0} ⊂ 𝔪 there exists a finitely generated submodule M_{0} ⊂ M such that 𝔪_{0}M ⊂ M_{0}.

(ii) The following conditions are equivalent: (a) M is almost finitely presented.

(b) for arbitrary ε, δ ∈ 𝔪 there exist positive integers n = n(ε), m = m(ε) and a three term complex A^{m} →^{ψ_{ε}} A^{n} →^{ϕ_{ε}} M with ε · Coker(ϕ_{ε}) = 0 and δ · Ker ϕ_{ε} ⊂ Im ψ_{ε}.

(c) For every finitely generated ideal 𝔪_{0} ⊂ 𝔪 there is a complex A^{m} →^{ψ} A^{n} →^{ϕ} M with 𝔪_{0} · Coker ϕ = 0 and 𝔪_{0} · Ker ϕ ⊂ Im ψ.

Proof. (i): Let M be an almost finitely generated A-module, and 𝔪_{0} ⊂ 𝔪 a finitely generated subideal. Choose a finitely generated subideal 𝔪_{1} ⊂ 𝔪 such that 𝔪_{0} ⊂ 𝔪_{1}^{3}; by hypothesis, there exist A-modules M′ and M′′, where M′′ is finitely generated, and morphisms f : M′ → M, g : M′ → M′′ whose kernels and cokernels are annihilated by 𝔪_{1}. We get morphisms 𝔪_{1}⊗_{V} M′′ → Im(g) and 𝔪_{1} ⊗_{V} Im(g) → M′, hence a composed morphism ϕ : 𝔪_{1} ⊗_{V} 𝔪_{1} ⊗_{V} M′′ → M′; it is easy to check that Coker(f ∘ ϕ) is annihilated by 𝔪_{1}^{3}, hence M_{0} := Im(f ∘ ϕ) will do.

To show (ii) we will need the following : Claim 2.3.11. Let F_{1} be a finitely generated A-module and suppose that we are given a, b ∈ V and a (not necessarily commutative) diagram

\xymatrix{F_{1} \ar[r]^{p} \ar[d]^{ϕ} & M \\ F_{2} \ar[u]^{ψ} \ar[ur]_{q}}

such that q ∘ ϕ = a · p, p ∘ ψ = b · q. Let I ⊂ V be an ideal such that Ker q has a finitely generated submodule containing I · Ker q. Then Ker p has a finitely generated submodule containing ab · I · Ker p. Proof of the claim: Let R be the submodule of Ker q given by the assumption. We have Im(ψ ∘ ϕ − ab · 1_{F_{1}}) ⊂ Ker p and ψ(R) ⊂ Ker p. We take R_{1} := Im(ψ ∘ ϕ − ab · 1_{F_{1}}) + ψ(R). Clearly ϕ(Ker p) ⊂ Ker q, so I · ϕ(Ker p) ⊂ R, hence I · ψ ∘ ϕ(Ker p) ⊂ ψ(R) and finally ab · I · Ker p ⊂ R_{1}.

Claim 2.3.12. If M satisfies condition (b) of the proposition, and ϕ : F → M is a morphism with F ≃ A^{n}, then for every finitely generated ideal 𝔪_{1} ⊂ 𝔪·Ann_{V} (Coker ϕ) there is a finitely generated submodule of Ker ϕ containing 𝔪_{1} · Ker ϕ. Proof of the claim: Now, let δ ∈ Ann_{V} (Coker ϕ) and ε_{1}, ε_{2}, ε_{3}, ε_{4} ∈ 𝔪. By assumption there is a complex A^{r} →^{t} A^{s} →^{q} M with ε_{1} · Coker q = 0, ε_{2} · Ker q ⊂ Im t. Letting F_{1} := F, F_{2} := A^{s}, a := ε_{1} ·ε_{3}, b := ε_{4} ·δ, one checks easily that ψ and ϕ can be given such that all the assumptions of claim 2.3.11 are fulfilled. So, with I := ε_{2} · V we see that ε_{1} · ε_{2} · ε_{3} · ε_{4} · δ · Ker ϕ lies in a finitely generated submodule of Ker ϕ. But 𝔪_{1} is contained in an ideal generated by finitely many such products ε_{1} · ε_{2} · ε_{3} · ε_{4} · δ.

Now, it is clear that (c) implies (a) and (b). To show that (b) implies (c), take a finitely generated ideal 𝔪_{1} ⊂ 𝔪 such that 𝔪_{0} ⊂ 𝔪 · 𝔪_{1}, pick a morphism ϕ : A^{n} → M whose cokernel is annihilated by 𝔪_{1}, and apply claim 2.3.12. We show that (a) implies (c). For a given finitely generated subideal 𝔪_{0} ⊂ 𝔪, pick another finitely generated 𝔪_{1} ⊂ 𝔪 such that 𝔪_{0} ⊂ 𝔪_{1}^{3}; find morphisms f : M′ → M and g : M′ → M′′ whose kernels and cokernels are annihilated by 𝔪_{1}, and such that M′′ is finitely presented. Let \underline{ε} := (ε_{1}, ..., ε_{r}) be a finite sequence of generators of 𝔪_{1}, and denote by K_{•} := K_{•}(\underline{ε}) the Koszul complex of V -modules associated to the sequence \underline{ε}. Set 𝔪′_{1} := Coker(K_{2} → K_{1}); we derive a natural surjection ∂ : 𝔪′_{1} → 𝔪_{1} and, for every i = 1, ..., r, maps e_{i} : V → 𝔪′_{1} such that the compositions

𝔪′_{1} →^{∂} 𝔪_{1} ↪ V →^{e_{i}} 𝔪′_{1} V →^{e_{i}} 𝔪′_{1} →^{∂} 𝔪_{1} ↪ V

are both scalar multiplication by ε_{i}. Hence, for every V^{a}-module N, the kernel and cokernel of the natural morphism 𝔪′_{1} ⊗_{V} N → N are annihilated by 𝔪_{1}. Let now ϕ be as in the proof of (i); notice that the diagram:

\xymatrix{𝔪_{1} ⊗_{V} 𝔪_{1} ⊗_{V} M′ \ar[r] \ar[d] & M′ \ar[d] \\ 𝔪_{1} ⊗_{V} 𝔪_{1} ⊗_{V} M′′ \ar[r] \ar[ur]^{ϕ} & M′′}

commutes. It follows that the composed morphism

𝔪′_{1} ⊗_{V} 𝔪′_{1} ⊗_{V} M′′ → 𝔪_{1} ⊗_{V} 𝔪_{1} ⊗_{V} M′′ →^{ϕ} M′ → M

has kernel and cokernel annihilated by 𝔪_{1}^{3}, so the claim follows. □

2.3.13. Suppose that 𝔪 = ⋃_{λ∈Λ} 𝔪_{λ}, where (𝔪_{λ} | λ ∈ Λ) is a filtered family of subideals such that 𝔪_{λ}^{2} = 𝔪_{λ} for every λ ∈ Λ. Let R be a V -algebra, M an R-module, and denote by R_{λ}^{a} (resp. R^{a}) the (V, 𝔪_{λ})^{a}-algebra (resp. (V, 𝔪)-algebra) associated to R; define similarly M^{a} and M_{λ}^{a}, for all λ ∈ Λ. Lemma 2.3.14. With the notation of (2.3.13), the following conditions are equivalent:

(i) M^{a} is an almost finitely generated (resp. almost finitely presented) R^{a}-module.

(ii) M_{λ}^{a} is an almost finitely generated (resp. almost finitely presented) R_{λ}^{a}-module for all λ ∈ Λ.

Proof. We only give the proof for the case of almost finitely presented modules, and let the reader spell out the analogous (and easier) argument for almost finitely generated modules.

(i) ⇒ (ii) : for given λ ∈ Λ and a finitely generated subideal 𝔪_{0} ⊂ 𝔪_{λ}, pick a complex (R^{a})^{m} → (R^{a})^{n} → M^{a} as in proposition 2.3.10(ii.c); after applying termwise the natural functor R^{a}-Mod → R_{λ}^{a}-Mod, we obtain another complex that satisfies again the condition of proposition 2.3.10(ii.c), so the claim follows.

(ii) ⇒ (i) : for finitely generated 𝔪_{0} ⊂ 𝔪 find λ ∈ Λ such that 𝔪_{0} ⊂ 𝔪_{λ}; then pick a complex (R_{λ}^{a})^{m} →^{ψ} (R_{λ}^{a})^{n} →^{ϕ} M_{λ}^{a} as in the foregoing. Set N := M_{λ∗}^{a}, let N^{a} be the image of N in R^{a}-Mod and notice that (M^{a}, N^{a}) ∈ E_{ℳ} (𝔪_{0}). Let ε_{1}, ..., ε_{k} be a set of generators for 𝔪_{0}; let α : R^{m} → (R_{λ∗}^{a})^{n} be the composition of the adjunction map R^{m} → (R_{λ∗}^{a})^{m} and the map ψ_{∗}. Let e_{1}, ..., e_{k} be a basis of V^{k} and f_{1}, ...f_{m} a basis of R^{m}; we define a map β : V^{k} ⊗_{V} R^{m} → R^{m} by the rule: e_{i} ⊗ f_{j} ↦ ε_{i} · f_{j}. The map α ∘ β lifts to an R-linear map γ : V^{k} ⊗_{V} R^{m} → R^{n} and ϕ_{∗} induces an R-linear map Coker γ → N whose kernel and cokernel are annihilated by 𝔪_{0}^{2} · 𝔪. All in all, this shows that (Coker γ^{a}, M^{a}) ∈ E_{ℳ} (𝔪_{0}^{4}), so the claim follows. □

The following proposition generalises a well-known characterization of finitely presented modules over usual rings. Proposition 2.3.15. Let M be an A-module.

(i) M is almost finitely generated if and only if, for every filtered system (N_{λ}, ϕ_{λμ}) (indexed by a directed set Λ) the natural morphism

(2.3.16)  M : colim_{Λ} alHom_{A}(M, N_{λ}) → alHom_{A}(M, colim_{Λ} N_{λ})

is a monomorphism. (ii) M is almost finitely presented if and only if for every filtered inductive sytem as above,

(2.3.16)is an isomorphism.

Proof. The “only if” part in (i) (resp. (ii)) is first checked when M is finitely generated (resp. finitely presented) and then extended to the general case. We leave the details to the reader and we proceed to verify the “if” part. For (i), choose a set I and an epimorphism p : A^{(I)} → M. Let Λ be the directed set of finite subsets of I, ordered by inclusion. For S ∈ Λ, let M_{S} := p(A^{S}). Then colim_{Λ} (M/M_{S}) = 0, so the assumption gives colim_{Λ} alHom_{A}(M, M/M_{S}) = 0, i.e. colim_{Λ} Hom_{A}(M, M/M_{S}) = 0 is almost zero, so, for every ε ∈ 𝔪, the image of ε · 1_{M} in the above colimit is 0, i.e. there exists S ∈ Λ such that εM ⊂ M_{S}, which proves the contention. For (ii), we present M as a filtered colimit colim_{Λ} M_{λ}, where each M_{λ} is finitely presented (this can be done e.g. by taking such a presentation of the A_{∗}-module M_{∗} and applying N ↦ N^{a}). The assumption of (ii) gives that colim_{Λ} Hom_{A}(M, M_{λ}) → Hom_{A}(M, M) is an almost isomorphism, hence, for every ε ∈ 𝔪 there is λ ∈ Λ and ϕ_{ε} : M → M_{λ} such that p_{λ} ∘ ϕ_{ε} = ε · 1_{M}, where p_{λ} : M_{λ} → M is the natural morphism to the colimit. If such a ϕ_{ε} exists for λ, then it exists for every μ ≥ λ. Hence, if 𝔪_{0} ⊂ 𝔪 is a finitely generated subideal, say 𝔪_{0} = ∑_{j}^{k} V ε_{j}, then there exist λ ∈ Λ and ϕ_{i} : M → M_{λ} such that p_{λ} ∘ ϕ_{i} = ε_{i} · 1_{M} for i = 1, ..., k. Hence Im(ϕ_{i} ∘p_{λ} −ε_{i} ·1_{M_{λ}}) is contained in Ker p_{λ} and contains ε_{i} ·Ker p_{λ}. Hence Ker p_{λ} has a finitely generated submodule L containing 𝔪_{0} · Ker p_{λ}. Choose a presentation A^{m} → A^{n} →^{π} M_{λ}. Then one can lift 𝔪_{0}L to a finitely generated submodule L′ of A^{n}. Then Ker(π) + L′ is a finitely generated submodule of Ker(p_{λ} ∘ π) containing 𝔪_{0}^{2} · Ker(p_{λ} ∘ π). Since we also have 𝔪_{0} · Coker(p_{λ} ∘ π) = 0 and 𝔪_{0} is arbitrary, the conclusion follows from proposition 2.3.10. □

Lemma 2.3.17. Let 0 → M′ → M → M′′ → 0 be an exact sequence of A-modules. Then: (i) If M′, M′′ are almost finitely generated (resp. presented) then so is M. (ii) If M is almost finitely presented, then M′′ is almost finitely presented if and only if M′

is almost finitely generated.

Proof. These facts can be deduced from proposition 2.3.15 and remark 2.4.12(iii), or proved directly. □

Lemma 2.3.18. Let (M_{n} ; ϕ_{n} : M_{n} → M_{n+1} | n ∈ ℕ) be a direct system of A-modules and suppose there exist sequences (ε_{n} | n ∈ ℕ) and (δ_{n} | n ∈ ℕ) of ideals of V such that

(i) lim_{n→∞}ε_{n}^{a} = V^{a} (for the uniform structure of definition 2.3.1) and ∏_{j=0}^{∞} δ_{j} is a Cauchy product (see (2.3.4);

(ii) for all n ∈ ℕ there exist integers N(n) and morphisms of A-modules ψ_{n} : A^{N(n)} → M_{n}

such that ε_{n} · Coker ψ_{n} = 0; (iii) δ_{n} · Coker ϕ_{n} = 0 for all n ∈ ℕ.

Then colim_{n∈ℕ}M_{n} is an almost finitely generated A-module.

Proof. Let M := colim_{n∈ℕ}M_{n}. For any n ∈ ℕ let a_{n} = ⋂_{m≥0}(∏_{j=n}^{n+m} δ_{j}). Then lim_{n→∞}a_{n} = V. For m > n set ϕ_{n,m} = ϕ_{m} ∘ ... ∘ ϕ_{n+1} ∘ ϕ_{n} : M_{n} → M_{m+1} and let ϕ_{n,∞} : M_{n} → M be the natural morphism. An easy induction shows that ∏_{j=n}^{m} δ_{j} · Coker ϕ_{n,m} = 0 for all m > n ∈ ℕ. Since Coker ϕ_{n,∞} = colim_{m∈ℕ}Coker ϕ_{n,m} we obtain a_{n} · Coker ϕ_{n,∞} = 0 for all n ∈ ℕ. Therefore

ε_{n} · a_{n} · Coker(ϕ_{n,∞} ∘ ψ_{n}) = 0 for all n ∈ ℕ. Since lim_{n→∞}ε_{n} · a_{n} = V, the claim follows. □

In the remaining of this section we assume that condition (B) of (2.1.6) holds. We wish to define the Fitting ideals of an arbitrary uniformly almost finitely generated A-module M. This will be achieved in two steps: first we will see how to define the Fitting ideals of a finitely generated module, then we will deal with the general case. We refer to [51, Ch.XIX] for the definition of the Fitting ideals F_{i}(M) of a finitely generated module over an arbitrary ring R. Lemma 2.3.19. Let R be a V -algebra and M, N two finitely generated R-modules with an isomorphism of R^{a}-modules M^{a} ≃ N^{a}. Then F_{i}(M)^{a} = F_{i}(N)^{a} for every i ≥ 0.

Proof. By the usual arguments, for every ε ∈ 𝔪 we have morphisms α : M → N, β : N → M with kernels and cokernels killed by ε^{2}. Then we have: F_{i}(N) ⊃ F_{i}(Im α) · F_{0}(Coker α). If N is generated by k elements, then the same holds for Coker α, whence Ann_{R}(Coker α)^{k} ⊂ F_{0}(Coker α), therefore ε^{2k}R ⊂ F_{0}(Coker α), and consequently F_{i}(N) ⊃ ε^{2k}F_{i}(Im α). Since Im(α) is a quotient of M, it is clear that F_{i}(M) ⊂ F_{i}(Im α), so finally ε^{2k}F_{i}(M) ⊂ F_{i}(N). Arguing symmetrically with β one has ε^{2k}F_{i}(N) ⊂ F_{i}(M). Since we assume (B), the claim follows. □

2.3.20. Let M be a finitely generated A-module. In light of lemma 2.3.19, the Fitting ideals F_{i}(M) are well defined as ideals in A.

Lemma 2.3.21. Let 𝔪_{0} ⊂ 𝔪 be a finitely generated subideal and n ∈ ℕ. Pick ε_{1}, ..., ε_{k} ∈ 𝔪 such that 𝔪_{0} ⊂ (ε_{1}^{3n}, ..., ε_{k}^{3n}) and set 𝔪_{1} := (ε_{1}, ..., ε_{k}). Then (F_{i}(M), F_{i}(M′)) ∈ E_{A}(𝔪_{0}) for every (M, M′) ∈ E_{ℳ} (𝔪_{1}) such that M and M′ are generated by at most n of their almost elements.

Proof. Let M, M′ be as in the lemma. By hypothesis, there exist an A-module N and morphisms ϕ : N → M and ψ : N → M′ such that 𝔪_{1} annihilates the kernel and cokernel of ϕ and ψ. By symmetry, it suffices to show that ε_{i}^{3n}F_{i}(M) ⊂ F_{i}(M′) for every i = 1, ..., k. Now, for every i ≤ k, the morphism M → M : x ↦ ε_{i} · x factors through a morphism α : M → ϕ(N), and similarly, scalar multiplication by ε_{i} on N factors through a morphism β : ϕ(N) → N. Then η := ψ ∘ β ∘ α : M → M′ has kernel and cokernel annihilated by ε_{i}^{3}. Pick finitely generated A_{∗}-modules L ⊂ M_{∗}, L′ ⊂ M′_{∗} such that L^{a} = M and L^{′a} = M′. Replacing L′ by L′ + η_{∗}(L) we can assume that η_{∗}(L) ⊂ L′. Then F_{i}(M) = F_{i}(L)^{a}, F_{i}(L′)^{a} = F_{i}(M′) and F_{i}(L′) ⊃ F_{i}(L/(L ∩ Ker η_{∗})) · F_{0}(L′/η_{∗}L). Since L′/η_{∗}(L) is generated by at most n elements and is annihilated by ε_{i}^{3} · 𝔪, we have ε_{i}^{3n} · 𝔪 ⊂ F_{0}(L′/(η_{∗}L)). Furthermore F_{i}(L/(L ∩ Ker η_{∗})) ⊃ F_{i}(L), so the claim follows. □

Proposition 2.3.22. For every i, n ≥ 0, the map F_{i} : ℳ_{n}(A) → ℐ_{A}(A) is uniformly continuous and therefore it extends uniquely to a uniformly continuous map F_{i} : 𝒰_{n}(A) → ℐ_{A}(A).

Proof. The uniform continuity follows readily from lemma 2.3.21. Since ℐ_{A}(A) is complete, it follows that F_{i} extends to the whole of 𝒰_{n}(A). Finally, the extension is unique because ℐ_{A}(A) is separated. □

Definition 2.3.23. Let M be a uniformly almost finitely generated A-module. We call F_{i}(M) the i-th Fitting ideal of M.

Proposition 2.3.24. (i) Let 0 → M′ →^{ϕ} M →^{ψ} M′′ → 0 be a short exact sequence of uniformly almost finitely generated A-modules. Then: ∑_{j+k=i} F_{j}(M′)·F_{k}(M′′) ⊂ F_{i}(M) for every i ≥ 0.

(ii) For every uniformly almost finitely generated A-module M, any A-algebra B and any i ≥ 0 we have F_{i}(B ⊗_{A} M) = F_{i}(M) · B.

Proof. (i): Let n be uniform bound for M and M′; by remark 2.3.2 we can find, for every

subideal 𝔪_{0} ⊂ 𝔪, A-modules M_{0}, M′_{0}, L, L′ and morphisms M →^{α} L ←^{β} M_{0}, M′ →^{α′} L′ ←^{β′} M′_{0}

whose kernels and cokernels are annihilated by 𝔪_{0}, and such that M_{0} and M′_{0} are generated by n almost elements. Let N be defined by the push-out diagram

\xymatrix{M′ \ar[r]^{ϕ} \ar[d]_{α′} & M \ar[r]^{α} & L \ar[d]^{γ} \\ L′ \ar[rr]^{γ′} &  & N.}

Furthermore set M′_{1} := Im(γ′∘β′ : M′_{0} → N), M_{1} := Im((γ∘β)⊕(γ′∘β′) : M_{0}⊕M′_{0} → N) and let M′′_{1} be the cokernel of the induced monomorphism M′_{1} → M_{1}. We deduce a commutative diagram with short exact rows:

(2.3.25)  \xymatrix{0 \ar[r] & M′ \ar[r]^{ϕ} \ar[d] & M \ar[r]^{ψ} \ar[d] & M′′ \ar[d] \ar[r] & 0 \\ 0 \ar[r] & Im γ′ \ar[r] & N \ar[r] & Coker γ′ \ar[r] & 0 \\ 0 \ar[r] & M′_{1} \ar[u] \ar[r] & M_{1} \ar[u] \ar[r] & M′′_{1} \ar[u] \ar[r] & 0.}

One checks easily that the kernels and cokernels of all the vertical arrows in (2.3.25) are annihilated by 𝔪_{0}^{2}, i.e. (M, M_{1}), (M′, M′_{1}), (M′′, M′′_{1}) ∈ E_{ℳ} (𝔪_{0}^{2}). Let x_{1}, ..., x_{n} ∈ M_{0∗} (resp. x′_{1}, ..., x′_{n} ∈ M′_{0∗}) be a set of generators for M_{0} (resp. for M′_{0}). For every i = 1, ..., n, let z_{i} := γ ∘ β(x_{i}) and z′_{i} := γ′ ∘ β′(x′_{i}). Let Q′ ⊂ N_{∗} (resp Q ⊂ N_{∗}) be the A_{∗}-module generated by the z′_{i} (resp. and by the z_{i}). It is clear that the bottom row of (2.3.25) is naturally isomorphic to the short exact sequence (0 → Q′ → Q → Q/Q′ → 0)^{a}. It is well-known that F_{i}(Q′) · F_{j}(Q/Q′) ⊂ F_{i+j}(Q) for every i, j ∈ ℕ; by lemma 2.3.5(ii.c) and proposition 2.3.22 all the operations under considerations are uniformly continuous, so we deduce F_{i}(M′) · F_{j}(M/M′) ⊂ F_{i+j}(M), which is (i).

(ii): since the identity is known for usual finitely generated modules over rings, the claim follows easily from proposition 2.3.22 and lemma 2.3.7(v). □

2.4. Almost homological algebra. In this section we fix an almost V -algebra A and we consider various constructions in the category of A-modules.

2.4.1. By corollary 2.2.15 any inverse system (M_{n} | n ∈ ℕ) of A-modules has an (inverse) limit lim_{n∈ℕ}M. As usual, we denote by lim^{1} the right derived functor of the inverse limit functor. Notice that [67, Cor. 3.5.4] holds in the almost case since axiom (AB4*) holds in A-Mod (on the other hand, [67, Lemma 3.5.3] does not hold under (AB4*), (the proof given there uses elements : for a counterexample in an exotic abelian category, see [57]). Lemma 2.4.2. Let (M_{n} ; ϕ_{n} : M_{n} → M_{n+1} | n ∈ ℕ) (resp. (N_{n} ; ψ_{n} : N_{n+1} → N_{n} | n ∈ ℕ)) be a direct (resp. inverse) system of A-modules and morphisms and (ε_{n} | n ∈ ℕ) a sequence of ideals of V^{a} converging to V^{a} (for the uniform structure of definition 2.3.1).

(i) If ε_{n} · M_{n} = 0 for all n ∈ ℕ then colim_{n∈ℕ}M_{n} ≃ 0.

(ii) If ε_{n} · N_{n} = 0 for all n ∈ ℕ then lim_{n∈ℕ}N_{n} ≃ 0 ≃ lim_{n∈ℕ}^{1}N_{n}.

(iii) If ε_{n} · Coker ψ_{n} = 0 for all n ∈ ℕ and ∏_{j=0}^{∞} ε_{j} is a Cauchy product, then lim_{n∈ℕ}^{1}N_{n} ≃ 0.

Proof. (i) and (ii) : we remark only that lim_{n∈ℕ}^{1}N_{n} ≃ lim_{n∈ℕ}^{1}N_{n+p} for all p ∈ ℕ and leave the details

to the reader. We prove (iii). From [67, Cor. 3.5.4] it follows easily that (lim_{n∈ℕ}^{1}N_{n∗})^{a} ≃ lim_{n∈ℕ}^{1}N_{n}.

It then suffices to show that lim_{n∈ℕ}^{1}N_{n∗} is almost zero. We have ε_{n}^{2} · Coker ψ_{n∗} = 0 and the

product ∏_{j=0}^{∞} ε_{j}^{2} is again a Cauchy product. Next let N′_{n} := ⋂_{p≥0} Im(N_{n+p∗} → N_{n∗}). If

J_{n} := ⋂_{p≥0}(ε_{n} · ε_{n+1} · ... · ε_{n+p})^{2} then J_{n}N_{n∗} ⊂ N′_{n} and lim_{n→∞}J_{n}^{a} = V^{a}. In view of (ii),

lim_{n∈ℕ}^{1}N_{n∗}/N′_{n} is almost zero, hence we reduce to showing that lim_{n∈ℕ}^{1}N′_{n} is almost zero. But

J_{n+p+q} · N′_{n} ⊂ Im(N′_{n+p+q} → N′_{n}) ⊂ Im(N′_{n+p} → N′_{n})

for all n, p, q ∈ ℕ. On the other hand, since the ideals J_{n}^{a} converge to V^{a}, we get ⋃_{q=0}^{∞} 𝔪 · J_{n+p+q} = 𝔪, hence 𝔪N′_{n} ⊂ Im(N′_{n+p} → N′_{n}) and finally 𝔪N′_{n} = 𝔪^{2}N′_{n} ⊂ Im(𝔪N′_{n+p} → 𝔪N′_{n}) which means that {𝔪N′_{n}} is a surjective inverse system, so its lim^{1} vanishes and the result follows. □

Example 2.4.3. Let (V, 𝔪) be as in example 2.1.2. Then every finitely generated ideal in V is principal, so in the situation of the lemma we can write ε_{j} = (x_{j}) for some x_{j} ∈ V. Then the hypothesis in (iii) can be stated by saying that there exists c ∈ ℕ such that x_{j} ≠ 0 for all j ≥ c and the sequence n ↦ ∏_{j=c}^{n} |x_{j}| is Cauchy in Γ. Definition 2.4.4. Let M be an A-module.

(i) We say that M is flat (resp. faithfully flat) if the functor N ↦ M ⊗_{A} N, from the category of A-modules to itself is exact (resp. exact and faithful).

(ii) We say that M is almost projective if the functor N ↦ alHom_{A}(M, N) is exact. For euphonic reasons, we will use the expression ”almost finitely generated projective” to denote an A-module which is almost projective and almost finitely generated. This convention does not give rise to ambiguities, since we will never consider projective almost modules : indeed, the following example 2.4.5 explains why the categorical notion of projectivity is useless in the setting of almost ring theory. Example 2.4.5. First of all we remark that the functor M ↦ M_{!} preserves (categorical) projectivity, since it is left adjoint to an exact functor. Moreover, if P_{!} is a projective V -module P is a projective V^{a}-module, as one checks easily using the fact the the functor M ↦ M_{!} is right exact. Hence one has an equivalence from the full subcategory of projective V^{a}-modules, to the full subcategory of projective V -modules P such that P = 𝔪̃ ⊗_{V} P. The latter condition is equivalent to P = 𝔪P; indeed, as P is flat, we have 𝔪P = 𝔪 ⊗_{V} P.

As an example, suppose that V is local; then every projective V -module is free, so if 𝔪 ≠ V the condition P = 𝔪P implies P = 0, ergo, there are no non-trivial projective V^{a}-modules. Lemma 2.4.6. Let M be an almost finitely generated A-module and B a flat A-algebra. Then Ann_{B}(B ⊗_{A} M) = B ⊗_{A} Ann_{A}(M).

Proof. Using lemma 2.3.7(i),(v) we reduce easily to the case of a finitely generated A-module M. Then, let x_{1}, ..., x_{k} ∈ M_{∗} be a finite set of generators for M; we have: Ann_{A}(M) = Ker(ϕ : A → M^{k}), where ϕ is defined by the rule: a ↦ (a · x_{1}, ..., a · x_{k}) for every a ∈ A_{∗}. Since B is flat, we have Ker(1_{B} ⊗_{A} ϕ) ≃ B ⊗_{A} Ker ϕ, whence the claim. □

Lemma 2.4.7. Let P be one of the properties : “flat”, “almost projective”, “almost finitely generated”, “almost finitely presented”. If B is a P A-algebra, and M is a P B-module, then M is P as an A-module.

Proof. Left to the reader. □

2.4.8. Let R be a V -algebra and M a flat (resp. faithfully flat) R-module (in the usual sense, see [54, p.45]). Then M^{a} is a flat (resp. faithfully flat) R^{a}-module. Indeed, the functor M ⊗_{R} − preserves the Serre subcategory of almost zero modules, so by general facts it induces an exact functor on the localized categories (cp. [38, p.369]). For the faithfullness we have to show that an R-module N is almost zero whenever M ⊗_{R} N is almost zero. However, M ⊗_{R} N is almost zero ⇔ M ⊗_{R} (𝔪 ⊗_{V} N) = 0 ⇔ 𝔪 ⊗_{V} N = 0 ⇔ N is almost zero. It is clear that A-Mod has enough almost projective (resp. flat) objects.

2.4.9. Let R be a V -algebra. The localisation functor induces a functor G : D(R-Mod) → D(R^{a}-Mod) and, in view of corollary 2.2.22, M ↦ M_{!} induces a functor F : D(R^{a}-Mod) → D(R-Mod). We have a natural isomorphism G ∘ F ≃ 1_{D(R^{a}-Mod)} and a natural transformation F ∘ G → 1_{D(R-Mod)}. These satisfy the triangular identities of [53, p.83] so F is a left adjoint to G. If Σ denotes the multiplicative set of morphisms in D(R-Mod) which induce almost isomorphisms on the cohomology modules, then the localised category Σ^{−1}D(R-Mod) exists (see e.g. [67, Th.10.3.7]) and by the same argument we get an equivalence of categories Σ^{−1}D(R-Mod) ≃ D(R^{a}-Mod).

2.4.10. Given an A-module M, we can derive the functors M ⊗_{A} − (resp. alHom_{A}(M, −), resp. alHom_{A}(−, M)) by taking flat (resp. injective, resp. almost projective) resolutions : one remarks that bounded above exact complexes of flat (resp. almost projective) A-modules are acyclic for the functor M ⊗_{A} − (resp. alHom_{A}(−, M)) (recall the standard argument: if F_{•} is a bounded above exact complex of flat A-modules, let Φ_{•} be a flat resolution of M; then Tot(Φ_{•} ⊗_{A} F_{•}) → M ⊗_{A} F_{•} is a quasi-isomorphism since it is so on rows, and Tot(Φ_{•} ⊗_{A} F_{•}) is acyclic since its columns are; similarly, if P_{•} is a complex of almost projective objects, one considers the double complex alHom_{A}(P_{•}, J^{•}) where J^{•} is an injective resolution of M; cp. [67, §2.7]); then one uses the construction detailed in [67, Th.10.5.9]. We denote by Tor_{i}^{A}(M, −) (resp. alExt_{A}^{i}(M, −), resp. alExt_{A}^{i}(−, M)) the corresponding derived functors. If A := R^{a} for some V -algebra R, we obtain easily natural isomorphisms

(2.4.11)  Tor_{i}^{R}(M, N)^{a} ≃ Tor_{i}^{A}(M^{a}, N^{a})

for all R-modules M, N. A similar result holds for Ext_{R}^{i}(M, N). Remark 2.4.12. (i) Clearly, an A-module M is flat (resp. almost projective) if and only if Tor_{i}^{A}(M, N) = 0 (resp. alExt_{A}^{i}(M, N) = 0) for all A-modules N and all i > 0. In particular, an almost projective A-module is flat, because for every ε ∈ 𝔪 the scalar multiplication by ε : M → M factors through a free module.

(ii) Let M, N be two flat (resp. almost projective) A-modules. Then M ⊗_{A} N is a flat (resp. almost projective) A-module and for any A-algebra B, the B-module B ⊗_{A} M is flat (resp. almost projective).

(iii) Resume the notation of proposition 2.3.15. If M is almost finitely presented, then one has also that the natural morphism colim_{Λ} alExt_{A}^{1}(M, N_{λ}) → alExt_{A}^{1}(M, colim_{Λ} N_{λ}) is a monomorphism. This is deduced from proposition 2.3.15(ii), using the fact that (N_{λ}) can be injected into an inductive system (J_{λ}) of injective A-modules (e.g. J_{λ} = E^{Hom_{A}(N_{λ},E)}, where E is an injective cogenerator for A-Mod), and by applying alExt sequences. Lemma 2.4.13. Resume the notation of (2.3.13). The following conditions are equivalent:

(i) M^{a} is a flat (resp. almost projective) R^{a}-module.

(ii) M_{λ}^{a} is a flat (resp. almost projective) R_{λ}^{a}-module for every λ ∈ Λ.

Proof. It is a straightforward consequence of (2.4.11)and its analogue for Ext_{R}^{i}(M, N). □

Lemma 2.4.14. Let R be a V -algebra.

(i) There is a natural isomorphism: Ext_{R}^{i}(M_{!}, N) ≃ Ext_{R^{a}}^{i}(M, N^{a}) for every R^{a}-module M, every R-module N and every integer i ∈ ℕ.

(ii) If P is an almost projective R^{a}-module, then hom.dim_{R}P_{!} ≤ hom.dim_{V}𝔪̃.

Proof. (here hom.dim denotes homological dimension). (i) is a straightforward consequence of the existence of the adjunction (F, G) of (2.4.9). Next we consider, for arbitrary R-modules M and N, the spectral sequence:

E_{2}^{p,q} := Ext_{V}^{p} (𝔪̃, Ext_{R}^{q}(M, N)) ⇒ Ext_{R}^{p+q}(𝔪̃ ⊗_{V} M, N).

(this spectral sequence is constructed e.g. from the double complex Hom_{V} (F_{p}, Hom_{R}(F′_{q}, N)) where F_{•} (resp. F′_{•}) is a projective resolution of 𝔪̃ (resp. of M)). If now we let M := P_{!}, we deduce from (i) that Ext_{V}^{p} (𝔪̃, Ext_{R}^{q}(P_{!}, N)) ≃ Ext_{V^{a}}^{p}(V^{a}, Ext_{R}^{q}(P_{!}, N)^{a}) ≃ 0 for every p ∈ ℕ and every q > 0. Since 𝔪̃ ⊗_{V} P_{!} ≃ P_{!}, assertion (ii) follows easily. □

Lemma 2.4.15. Let M be an almost finitely generated A-module. Then M is almost projective if and only if, for arbitrary ε ∈ 𝔪, there exist n(ε) ∈ ℕ and A-linear morphisms

(2.4.16)  M ⟶^{u_{ε}} A^{n(ε)} ⟶^{v_{ε}} M

such that v_{ε} ∘ u_{ε} = ε · 1_{M}.

Proof. Let morphisms as in (2.4.16) be given. Pick any A-module N and apply the functor alExt_{A}^{i}(−, N) to (2.4.16)to get morphisms

alExt_{A}^{i}(M, N) → alExt_{A}^{i}(A^{n(ε)}, N) → alExt_{A}^{i}(M, N)

whose composition is again the scalar multiplication by ε; hence ε · alExt_{A}^{i}(M, N) = 0 for all i > 0. Since ε is arbitrary, it follows from remark 2.4.12(i) that M is almost projective. Conversely, suppose that M is almost projective; by hypothesis, for arbitrary ε ∈ 𝔪 we can find n := n(ε) and a morphism ϕ_{ε} : A^{n} → M such that ε · Coker ϕ_{ε} = 0. Let M_{ε} be the image of ϕ_{ε}, so that ϕ_{ε} factors as A^{n(ε)} ⟶^{ψ_{ε}} M_{ε} ⟶^{j_{ε}} M. Also ε · 1_{M} : M → M factors as M ⟶^{γ_{ε}} M_{ε} ⟶^{j_{ε}} M. Since by hypothesis M is almost projective, the natural morphism induced by ψ_{ε} :

alHom_{A}(M, A^{n}) ⟶^{ψ_{ε}^{∗}} alHom_{A}(M, M_{ε})

is an epimorphism. Then for arbitrary δ ∈ 𝔪 the morphism δ · γ_{ε} is in the image of ψ_{ε}^{∗}, in other words, there exists an A-linear morphism u_{εδ} : M → A^{n} such that ψ_{ε} ∘ u_{εδ} = δ · γ_{ε}. If now we take v_{εδ} := ϕ_{ε}, it is clear that v_{εδ} ∘ u_{εδ} = ε · δ · 1_{M}. This proves the claim. □

Lemma 2.4.17. Let R be any ring, M any R-module and C := Coker(ϕ : R^{n} → R^{m}) any finitely presented (left) R-module. Let C′ := Coker(ϕ^{∗} : R^{m} → R^{n}) be the cokernel of the transpose of the map ϕ. Then there is a natural isomorphism

Tor_{1}^{R}(C′, M) ≃ Hom_{R}(C, M)/Im(Hom_{R}(C, R) ⊗_{R} M).

Proof. We have a spectral sequence :

E_{ij}^{2} := Tor_{i}^{R}(H_{j}(Cone ϕ^{∗}), M) ⇒ H_{i+j}(Cone(ϕ^{∗}) ⊗_{R} M).

On the other hand we have also natural isomorphisms

Cone(ϕ^{∗}) ⊗_{R} M ≃ Hom_{R}(Cone ϕ, R)[1] ⊗_{R} M ≃ Hom_{R}(Cone ϕ, M)[1].

Hence :

E_{10}^{2} ≃ E_{10}^{∞} ≃ H_{1}(Cone(ϕ^{∗}) ⊗_{R} M)/E_{01}^{∞} ≃ H^{0}(Hom_{R}(Cone ϕ, M))/Im(E_{01}^{2})
≃ Hom_{R}(C, M)/Im(Hom_{R}(C, R) ⊗_{R} M)

which is the claim. □

Proposition 2.4.18. Let A be a V^{a}-algebra.

(i) Every almost finitely generated projective A-module is almost finitely presented. (ii) Every almost finitely presented flat A-module is almost projective.

Proof. (ii) : let M be such an A-module. Let ε, δ ∈ 𝔪 and pick a three term complex

A^{m} ⟶^{ψ} A^{n} ⟶^{ϕ} M

such that ε · Coker ϕ = δ · Ker(ϕ)/Im(ψ) = 0. Set P := Coker ψ_{∗}; this is a finitely presented A_{∗}-module and ϕ_{∗} factors through a morphism \overline{ϕ}_{∗} : P → M_{∗}. Let γ ∈ 𝔪; from lemma 2.4.17 we see that γ · \overline{ϕ} is the image of some element ∑_{j=1}^{n} ϕ_{j} ⊗ m_{j} ∈ Hom_{A_{∗}}(P, A_{∗}) ⊗_{A_{∗}} M_{∗}. If we define L := A_{∗}^{n} and v : P → L, w : L → M_{∗} by v(x) := (ϕ_{1}(x), ..., ϕ_{n}(x)) and w(y_{1}, ..., y_{n}) := ∑_{j=1}^{n} y_{j} ·m_{j}, then clearly γ ·\overline{ϕ} = w ∘v. Let K := Ker \overline{ϕ}_{∗}. Then δ ·K^{a} = 0 and the map δ ·1_{P^{a}} factors through a morphism σ : (P/K)^{a} → P^{a}. Similarly the map ε·1_{M} factors through a morphism λ : M → (P/K)^{a}. Let α := v^{a} ∘σ ∘λ : M → L^{a} and β := w^{a} : L^{a} → M. The reader can check that β ∘ α = ε · δ · γ · 1_{M}. By lemma 2.4.15 the claim follows.

(i) : let P be such an almost finitely generated projective A-module. For any finitely generated ideal 𝔪_{0} ⊂ 𝔪 pick a morphism ϕ : A^{r} → P such that 𝔪_{0} · Coker ϕ = 0. If ε_{1}, ..., ε_{k} is a set of generators for 𝔪_{0}, a standard argument shows that, for any i ≤ k, ε_{i} · 1_{P} lifts to a morphism ψ_{i} : P → A^{r}/Ker ϕ; then, since P is almost projective, ε_{j}ψ_{i} lifts to a morphism ψ_{ij} : P → A^{r}. Now claim 2.3.11 applies with F_{1} := A^{r}, F_{2} := M = P, p := ϕ, q := 1_{P} and ψ := ψ_{ij} and shows that Ker ϕ has a finitely generated submodule M_{ij} containing ε_{i} · ε_{j} · Ker ϕ. Then the span of all such M_{ij} is a finitely generated submodule of Ker ϕ containing 𝔪_{0}^{2} · Ker ϕ. By proposition 2.3.10(ii), the claim follows. □

In general a flat almost finitely generated A-module is not necessarily almost finitely presented, but one can give the following criterion, which extends [17, §1, Exerc.13]. Proposition 2.4.19. If A → B is a monomorphism of V^{a}-algebras and M is an almost finitely generated flat A-module such that B⊗_{A}M is almost finitely presented over B, then M is almost finitely presented over A.

Proof. Let 𝔪_{0} be a finitely generated subideal of 𝔪 and an A-linear morphism ϕ : A^{n} → M such that 𝔪_{0} · Coker ϕ = 0. By assumption and claim 2.3.12, we can find a finitely generated B-submodule R of Ker(1_{B} ⊗_{A} ϕ) such that

(2.4.20)  𝔪_{0}^{2} · Ker(1_{B} ⊗_{A} ϕ) ⊂ R.

By a Tor sequence we have 𝔪_{0} · Coker(1_{B} ⊗_{A} Ker(ϕ) → Ker(1_{B} ⊗_{A} ϕ)) = 0, hence 𝔪 · 𝔪_{0} · Coker(1_{B_{∗}} ⊗_{A_{∗}} Ker(ϕ)_{∗} → Ker(1_{B} ⊗_{A} ϕ)_{∗}) = 0, therefore there exists a finitely generated submodule R_{0} of Ker ϕ such that

(2.4.21)  𝔪_{0}^{2}R ⊂ B · Im(R_{0} → Ker(1_{B} ⊗_{A} ϕ)).

By lemma 2.4.17, for every ε ∈ 𝔪 the morphism ε · \overline{ϕ} : A^{n}/R_{0} → M factors through a morphism ψ : A^{n}/R_{0} → F, where F is a finitely generated free A-module. Since F ⊂ B ⊗_{A} F, we deduce easily Ker(A^{n}/R_{0} → B^{n}/B · R_{0}) ⊂ Ker ψ; on the other hand, by (2.4.20) and (2.4.21) we derive 𝔪_{0}^{4} · Ker ϕ ⊂ Ker(A^{n}/R_{0} → B^{n}/B · R_{0}). Thus ψ factors through M′ := A^{n}/(R_{0} + 𝔪_{0}^{4} · Ker ϕ). Clearly 𝔪_{0}^{4} · Ker(M′ → M) = 𝔪_{0} · Coker(M′ → M) = 0; hence, for every A-module N, the kernel of the induced morphism

(2.4.22)  alExt_{A}^{1}(M, N) → alExt_{A}^{1}(M′, N)

is annihilated by 𝔪_{0}^{5}; however (2.4.22) factors through alExt_{A}^{1}(F, N) = 0, therefore 𝔪_{0}^{5} · alExt_{A}^{1}(M, N) = 0 for every A-module N. This shows that M is almost projective, which is equivalent to the conclusion, in view of proposition 2.4.18(i). □

Definition 2.4.23. Let M be an A-module, f : A → M an almost element of M.

(i) The dual A-module of M is the A-module M^{∗} := alHom_{A}(M, A).

(ii) The evaluation morphism is the morphism ev_{M/A} : M ⊗_{A} M^{∗} → A : m⊗ϕ ↦ ϕ(m). (iii) The evaluation ideal of M is the ideal ℰ_{M/A} := Im ev_{M/A}. (iv) The evaluation ideal of f is the ideal ℰ_{M/A}(f) := Im(ev_{M/A} ∘ (f ⊗_{A} 1_{M^{∗}})).

(v) We say that M is reflexive if the natural morphism

(2.4.24)  M → (M^{∗})^{∗}  m ↦ (f ↦ f(m))

is an isomorphism of A-modules. (vi) We say that M is invertible if M ⊗_{A} M^{∗} ≃ A.

Remark 2.4.25. Notice that if B is an A-algebra and M any B-module, then by “restriction of scalars” M is also an A-module and the dual A-module M^{∗} has a natural structure of Bmodule. This is defined by the rule (b · f)(m) := f(b · m) (b ∈ B_{∗}, m ∈ M_{∗} and f ∈ M_{∗}^{∗}). With respect to this structure (2.4.24) becomes a B-linear morphism. Incidentally, notice that the two meanings of “M_{∗}^{∗}” coincide, i.e. (M_{∗})^{∗} ≃ (M^{∗})_{∗}.

2.4.26. If E, F and N are A-modules, there is a natural morphism :

(2.4.27)  E ⊗_{A} alHom_{A}(F, N) → alHom_{A}(F, E ⊗_{A} N).

Let P be an A-module. As a special case of (2.4.27)we have the morphism:

ω_{P/A} : P ⊗_{A} P^{∗} → End_{A}(P)^{a} := alHom_{A}(P, P)

such that ω_{P/A}(p ⊗ ϕ)(q) := p · ϕ(q) for every p, q ∈ P_{∗} and ϕ : P → A. Proposition 2.4.28. Let P be an almost projective A-module.

(i) For every morphism of algebras A → B we have ℰ_{B⊗_{A}P/B} = ℰ_{P/A} · B.

(ii) ℰ_{P/A} = ℰ_{P/A}^{2}.

(iii) P = 0 if and only if ℰ_{P/A} = 0.

(iv) P is faithfully flat if and only if ℰ_{P/A} = A.

(v) ℰ_{P/A}(f) is the smallest of the ideals J ⊂ A such that f ∈ (JP)_{∗}.

Proof. Pick an indexing set I large enough, and an epimorphism ϕ : F := A^{(I)} → P. For every i ∈ I we have the standard morphisms A →^{e_{i}} F →^{π_{i}} A such that π_{i} ∘ e_{j} = δ_{ij} · 1_{A} and ∑_{i∈I} e_{i} ∘ π_{i} = 1_{F}. For every x ∈ 𝔪 choose ψ_{x} ∈ Hom_{A}(P, F) such that ϕ ∘ ψ_{x} = x · 1_{P}. It is easy to check that ℰ_{P/A} is generated by the almost elements π_{i} ∘ ψ_{x} ∘ ϕ ∘ e_{j} (i, j ∈ I, x ∈ 𝔪). (i) follows already. For (iii), the “only if” is clear; if ℰ_{P/A} = 0, then ψ_{x}∘ϕ = 0 for all x ∈ 𝔪, hence ψ_{x} = 0 and therefore x · 1_{P} = 0, i.e. P = 0. Next, notice that (i) and (iii) imply P/ℰ_{P/A}P = 0, i.e. P = ℰ_{P/A}P, so (ii) follows directly from the definition of ℰ_{P/A}. Since P is flat, to show (iv) we have only to verify that the functor M ↦ P ⊗_{A} M is faithful. To this purpose, it suffices to check that P ⊗_{A} A/J ≠ 0 for every proper ideal J of A. This follows easily from (i) and (iii). Finally, it clear that ℰ_{P/A}(f) ⊂ J for every ideal J ⊂ A with f ∈ (JP)_{∗}. Conversely, for given ε ∈ 𝔪 pick a sequence P →^{u} A^{n} →^{v} P as in (2.4.16); we have u(f) = (a_{1}, ..., a_{n}) with a_{1}, ..., a_{n} ∈ ℰ_{P/A}(f)_{∗}. (v) follows from the identity εf = ∑_{i=1}^{n} a_{i} · v(e_{i}) (where e_{1}, ..., e_{n} is the standard basis of the free A_{∗}-module A_{∗}^{n}). □

Lemma 2.4.29. Let E, F, N be three A-modules.

(i) The morphism (2.4.27)is an isomorphism in the following cases : (a) when E is flat and F is almost finitely presented;

(b) when either E or F is almost finitely generated projective;

(c) when F is almost projective and E is almost finitely presented;

(d) when E is almost projective and F is almost finitely generated. (ii) The morphism (2.4.27)is a monomorphism in the following cases :

(a) when E is flat and F is almost finitely generated;

(b) when E is almost projective. (iii) The morphism (2.4.27)is an epimorphism when F is almost projective and E is almost

finitely generated.

Proof. If F ≃ A^{(I)} for some finite set I, then alHom_{A}(F, N) ≃ N^{(I)} and the claims are obvious. More generally, if F is almost finitely generated projective, for any ε ∈ 𝔪 there exists a finite set I := I(ε) and morphisms

(2.4.30)  F ⟶^{u_{ε}} A^{(I)} ⟶^{v_{ε}} F

such that v_{ε} ∘ u_{ε} = ε · 1_{F}. We apply the natural transformation

E ⊗_{A} alHom_{A}(−, N) → alHom_{A}(−, E ⊗_{A} N)

to (2.4.30) : an easy diagram chase allows then to conclude that the kernel and cokernel of (2.4.27)are killed by ε. As ε is arbitrary, it follows that (2.4.27)is an isomorphism in this case. An analogous argument works when E is almost finitely generated projective, so we get (i.b). If F is only almost projective, then we still have morphisms of the type (2.4.30), but now I(ε) is no longer necessarily finite. However, the cokernels of the induced morphisms 1_{E} ⊗ u_{ε} and alHom_{A}(v_{ε}, E ⊗_{A} N) are still annihilated by ε. Hence, to show (iii) (resp. (i.c)) it suffices to consider the case when F is free and E is almost finitely generated (resp. presented). By passing to almost elements, we can further reduce to the analogous question for usual rings and modules, and by the usual juggling we can even replace E by a finitely generated (resp. presented) A_{∗}-module and F by a free A_{∗}-module. This case is easily dealt with, and (iii) and (i.c) follow. Case (i.d) (resp. (ii.b)) is similar : one considers almost elements and replaces E_{∗} by a free A_{∗}-module (resp. and F_{∗} by a finitely generated A_{∗}-module). In case (ii.a) (resp. (i.a)), for every finitely generated submodule 𝔪_{0} of 𝔪 we can find, by proposition 2.3.10, a finitely generated (resp. presented) A-module F_{0} and a morphism F_{0} → F whose kernel and cokernel are annihilated by 𝔪_{0}. It follows easily that we can replace F by F_{0} and suppose that F is finitely generated (resp. presented). Then the argument in [16, Ch.I §2 Prop.10] can be taken over verbatim to show (ii.a) (resp. (i.a)). □

Lemma 2.4.31. Let B be an A-algebra.

(i) Let P be an A-module. If either P or B is almost finitely generated projective as an A-module, the natural morphism

(2.4.32)  B ⊗_{A} alHom_{A}(P, N) → alHom_{B}(B ⊗_{A} P, B ⊗_{A} N)

is an isomorphism for all A-modules N. (ii) Every almost finitely generated projective A-module is reflexive. (iii) If P is an almost finitely generated projective B-module, the natural morphism

(2.4.33)  alHom_{B}(P, B) ⊗_{B} alHom_{A}(B, A) → alHom_{A}(P, A) : ϕ ⊗ ψ ↦ ψ ∘ ϕ

is an isomorphism of B-modules.

Proof. (i) is an easy consequence of lemma 2.4.29(i.b). To prove (ii), we apply the natural transformation (2.4.24) to (2.4.30) : by diagram chase one sees that the kernel and cokernel of the morphism F → (F^{∗})^{∗} are killed by ε. (iii) is analogous : one applies the natural transformation (2.4.33)to (2.4.30). □

Lemma 2.4.34. Let (M_{n} ; ϕ_{n} : M_{n} → M_{n+1} | n ∈ ℕ) be a direct system of A-modules and suppose there exist sequences (ε_{n} | n ∈ ℕ) and (δ_{n} | n ∈ ℕ) of ideals of V such that

(i) lim_{n→∞}ε_{n} = V and ∏_{j=0}^{∞} δ_{j} is a Cauchy product (see (2.3.4));

(ii) ε_{n} · alExt_{A}^{i}(M_{n}, N) = δ_{n} · alExt_{A}^{i}(Coker ϕ_{n}, N) = 0 for all A-modules N, all i > 0

and all n ∈ ℕ; (iii) δ_{n} · Ker ϕ_{n} = 0 for all n ∈ ℕ.

Then colim_{n∈ℕ}M_{n} is an almost projective A-module.

Proof. Let M = colim_{n∈ℕ}M_{n}. By remark 2.4.12(i) it suffices to show that alExt_{A}^{i}(M, N) vanishes for all i > 0 and all A-modules N. The maps ϕ_{n} define a map ϕ : ⊕_{n}M_{n} → ⊕_{n}M_{n} such that we have a short exact sequence 0 → ⊕_{n}M_{n} ⟶^{1−ϕ} ⊕_{n}M_{n} ⟶ M → 0. Applying the long exact alExt sequence one obtains a short exact sequence (cp. [67, 3.5.10])

0 → lim_{n∈ℕ}^{1}alExt_{A}^{i−1}(M_{n}, N) → alExt_{A}^{i}(M, N) → lim_{n∈ℕ}alExt_{A}^{i}(M_{n}, N) → 0.

Then lemma 2.4.2(ii) implies that alExt_{A}^{i}(M, N) ≃ 0 for all i > 1 and moreover alExt_{A}^{1}(M, N) is isomorphic to lim_{n∈ℕ}^{1}alHom_{A}(M_{n}, N). Let

ϕ_{n}^{∗} : alHom_{A}(M_{n+1}, N) → alHom_{A}(M_{n}, N) f ↦ f ∘ ϕ_{n}

be the transpose of ϕ_{n} and write ϕ_{n} as a composition M_{n} ⟶^{p_{n}} Im(ϕ_{n}) ↪^{q_{n}} M_{n+1}, so that ϕ_{n}^{∗} = q_{n}^{∗}∘p_{n}^{∗}, the composition of the respective transposed morphims. We have monomorphisms

Coker p_{n}^{∗} ↪ alHom_{A}(Ker ϕ_{n}, N)
Coker q_{n}^{∗} ↪ alExt_{A}^{1}(Coker ϕ_{n}, N)

for all n ∈ ℕ. Hence δ_{n}^{2} · Coker ϕ_{n}^{∗} = 0 for all n ∈ ℕ. Since ∏_{n=0}^{∞} δ_{n}^{2} is a Cauchy product, lemma 2.4.2(iii) shows that lim_{n∈ℕ}^{1}alHom_{A}(M_{n}, N) ≃ 0 and the assertion follows. □

Proposition 2.4.35. Suppose that 𝔪̃ is a flat V -module. Then for any V -algebra R the functor M ↦ M_{!} commutes with tensor products and takes flat R^{a}-modules to flat R-modules.

Proof. Let M be a flat R^{a}-module and N ↪ N′ an injective map of R-modules. Denote by K the kernel of the induced map M_{!} ⊗_{R} N → M_{!} ⊗_{R} N′; we have K^{a} ≃ 0. We obtain an exact sequence 0 → 𝔪̃ ⊗_{V} K → 𝔪̃ ⊗_{V} M_{!} ⊗_{R} N → 𝔪̃ ⊗_{V} M_{!} ⊗_{R} N′. But one sees easily that 𝔪̃ ⊗_{V} K = 0 and 𝔪̃ ⊗_{V} M_{!} ≃ M_{!}, which shows that M_{!} is a flat R-module. Similarly, let M, N be two R^{a}-modules. Then the natural map M_{∗} ⊗_{R} N_{∗} → (M ⊗_{R^{a}} N)_{∗} is an almost isomorphism and the assertion follows from remark 2.1.4(i). □

2.5. Almost homotopical algebra. The formalism of abelian tensor categories provides a minimal framework wherein the rudiments of deformation theory can be developed.

2.5.1. Let (𝒞 , ⊗, U) be an abelian tensor category; we assume henceforth that ⊗ is a right exact functor. Let A be a given 𝒞 -monoid. Then, for any two-sided ideal I of A, the quotient A/I in the underlying abelian category 𝒞 has a unique 𝒞 -monoid structure such that A → A/I is a morphism of monoids. A/I is unitary if A is. If I is a two-sided ideal of A such that I^{2} = 0, then, using the right exactness of ⊗ one checks that I has a natural structure of an A/I-bimodule, unitary when A is. Definition 2.5.2. A 𝒞 -extension of a 𝒞 -monoid B by a B-bimodule I is a short exact sequence of objects of 𝒞

(2.5.3)  X :  0 → I → C →^{p} B → 0

such that C is a 𝒞 -monoid, p is a morphism of 𝒞 -monoids, I is a square zero two-sided ideal in C and the E/I-bimodule structure on I coincides with the given bimodule structure on I. The 𝒞 -extensions form a category Exmon_{𝒞} . The morphisms are commutative diagrams with exact rows

\xymatrix{X : \ar[d] & 0 \ar[r] & I \ar[r] \ar[d]^{f} & E \ar[r]^{p} \ar[d]^{g} & B \ar[r] \ar[d]^{h} & 0 \\ X′ : & 0 \ar[r] & I′ \ar[r] & E′ \ar[r]^{p′} & B′ \ar[r] & 0}

such that g and h are morphisms of 𝒞 -monoids. We let Exmon_{𝒞} (B, I) be the subcategory of Exmon_{𝒞} consisting of all 𝒞 -extensions of B by I, where the morphisms are all short exact sequences as above such that f := 1_{I} and h := 1_{B}.

2.5.4. We have also the variant in which all the 𝒞 -monoids in (2.5.3)are required to be unitary (resp. to be algebras) and I is a unitary B-bimodule (resp. whose left and right B-module actions coincide, i.e. are switched by composition with the “commutativity constraints” θ_{B|I} and θ_{I|B}, see (2.2.5)); we will call Exun_{𝒞} (resp. Exal_{𝒞} ) the corresponding category.

2.5.5. For a morphism ϕ : C → B of 𝒞 -monoids, and a 𝒞 -extension X in Exmon_{𝒞} (B, I), we can pullback X via ϕ to obtain an exact sequence X ∗ ϕ with a morphism ϕ^{∗} : X ∗ ϕ → X; one checks easily that there exists a unique structure of 𝒞 -extension on X ∗ ϕ such that ϕ^{∗} is a morphism of 𝒞 -extension; then X ∗ ϕ is an object in Exmon_{𝒞} (C, I). Similarly, given a B-linear morphism ψ : I → J, we can push out X and obtain a well defined object ψ ∗ X in Exmon_{𝒞} (B, J) with a morphism X → ψ ∗ X of Exmon_{𝒞} . In particular, if I_{1} and I_{2} are two B-bimodules, the functors p_{i}∗ (i = 1, 2) associated to the natural projections p_{i} : I_{1} ⊕ I_{2} → I_{i} establish an equivalence of categories

(2.5.6)  Exmon_{𝒞} (B, I_{1} ⊕ I_{2}) →^{∼} Exmon_{𝒞} (B, I_{1}) × Exmon_{𝒞} (B, I_{2})

whose quasi-inverse is given by (E_{1}, E_{2}) ↦ (E_{1} ⊕ E_{2}) ∗ δ, where δ : B → B ⊕ B is the diagonal morphism. A similar statement holds for Exal and Exun. These operations can be used to induce an abelian group structure on the set Exmon_{𝒞} (B, I) of isomorphism classes of objects of Exmon_{𝒞} (B, I) as follows. For any two objects X, Y of Exmon_{𝒞} (B, I) we can form X ⊕ Y which is an object of Exmon_{𝒞} (B ⊕ B, I ⊕ I). Let α : I ⊕ I → I be the addition morphism of I. Then we set X +Y := α∗(X ⊕Y)∗δ. One can check that X +Y ≃ Y +X for any X, Y and that the trivial split 𝒞 -extension B ⊕I is a neutral element for +. Moreover every isomorphism class has an inverse −X. The functors X ↦ X ∗ ϕ and X ↦ ψ ∗ X commute with the operation thus defined, and induce group homomorphisms

∗ϕ : Exmon_{𝒞} (B, I) → Exmon_{𝒞} (C, I)

ψ∗ : Exmon_{𝒞} (B, I) → Exmon_{𝒞} (B, J).

2.5.7. We will need the variant Exal_{𝒞} (B, I) defined in the same way, starting from the category Exal_{𝒞} (B, I). For instance, if A is an almost algebra (resp. a commutative ring), we can consider the abelian tensor category 𝒞 = A-Mod. In this case the 𝒞 -extensions will be called simply A-extensions, and we will write Exal_{A} rather than Exal_{𝒞} . In fact the commutative unitary case will soon become prominent in our work, and the more general setup is only required for technical reasons, in the proof of proposition 2.5.13 below, which is the abstract version of a well-known result on the lifting of idempotents over nilpotent ring extensions.

2.5.8. Let A be a 𝒞 -monoid. We form the biproduct A^{†} := U ⊕ A in 𝒞. We denote by p_{1}, p_{2} the associated projections from A^{†} onto U and respectively A. Also, let i_{1}, i_{2} be the natural monomorphisms from U, resp. A to A^{†}. A^{†} is equipped with a unitary monoid structure

μ^{†} := i_{2} ∘ μ ∘ (p_{2} ⊗ p_{2}) + i_{2} ∘ ℓ_{A}^{−1} ∘ (p_{1} ⊗ p_{2}) + i_{2} ∘ r_{A}^{−1} ∘ (p_{2} ⊗ p_{1}) + i_{1} ∘ u^{−1} ∘ (p_{1} ⊗ p_{1})

where ℓ_{A}, r_{A} are the natural isomorphisms provided by [24, Prop. 1.3] and u : U → U ⊗ U is as in [24, §1]. In terms of the ring A_{∗}^{†} ≃ U_{∗} ⊕ A_{∗} this is the multiplication (u_{1}, b_{1}) · (u_{2}, b_{2}) := (u_{1} · u_{2}, b_{1} · b_{2} + b_{1} · u_{2} + u_{1} · b_{2}). Then i_{2} is a morphism of monoids and one verifies that the “restriction of scalars” functor i_{2}^{∗} defines an equivalence from the category A^{†}-Uni.Mod of unitary A^{†}-modules to the category A-Mod of all A-modules; let j denote the inverse functor. A similar discussion applies to bimodules.

2.5.9. Similarly, we derive equivalences of categories

\xymatrix{Exun_{𝒞} (A^{†}, j(M)) \ar[r]^{∗i_{2}} & Exmon_{𝒞} (A, M) \ar[l]^{(−)^{†}}}

for all A-bimodules M.

2.5.10. Next we specialise to A := U : for a given U-module M let e_{M} := σ_{M/U} ∘ ℓ_{M} : M → M; working out the definitions one finds that the condition that (M, σ_{M/U}) is a module structure is equivalent to e_{M}^{2} = e_{M}. Let U × U be the product of U by itself in the category of 𝒞 -monoids. There is an isomorphism of unitary 𝒞 -monoids ζ : U^{†} → U × U given by ζ := i_{1} ∘ p_{1} + i_{2} ∘ p_{1} + i_{2} ∘ p_{2}. Another isomorphism is τ ∘ ζ, where τ is the flip i_{1} ∘ p_{2} + i_{2} ∘ p_{1}. Hence we get equivalences of categories

\xymatrix{U-Mod \ar[r]^{j} & U^{†}-Uni.Mod \ar[r]^{(ζ^{−1})^{∗}} \ar[l]^{i_{2}^{∗}} & (U × U)-Uni.Mod. \ar[l]^{(τ∘ζ)^{∗}}}

The composition i_{2}^{∗} ∘ (ζ^{−1} ∘ τ ∘ ζ)^{∗} ∘ j defines a self-equivalence of U-Mod which associates to a given U-module M the new U-module M^{flip} whose underlying object in 𝒞 is M and such that e_{M^{flip}} = 1_{M} − e_{M}. The same construction applies to U-bimodules and finally we get equivalences

(2.5.11)  Exmon_{𝒞} (U, M) →^{∼} Exmon_{𝒞} (U, M^{flip})  X ↦ X^{flip}

for all U-bimodules M. If X := (0 → M → E →^{π} U → 0) is an extension and X^{flip} := (0 → M^{flip} → E^{flip} → U → 0), then one verifies that there is a natural isomorphism X^{flip} → X of complexes in 𝒞 inducing −1_{M} on M, the identity on U and carrying the multiplication morphism on E^{flip} to

−μ_{E} + ℓ_{E}^{−1} ∘ (π ⊗ 1_{E}) + r_{E}^{−1} ∘ (1_{E} ⊗ π) : E ⊗ E → E.

In terms of the associated rings, this corresponds to replacing the given multiplication (x, y) ↦ x · y of E_{∗} by the new operation (x, y) ↦ π_{∗}(x) · y + π_{∗}(y) · x − x · y. Lemma 2.5.12. If M is a U-bimodule whose left and right actions coincide, then every extension of U by M splits uniquely.

Proof. Using the idempotent e_{M} we get a U-linear decomposition M ≃ M_{1} ⊕ M_{2} where the bimodule structure on M_{1} is given by the zero morphisms and the bimodule structure on M_{2} is given by ℓ_{M}^{−1} and r_{M}^{−1}. We have to prove that Exmon_{𝒞} (U, M) is equivalent to a onepoint category. By (2.5.6) we can assume that M = M_{1} or M = M_{2}. By (2.5.11) we have Exmon_{𝒞} (U, M_{2}) ≃ Exmon_{𝒞} (U, M_{2}^{flip}) and on M_{2}^{flip} the bimodule actions are the zero morphisms. So it is enough to consider M = M_{1}. In this case, if X := (0 → M → E → U → 0) is any extension, μ_{E} : E ⊗ E → E factors through a morphism U ⊗ U → E and composing with u : U → U ⊗ U we get a right inverse of E → U, which shows that X is the split extension. Then it is easy to see that X does not have any non-trivial automorphisms, which proves the assertion. □

Proposition 2.5.13. Let X := (0 → I → A →^{p} A′ → 0) be a 𝒞 -extension.

(i) Let e′ ∈ A′_{∗} be an idempotent element whose left action on the A′-bimodule I coincides with its right action. Then there exists a unique idempotent e ∈ A_{∗} such that p_{∗}(e) = e′. (ii) Especially, if A′ is unitary and I is a unitary A′-bimodule, then every extension of A′

by I is unitary.

Proof. (i) : the hypothesis e^{′2} = e′ implies that e′ : U → A′ is a morphism of (non-unitary) 𝒞 - monoids. We can then replace X by X ∗e′ and thereby assume that A′ = U, p : A → U and I is a (non-unitary) U-bimodule and the right and left actions on I coincide. The assertion to prove is that \underline{1}_{U} lifts to a unique idempotent e ∈ A_{∗}. However, this follows easily from lemma 2.5.12. To show (ii), we observe that, by (i), the unit \underline{1}_{A′} of A′_{∗} lifts uniquely to an idempotent e ∈ A_{∗}. We have to show that e is a unit for A_{∗}. Let us show the left unit property. Via e : U → A we can view the extension X as an exact sequence of left U-modules. We can then split X as the direct sum X_{1} ⊕ X_{2} where X_{1} is a sequence of unitary U-modules and X_{2} is a sequence of U-modules with trivial actions. But by hypothesis, on I and on A the U-module structure is unitary, so X = X_{1} and this is the left unit property. □

2.5.14. So much for the general nonsense; we now return to almost algebras. As already announced, from here on, we assume throughout that 𝔪̃ is a flat V -module. As an immediate consequence of proposition 2.5.13 we get natural equivalences of categories

(2.5.15)  Exal_{A_{1}}(B_{1}, M_{1}) × Exal_{A_{2}}(B_{2}, M_{2}) →^{∼} Exal_{A_{1}×A_{2}}(B_{1} × B_{2}, M_{1} ⊕ M_{2})

whenever A_{1}, A_{2} are V^{a}-algebras, B_{i} is a A_{i}-algebra and M_{i} is a (unitary) B_{i}-module, i = 1, 2.

2.5.16. Notice that, if A := R^{a} for some V -algebra R, S (resp. J) is a R-algebra (resp. an Smodule) and X is any object of Exal_{R}(S, J), then by applying termwise the localisation functor we get an object X^{a} of Exal_{A}(S^{a}, J^{a}). With this notation we have the following lemma. Lemma 2.5.17. Let B be any A-algebra and I a B-module.

(i) The natural functor

(2.5.18)  Exal_{A_{!!}}(B_{!!}, I_{∗}) → Exal_{A}(B, I)  X ↦ X^{a}

is an equivalence of categories. (ii) The equivalence (2.5.18)induces a group isomorphism Exal_{A_{!!}}(B_{!!}, I_{∗}) →^{∼} Exal_{A}(B, I)

functorial in all arguments.

Proof. Of course (ii) is an immediate consequence of (i). To show (i), let X := (0 → I → E → B → 0) be any object of Exal_{A}(B, I). Using corollary 2.2.22 one sees easily that the sequence X_{!} := (0 → I_{!} → E_{!!} → B_{!!} → 0) is right exact; X_{!} won’t be exact in general, unless B (and therefore E) is an exact algebra. In any case, the kernel of I_{!} → E_{!!} is almost zero, so we get an extension of B_{!!} by a quotient of I_{!} which maps to I_{∗}. In particular we get by pushout an extension X_{!∗} by I_{∗}, i.e. an object of Exal_{A_{!!}}(B_{!!}, I_{∗}) and in fact the assignment X ↦ X_{!∗} is a quasi-inverse for the functor (2.5.18). □

Remark 2.5.19. By inspecting the proof, we see that one can replace I_{∗} by I_{!∗} := Im(I_{!} → I_{∗}) in (i) and (ii) above. When B is exact, also I_{!} will do.

In [46, II.1.2] it is shown how to associate to any ring homomorphism R → S a natural simplicial complex of S-modules denoted 𝕃_{S/R} and called the cotangent complex of S over R. Definition 2.5.20. Let A → B be a morphism of almost V -algebras. The almost cotangent complex of B over A is the simplicial B_{!!}-module

𝕃_{B/A} := B_{!!} ⊗_{(V^{a}×B)_{!!}} 𝕃_{(V^{a}×B)_{!!}/(V^{a}×A)_{!!}}.

2.5.21. Usually we will want to view 𝕃_{B/A} as an object of the derived category D(s.B_{!!}-Mod) of simplicial B_{!!}-modules. Indeed, the hyperext functors computed in this category relate the cotangent complex to a number of important invariants. Recall that, for any simplicial ring R and any two R-modules E, F the hyperext of E and F is the abelian group defined as

𝔼xt_{R}^{p}(E, F) := colim_{n≥−p} Hom_{D(R-Mod)}(σ^{n}E, σ^{n+p}F)

(where σ is the suspension functor of [46, I.3.2.1.4]).

Let us fix an almost algebra A. First we want to establish the relationship with differentials. Definition 2.5.22. Let B be any A-algebra, M any B-module.

(i) An A-derivation of B with values in M is an A-linear morphism ∂ : B → M such that ∂(b_{1} ·b_{2}) = b_{1} ·∂(b_{2})+b_{2} ·∂(b_{1}) for b_{1}, b_{2} ∈ B_{∗}. The set of all M-valued A-derivations of B forms a V -module Der_{A}(B, M) and the almost V -module Der_{A}(B, M)^{a} has a natural structure of B-module.

(ii) We reserve the notation I_{B/A} for the ideal Ker(μ_{B/A} : B ⊗_{A} B → B). The module of relative differentials of ϕ is defined as the (left) B-module Ω_{B/A} := I_{B/A}/I_{B/A}^{2}. It is endowed with a natural A-derivation δ : B → Ω_{B/A} defined by b ↦ \underline{1} ⊗ b − b ⊗ \underline{1} for all b ∈ B_{∗}. The assignment (A → B) ↦ Ω_{B/A} defines a functor

Ω : V^{a}-Alg.Morph → V^{a}-Alg.Mod

from the category of morphisms A → B of almost V -algebras to the category denoted V^{a}-Alg.Mod, consisting of all pairs (B, M) where B is an almost V -algebra and M is a B-module. The morphisms in V^{a}-Alg.Morph are the commutative squares; the morphisms (B, M) → (B′, M′) in V^{a}-Alg.Mod are all pairs (ϕ, f) where ϕ : B → B′ is a morphism of almost V -algebras and f : B′ ⊗_{B} M → M′ is a morphism of B′-modules.

2.5.23. The module of relative differentials enjoys the familiar universal properties that one expects. In particular Ω_{B/A} represents the functor Der_{A}(B, −), i.e. for any (left) B-module M the morphism

(2.5.24)  Hom_{B}(Ω_{B/A}, M) → Der_{A}(B, M)  f ↦ f ∘ δ

is an isomorphism. As an exercise, the reader can supply the proof for this claim and for the following standard proposition. Proposition 2.5.25. Let B and C be two A-algebras.

(i) There is a natural isomorphism:

Ω_{C⊗_{A}B/C} ≃ C ⊗_{A} Ω_{B/A}.

(ii) Suppose that C is a B-algebra. Then there is a natural exact sequence of C-modules:

C ⊗_{B} Ω_{B/A} → Ω_{C/A} → Ω_{C/B} → 0.

(iii) Let I be an ideal of B and let C := B/I be the quotient A-algebra. Then there is a

natural exact sequence: I/I^{2} → C ⊗_{B} Ω_{B/A} → Ω_{C/A} → 0. (iv) The functor Ω : V^{a}-Alg.Morph → V^{a}-Alg.Mod commutes with all colimits. □ We supplement these generalities with one more statement which is in the same vein as

lemma 2.3.21 and which will be useful in section 6.3 to calculate the Fitting ideals of modules of differentials. Lemma 2.5.26. Let ϕ : B → B′ be a morphism of A-algebras such that I · Ker(ϕ) = I · Coker(ϕ) = 0 for an ideal I ⊂ A. Let dϕ : Ω_{B/A} ⊗_{B} B′ → Ω_{B′/A} be the natural morphism. Then I · Coker dϕ = 0 and I^{4} · Ker dϕ = 0.

Proof. We will use the standard presentation

(2.5.27)  H(B/A) : B ⊗_{A} B ⊗_{A} B →^{∂} B ⊗_{A} B →^{d} Ω_{B/A} → 0

where d is defined by : b_{1} ⊗ b_{2} ↦ b_{1} · db_{2} and ∂ is the differential of the Hochschild complex :

b_{1} ⊗ b_{2} ⊗ b_{3} ↦ b_{1}b_{2} ⊗ b_{3} − b_{1} ⊗ b_{2}b_{3} + b_{1}b_{3} ⊗ b_{2}.

By naturality of H(B), we deduce a morphism of complexes : B′ ⊗_{B} H(B/A) → H(B′/A). Then, by snake lemma, we derive an exact sequence : Ker(1_{B′} ⊗_{A}ϕ) → Ker dϕ → X, where X is a quotient of B′⊗_{A}Coker(ϕ⊗_{A}ϕ). Using the Tor exact sequences we see that Ker(1_{B′}⊗_{A}ϕ) is annihilated by I^{2}. It follows easily that I^{4} annihilates Ker dϕ. Similarly, Coker dϕ is a quotient of Coker(1_{B′} ⊗_{A} ϕ), so I · Coker dϕ = 0. □

Lemma 2.5.28. For any A-algebra B there is a natural isomorphism of B_{!!}-modules

(Ω_{B/A})_{!} ≃ Ω_{B_{!!}/A_{!!}}.

Proof. Using the adjunction (2.5.24)we are reduced to showing that the natural map

ϕ_{M} : Der_{A_{!!}}(B_{!!}, M) → Der_{A}(B, M^{a})

is a bijection for all B_{!!}-modules M. Given ∂ : B → M^{a} we construct ∂_{!} : B_{!} → M_{!}^{a} → M. We extend ∂_{!} to V ⊕ B_{!} by setting it equal to zero on V. Then it is easy to check that the resulting map descends to B_{!!}, hence giving an A-derivation B_{!!} → M. This procedure yields a right inverse ψ_{M} to ϕ_{M}. To show that ϕ_{M} is injective, suppose that ∂ : B_{!!} → M is an almost zero A-derivation. Composing with the natural A-linear map B_{!} → B_{!!} we obtain an almost zero map ∂′ : B_{!} → M. But 𝔪 · B_{!} = B_{!}, hence ∂′ = 0. This implies that in fact ∂ = 0, and the assertion follows. □

Proposition 2.5.29. Let M be a B-module. There exists a natural isomorphism of B_{!!}-modules

𝔼xt_{B_{!!}}^{0}(𝕃_{B/A}, M_{!}) ≃ Der_{A}(B, M).

Proof. To ease notation, set Ã:= V^{a} × A and B̃ := V^{a} × B. We have natural isomorphisms :

𝔼xt_{B_{!!}}^{0}(𝕃_{B/A}, M_{!}) ≃ 𝔼xt0 ; B̃_{!!}(𝕃_{B̃_{!!}/Ã_{!!}}, M_{!}) by [46, I.3.3.4.4]
≃ DerÃ_{!!}(B̃_{!!}, M_{!})  by [46, II.1.2.4.2]
≃ DerÃ(B̃, M)  by lemma 2.5.28.

But it is easy to see that the natural map Der_{A}(B, M) → DerÃ(B̃, M) is an isomorphism. □

Theorem 2.5.30. There is a natural isomorphism

(2.5.31)  Exal_{A}(B, M) →^{∼} 𝔼xt_{B_{!!}}^{1}(𝕃_{B/A}, M_{!}).

Proof. With the notation of the proof of proposition 2.5.29 we have natural isomorphisms

𝔼xt_{B_{!!}}^{1}(𝕃_{B/A}, M_{!}) ≃ 𝔼xt1 ; B̃_{!!}(𝕃_{B̃_{!!}/Ã_{!!}}, M_{!}) by [46, I.3.3.4.4]
≃ ExalÃ_{!!}(B̃_{!!}, M_{!})  by [46, III.1.2.3]
≃ ExalÃ(B̃, M)

where the last isomorphism follows directly from lemma 2.5.17(ii) and the subsequent remark 2.5.19. Finally, (2.5.15)shows that ExalÃ(B̃, M) ≃ Exal_{A}(B, M), as required. □

Moreover we have the following transitivity theorem as in [46, II.2.1.2]. Theorem 2.5.32. Let A → B → C be a sequence of morphisms of almost V -algebras. There exists a natural distinguished triangle of D(s.C_{!!}-Mod)

C_{!!} ⊗_{B_{!!}} 𝕃_{B/A} →^{u} 𝕃_{C/A} →^{v} 𝕃_{C/B} → C_{!!} ⊗_{B_{!!}} σ𝕃_{B/A}

where the morphisms u and v are obtained by functoriality of 𝕃.

Proof. It follows directly from loc. cit. □

Proposition 2.5.33. Let (A_{λ} → B_{λ})_{λ∈I} be a system of almost V -algebra morphisms indexed by a small filtered category I. Then there is a natural isomorphism in D(s.colim_{λ∈I} B_{λ!!}-Mod)

colim_{λ∈I} 𝕃_{B_{λ}/A_{λ}} ≃ 𝕃_{colim_{λ∈I} B_{λ}/colim_{λ∈I} A_{λ}}.

Proof. Remark 2.2.28(i) gives an isomorphism : colim_{λ∈I} A_{λ!!} →^{∼} (colim_{λ∈I} A_{λ})_{!!} (and likewise for colim_{λ∈I} B_{λ}). Then the claim follows from [46, II.1.2.3.4]. □

Next we want to prove the almost version of the flat base change theorem [46, II.2.2.1]. To this purpose we need some preparation. Proposition 2.5.34. Let B and C be two A-algebras and set T_{i} := Tor_{i}^{A_{!!}}(B_{!!}, C_{!!}).

(i) If A, B, C and B ⊗_{A} C are all exact, then for every i > 0 the natural morphism 𝔪̃ ⊗_{V} T_{i} → T_{i} is an isomorphism.

(ii) If, furthermore, Tor_{i}^{A}(B, C) ≃ 0 for some i > 0, then the corresponding T_{i} vanishes.

Proof. (i): for any almost V -algebra D we let k_{D} denote the complex of D_{!!}-modules [𝔪̃ ⊗_{V} D_{!!} → D_{!!}] placed in degrees −1, 0; we have a distiguished triangle

𝒯 (D) : 𝔪̃ ⊗_{V} D_{!!} → D_{!!} → k_{D} → 𝔪̃ ⊗_{V} D_{!!}[1].

By assumption, the natural map k_{A} → k_{B} is a quasi-isomorphism and 𝔪̃ ⊗_{V} B_{!!} ≃ B_{!}. On the other hand, for all i ∈ ℕ we have

Tor_{i}^{A_{!!}}(k_{B}, C_{!!}) ≃ Tor_{i}^{A_{!!}}(k_{A}, C_{!!}) ≃ H^{−i}(k_{A} ⊗_{A_{!!}} C_{!!}) = H^{−i}(k_{C}).

In particular Tor_{i}^{A_{!!}}(k_{B}, C_{!!}) = 0 for all i > 1. As 𝔪̃ is flat over V, we have 𝔪̃ ⊗_{V} T_{i} ≃

Tor_{i}^{A_{!!}}(𝔪̃ ⊗_{V} B_{!!}, C_{!!}). Then by the long exact Tor sequence associated to 𝒯 (B) ⊗^{L}_{A_{!!}} C_{!!} we get the assertion for all i > 1. Next we consider the natural map of distinguished triangles

𝒯 (A) ⊗^{L}_{A_{!!}} A_{!!} → 𝒯 (B) ⊗^{L}_{A_{!!}} C_{!!}; writing down the associated morphism of long exact Tor sequences, we obtain a diagram with exact rows :

\xymatrix{0 \ar[r] & Tor_{1}^{A_{!!}}(k_{A}, A_{!!}) \ar[r]^{∂} \ar[d] & (𝔪̃ ⊗_{V} A_{!!}) ⊗_{A_{!!}} A_{!!} \ar[r]^{i} \ar[d] & A_{!!} ⊗_{A_{!!}} A_{!!} \ar[d] \\  & Tor_{1}^{A_{!!}}(k_{B}, C_{!!}) \ar[r]^{∂′} & (𝔪̃ ⊗_{V} B_{!!}) ⊗_{A_{!!}} C_{!!} \ar[r]^{i′} & B_{!!} ⊗_{A_{!!}} C_{!!}.}

By the above, the leftmost vertical map is an isomorphism; moreover, the assumption gives Ker i ≃ Ker(𝔪̃ → V) ≃ Ker i′. Then, since ∂ is injective, also ∂′ must be injective, which implies our assertion for the remaining case i = 1. (ii): follows directly from (i). □

Theorem 2.5.35. Let B, A′ be two A-algebras. Suppose that the natural morphism B ⊗^{L}_{A} A′ → B′ := B ⊗_{A} A′ is an isomorphism in D(s.A-Mod). Then the natural morphisms

\begin{matrix}B′_{!!} ⊗_{B_{!!}} 𝕃_{B/A} → 𝕃_{B′/A′} \\ (B′_{!!} ⊗_{B_{!!}} 𝕃_{B/A}) ⊕ (B′_{!!} ⊗_{A′_{!!}} 𝕃_{A′/A}) → 𝕃_{B′/A} \end{matrix}

are quasi-isomorphisms.

Proof. Let us remark that the functor D ↦ V^{a} × D : A-Alg → (V^{a} × A)-Alg commutes with tensor products; hence the same holds for the functor D ↦ (V^{a} × D)_{!!} (see remark 2.2.28(i)). Then, in view of proposition 2.5.34(ii), the theorem is reduced immediately to [46, II.2.2.1]. □

As an application we obtain the vanishing of the almost cotangent complex for a certain class of morphisms. Theorem 2.5.36. Let R → S be a morphism of almost algebras such that

Tor_{i}^{R}(S, S) ≃ 0 ≃ Tor_{i}^{S⊗_{R}S}(S, S) for all i > 0

(for the natural S ⊗_{R} S-module structure induced by μ_{S/R}). Then 𝕃_{S/R} ≃ 0 in D(S_{!!}-Mod).

Proof. Since Tor_{i}^{R}(S, S) = 0 for all i > 0, theorem 2.5.35 applies (with A := R and B := A′ := S), giving the natural isomorphisms

(2.5.37)  \begin{matrix}(S ⊗_{R} S)_{!!} ⊗_{S_{!!}} 𝕃_{S/R} ≃ 𝕃_{S⊗_{R}S/S} \\ ((S ⊗_{R} S)_{!!} ⊗_{S_{!!}} 𝕃_{S/R}) ⊕ ((S ⊗_{R} S)_{!!} ⊗_{S_{!!}} 𝕃_{S/R}) ≃ 𝕃_{S⊗_{R}S/R}. \end{matrix}

Since Tor_{i}^{S⊗_{R}S}(S, S) = 0, the same theorem also applies with A := S ⊗_{R} S, B := S, A′ := S, and we notice that in this case B′ ≃ S; hence we have

(2.5.38)  𝕃_{S/S⊗_{R}S} ≃ S_{!!} ⊗_{S_{!!}} 𝕃_{S/S⊗_{R}S} ≃ 𝕃_{S/S} ≃ 0.

Next we apply transitivity to the sequence R → S ⊗_{R} S → S, to obtain (thanks to (2.5.38))

(2.5.39)  S_{!!} ⊗_{S⊗_{R}S_{!!}} 𝕃_{S⊗_{R}S/R} ≃ 𝕃_{S/R}.

Applying S_{!!} ⊗_{S⊗_{R}S_{!!}} − to the second isomorphism (2.5.37)we obtain

(2.5.40)  𝕃_{S/R} ⊕ 𝕃_{S/R} ≃ S_{!!} ⊗_{S⊗_{R}S_{!!}} 𝕃_{S⊗_{R}S/R}.

Finally, composing (2.5.39)and (2.5.40)we derive

(2.5.41)  𝕃_{S/R} ⊕ 𝕃_{S/R} →^{∼} 𝕃_{S/R}.

However, by inspection, the isomorphism (2.5.41) is the sum map. Consequently 𝕃_{S/R} ≃ 0, as claimed. □

The following proposition shows that 𝕃_{B/A} is already determined by 𝕃_{B/A}^{a}. Proposition 2.5.42. Let A → B be a morphism of exact almost V -algebras. Then the natural map 𝔪̃ ⊗_{V} 𝕃_{B_{!!}/A_{!!}} → 𝕃_{B_{!!}/A_{!!}} is a quasi-isomorphism.

Proof. By transitivity we may assume A = V^{a}. Let P_{•} := P_{V} (B_{!!}) be the standard resolution of B_{!!} (see [46, II.1.2.1]). Each P [n]^{a} contains V as a direct summand, hence it is exact, so that we have an exact sequence of simplicial V -modules 0 → s.𝔪̃ → s.V ⊕ (P_{•}^{a})_{!} → (P_{•}^{a})_{!!} → 0. The augmentation (P_{•}^{a})_{!} → (B_{!!}^{a})_{!} ≃ B_{!} is a quasi-isomorphism and we deduce that (P_{•}^{a})_{!!} → B_{!!} is a quasi-isomorphism; hence (P_{•}^{a})_{!!} → P_{•} is a quasi-isomorphism as well. We have P [n] ≃ Sym(F_{n}) for a free V -module F_{n} and the map (P [n]^{a})_{!!} → P [n] is identified with Sym(𝔪̃ ⊗_{V} F_{n}) → Sym(F_{n}), whence Ω_{P [n]_{!!}^{a}/V} ⊗_{P [n]_{!!}^{a}} P [n] → Ω_{P [n]/V} is identified with 𝔪̃ ⊗_{V} Ω_{P [n]/V} → Ω_{P [n]/V}. By [46, II.1.2.6.2] the map 𝕃_{(P_{•}^{a})_{!!}/V}^{Δ} → 𝕃_{P_{•}/V}^{Δ} is a quasi-isomorphism. In view of [46, II.1.2.4.4] we derive that Ω_{(P_{•}^{a})_{!!}/V} → Ω_{P_{•}/V} is a quasi-isomorphism, i.e. 𝔪̃⊗_{V} Ω_{P_{•}/V} → Ω_{P_{•}/V} is a quasi-isomorphism. Since 𝔪̃ is flat and Ω_{P_{•}/V} → Ω_{P_{•}/V} ⊗_{P_{•}} B_{!!} = 𝕃_{B_{!!}/V} is a quasiisomorphism, we get the desired conclusion. □

Finally we have a fundamental spectral sequence as in [46, III.3.3.2]. Theorem 2.5.43. Let ϕ : A → B be a morphism of almost algebras such that B ⊗_{A} B ≃ B (e.g. such that B is a quotient of A). Then there is a first quadrant homology spectral sequence of bigraded almost algebras

E_{pq}^{2} := H_{p+q}(Sym_{B}^{q}(𝕃_{B/A}^{a})) ⇒ Tor_{p+q}^{A}(B, B).

Proof. We replace ϕ by 1_{V^{a}} × ϕ and apply the functor B ↦ B_{!!} (which commutes with tensor products by remark 2.2.28(i)) thereby reducing the assertion to [46, III.3.3.2]. □

3. Almost ring theory

With this chapter we begin in earnest the study of almost commutative algebra: in section 3.1 the classes of flat, unramified and étale morphisms are defined, together with some variants. In section 3.2 we derive the infinitesimal lifting theorems for étale algebras (theorem 3.2.18) and for almost projective modules (theorem 3.2.28).

In section 3.4 we turn to study some cases of non-flat descent; when we specialize to usual rings, we recover known theorems (of course, standard commutative algebra is a particular case of almost ring theory). But if the result is not new, the argument is : indeed, we believe that our treatment, even when specialized to usual rings, improves upon the method found in the literature.

The last section of chapter 3 calls on stage the Frobenius endomorphism of an almost algebra of positive characteristic. The main results are invariance of étale morphisms under pull-back by Frobenius maps (theorem 3.5.13) and theorem 3.5.28, which can be interpreted as a purity theorem. Perhaps the most remarkable aspect of the latter result is how cheap it is : in positive characteristic, the availability of the Frobenius map allows for a quick and easy proof. Philosophically, this proof is not too far removed from the method devised by Faltings for his more recent proof of purity in mixed characteristic.

Taken together, the above-mentioned four sections leave us with a decent understanding of the morphisms “of relative dimension zero” (this expression should be taken with a grain of salt, since we do not try to define the dimension of an almost algebra). On one hand, a good hold on the case of relative dimension zero is all that is required for the applications currently in sight (especially for the proof of the almost purity theorem, but also for the needs of our chapters 6 and 7); on the other hand, having reached this stage, one cannot help wondering what lies ahead, for instance whether there is a good notion of smooth morphism of almost algebras. The full answer to this question shall be delayed until chapter 5 : there we will introduce a class of morphisms that generalize “in higher dimension” the class of weakly étale morphisms, and that specialize to formally smooth morphisms in the “classical limit” V = 𝔪. We will present evidence that our notion of smoothness is well behaved and worthwhile; however we shall also see that smoothness “in higher dimension” exhibits some extra twists that have no analogue in standard commutative algebra, and cannot be easily guessed just by extrapolating from the case of étale morphisms of V^{a}-algebras (which, after all, reproduce very faithfully the behaviour of the classical notion defined in EGA).

Such extra twists are already foreshadowed by our results on the nilpotent deformation of almost projective modules : we show that such deformations exist, but are not unique; however, any two such deformations are “very close” in a precise sense (proposition 3.2.30).

In (usual) algebraic geometry one can also regard projective modules of finite rank n as GL_{n}- torsors (say for the Zariski topology); in almost ring theory this description carries through at least for the class of almost projective modules of finite rank (to be defined in section 4.3). From this vantage, one is naturally led to ask how much of the standard deformation theory for torsors over arbitrary group schemes generalizes to the almost world. We take up this question in section 3.3, focusing especially on the case of smooth affine almost group schemes, as a warm up to the later study of general smooth morphisms. Having committed seriously to almost group schemes and almost torsors, it is only a short while before one grows impatient at the limited expressive range afforded by the purely algebraic terminology introduced thus far, which by this point starts feeling a little like a linguistic straightjacket. That is why we find ourselves compelled to introduce a more geometric language : for our purpose an affine almost scheme is just an object of the opposite of the category of V^{a}-algebras; similarly we define quasi-coherent modules on affine almost schemes, as well as some suggestive notation to go with it, mimicking the standard usage in algebraic geometry.

Once these preliminaries are in place, the theory proceeds as in [47] : the techniques are rather sophisticated, but all the hard work has already been done in the previous sections, and we can just adapt Illusie’s treatise without much difficulty.

3.1. Flat, unramified andétale morphisms. Let A → B be a morphism of almost V -algebras. Using the natural “multiplication” morphism of A-algebras μ_{B/A} : B ⊗_{A} B → B we can view B as a B ⊗_{A} B-algebra. Definition 3.1.1. Let ϕ : A → B be a morphism of almost V -algebras.

(i) We say that ϕ is a flat (resp. faithfully flat, resp. almost projective) morphism if B is a flat (resp. faithfully flat, resp. almost projective) A-module.

(ii) We say that ϕ is (uniformly) almost finite (resp. finite) if B is a (uniformly) almost

finitely generated (resp. finitely generated) A-module. (iii) We say that ϕ is weakly unramified (resp. unramified) if B is a flat (resp. almost

projective) B ⊗_{A} B-module (via the morphism μ_{B/A} defined above). (iv) ϕ is weakly étale (resp. étale) if it is flat and weakly unramified (resp. unramified).

Furthermore, in analogy with definition 2.4.4, we shall write “(uniformly) almost finite projective” to denote a morphism ϕ which is both (uniformly) almost finite and almost projective. Lemma 3.1.2. Let ϕ : A → B and ψ : B → C be morphisms of almost V -algebras.

(i) Let A → A′ be any morphism of V^{a}-algebras; if ϕ is flat (resp. almost projective, resp. faithfully flat, resp. almost finite, resp. weakly unramified, resp. unramified, resp. weakly étale, resp. étale) then the same holds for ϕ ⊗_{A} 1_{A′}.

(ii) If both ϕ and ψ are flat (resp. almost projective, resp. faithfully flat, resp. almost finite, resp. weakly unramified, resp. unramified, resp. weakly étale, resp. étale), then so is ψ ∘ ϕ.

(iii) If ϕ is flat and ψ ∘ ϕ is faithfully flat, then ϕ is faithfully flat.

(iv) If ϕ is weakly unramified and ψ ∘ ϕ is flat (resp. weakly étale), then ψ is flat (resp.

weakly étale).

(v) If ϕ is unramified and ψ ∘ ϕ is étale, then ψ is étale.

(vi) ϕ is faithfully flat if and only if it is a monomorphism and B/A is a flat A-module. (vii) If ϕ is almost finite and weakly unramified, then ϕ is unramified. (viii) If ψ is faithfully flat and ψ ∘ ϕ is flat (resp. weakly unramified), then ϕ is flat (resp.

weakly unramified).

Proof. For (vi) use the Tor sequences. In view of proposition 2.4.18(ii), to show (vii) it suffices to know that B is an almost finitely presented B ⊗_{A} B-module; but this follows from the existence of an epimorphism of B ⊗_{A} B-modules (B ⊗_{A} B) ⊗_{A} B → Ker μ_{B/A} defined by x ⊗ b ↦ x · (\underline{1} ⊗ b − b ⊗ \underline{1}). Of the remaining assertions, only (iv) and (v) are not obvious, but the proof is just the “almost version” of a well-known argument. Let us show (v); the same argument applies to (iv). We remark that μ_{B/A} is an étale morphism, since ϕ is unramified. Define Γ_{ψ} := 1_{C} ⊗_{B} μ_{B/A}. By (i), Γ_{ψ} is étale. Define also p := (ψ ∘ ϕ) ⊗_{A} 1_{B}. By (i), p is flat (resp. étale). The claim follows by remarking that ψ = Γ_{ψ} ∘ p and applying (ii). □

Remark 3.1.3. (i) Suppose we work in the classical limit case, that is, 𝔪 := V (cp. example 2.1.2(ii)). Then we caution the reader that our notion of “étale morphism” is more general than the usual one, as defined in [40]. The relationship between the usual notion and ours is discussed in the digression (3.4.44).

(ii) The naive hope that the functor A ↦ A_{!!} might preserve flatness is crushed by the following counterexample. Let (V, 𝔪) be as in example 2.1.2(i) and let k be the residue field of V. Consider the flat map V × V → V defined as (x, y) ↦ x. We get a flat morphism V^{a} × V^{a} → V^{a} in V^{a}-Alg; applying the left adjoint to localisation yields a map V ×_{k} V → V that is not flat. On the other hand, faithful flatness is preserved. Indeed, let ϕ : A → B be a morphism of almost algebras. Then ϕ is a monomorphism if and only if ϕ_{!!} is injective; moreover, B_{!!}/Im(A_{!!}) ≃ B_{!}/A_{!}, which is flat over A_{!!} if and only if B/A is flat over A, by proposition 2.4.35.

We will find useful to study certain “almost idempotents”, as in the following proposition. Proposition 3.1.4. A morphism ϕ : A → B is unramified if and only if there exists an almost element e_{B/A} ∈ B ⊗_{A} B_{∗} such that

(i) e_{B/A}^{2} = e_{B/A}; (ii) μ_{B/A}(e_{B/A}) = \underline{1}; (iii) x · e_{B/A} = 0 for all x ∈ I_{B/A∗}.

Proof. Suppose that ϕ is unramified. We start by showing that for every ε ∈ 𝔪 there exist almost elements e_{ε} of B ⊗_{A} B such that

(3.1.5)  e_{ε}^{2} = ε · e_{ε}  μ_{B/A}(e_{ε}) = ε · \underline{1}  I_{B/A∗} · e_{ε} = 0.

Since B is an almost projective B ⊗_{A} B-module, for every ε ∈ 𝔪 there exists an “approximate splitting” for the epimorphism μ_{B/A} : B ⊗_{A} B → B, i.e. a B ⊗_{A} B-linear morphism u_{ε} : B → B ⊗_{A} B such that μ_{B/A} ∘ u_{ε} = ε · 1_{B}. Set e_{ε} := u_{ε} ∘ \underline{1} : A → B ⊗_{A} B. We see that μ_{B/A}(e_{ε}) = ε · \underline{1}. To show that e_{ε}^{2} = ε · e_{ε} we use the B ⊗_{A} B-linearity of u_{ε} to compute

e_{ε}^{2} = e_{ε} · u_{ε}(\underline{1}) = u_{ε}(μ_{B/A}(e_{ε}) · \underline{1}) = u_{ε}(μ_{B/A}(e_{ε})) = ε · e_{ε}.

Next take any almost element x of I_{B/A} and compute

x · e_{ε} = x · u_{ε}(\underline{1}) = u_{ε}(μ_{B/A}(x) · \underline{1}) = 0.

This establishes (3.1.5). Next let us take any other δ ∈ 𝔪 and a corresponding almost element e_{δ}. Both ε·\underline{1}−e_{ε} and δ·\underline{1}−e_{δ} are elements of I_{B/A∗}, hence we have (δ·\underline{1}−e_{δ})·e_{ε} = 0 = (ε·\underline{1}−e_{ε})·e_{δ} which implies

(3.1.6)  δ · e_{ε} = ε · e_{δ}  for all ε, δ ∈ 𝔪.

Let us define a map e_{B/A} : 𝔪 ⊗_{V} 𝔪 → B ⊗_{A} B_{∗} by the rule

(3.1.7)  ε ⊗ δ ↦ δ · e_{ε}  for all ε, δ ∈ 𝔪.

To show that (3.1.7) does indeed determine a well defined morphism, we need to check that δ · v · e_{ε} = δ · e_{v·ε} and δ · e_{ε+ε′} = δ · (e_{ε} + e_{ε′}) for all ε, ε′, δ ∈ 𝔪 and all v ∈ V. However, both identities follow easily by a repeated application of (3.1.6). It is easy to see that e_{B/A} defines an almost element with the required properties.

Conversely, suppose an almost element e_{B/A} of B ⊗_{A} B is given with the stated properties. We define u : B → B ⊗_{A} B by b ↦ e_{B/A} · (1 ⊗ b) (b ∈ B_{∗}) and v := μ_{B/A}. Then (iii) says that u is a B ⊗_{A} B-linear morphism and (ii) shows that v ∘ u = 1_{B}. Hence, by lemma 2.4.15, ϕ is unramified. □

Remark 3.1.8. The proof of proposition 3.1.4 shows that if I is an ideal in an almost V -algebra A, then A/I is almost projective over A if and only if I is generated by an idempotent of A_{∗}. This idempotent is uniquely determined. Corollary 3.1.9. Under the hypotheses and notation of the proposition, the ideal I_{B/A} has a natural structure of B ⊗_{A} B-algebra, with unit morphism given by \underline{1} := \underline{1}_{B⊗_{A}B/A} − e_{B/A} and whose multiplication is the restriction of μ_{B⊗_{A}B/A} to I_{B/A}. Moreover the natural morphism

B ⊗_{A} B → I_{B/A} ⊕ B x ↦ (x · \underline{1} ⊕ μ_{B/A}(x))

is an isomorphism of B ⊗_{A} B-algebras.

Proof. Left to the reader as an exercise. □

3.2. Nilpotent deformations of almost algebras and modules. Throughout the following, the terminology “epimorphism of V^{a}-algebras” will refer to a morphism of V^{a}-algebras that induces an epimorphism on the underlying V^{a}-modules. Lemma 3.2.1. Let A → B be an epimorphism of almost V -algebras with kernel I. Let U be the A-extension 0 → I/I^{2} → A/I^{2} → B → 0. Then the assignment f ↦ f ∗ U defines a natural isomorphism

(3.2.2)  Hom_{B}(I/I^{2}, M) →^{∼} Exal_{A}(B, M).

Proof. Let X := (0 → M → E →^{p} B → 0) be any A-extension of B by M. The composition g : A → E →^{p} B of the structural morphism for E followed by p coincides with the projection A → B. Therefore g(I) ⊂ M and g(I^{2}) = 0. Hence g factors through A/I^{2}; the restriction of g to I/I^{2} defines a morphism f ∈ Hom_{B}(I/I^{2}, M) and a morphism of A-extensions f ∗U → X. In this way we obtain an inverse for (3.2.2). □

3.2.3. Now consider any morphism of A-extensions

(3.2.4)  \xymatrix{B̃ : \ar[d]^{f̃} & 0 \ar[r] & I \ar[r] \ar[d]^{u} & B \ar[r] \ar[d]^{f} & B_{0} \ar[r] \ar[d]^{f_{0}} & 0 \\ C̃ : & 0 \ar[r] & J \ar[r] & C \ar[r] & C_{0} \ar[r] & 0.}

The morphism u induces by adjunction a morphism of C_{0}-modules

(3.2.5)  C_{0} ⊗_{B_{0}} I → J

whose image is the ideal I · C, so that the square diagram of almost algebras defined by f̃ is cofibred (i.e. C_{0} ≃ C ⊗_{B} B_{0}) if and only if (3.2.5)is an epimorphism. Lemma 3.2.6. Let f̃: B̃ → C̃ be a morphism of A-extensions as above, such that the corresponding square diagram of almost algebras is cofibred. Then the morphism f : B → C is flat if and only if f_{0} : B_{0} → C_{0} is flat and (3.2.5)is an isomorphism.

Proof. It follows directly from the (almost version of the) local flatness criterion (see [54, Th. 22.3]). □

We are now ready to put together all the work done so far and begin the study of deformations of almost algebras.

3.2.7. The morphism u : I → J is an element in Hom_{B_{0}}(I, J); by lemma 3.2.1 the latter group is naturally isomorphic to Exal_{B}(B_{0}, J). On the other hand, in view of proposition 2.5.42 and lemma 2.4.14(i) we have natural isomorphisms:

(3.2.8)  𝔼xt_{C_{0!!}}^{i}(𝕃_{C_{0}/B_{0}}, M_{!}) ≃ 𝔼xt_{C_{0}}^{i}(𝕃_{C_{0}/B_{0}}^{a}, M)

for every i ∈ ℕ and every C_{0}-module M. By applying transitivity (theorem 2.5.32) to the sequence of morphisms B → B_{0} →^{f_{0}} C_{0} we deduce an exact sequence of abelian groups

Exal_{B_{0}}(C_{0}, J) → Exal_{B}(C_{0}, J) → Hom_{B_{0}}(I, J) →^{∂} 𝔼xt_{C_{0}}^{2}(𝕃_{C_{0}/B_{0}}^{a}, J).

Hence we can form the element ω(B̃, f_{0}, u) := ∂(u) ∈ 𝔼xt_{C_{0}}^{2}(𝕃_{C_{0}/B_{0}}^{a}, J). The proof of the next result goes exactly as in [46, III.2.1.2.3]. Proposition 3.2.9. Let the A-extension B̃, the B_{0}-linear morphism u : I → J and the morphism of A-algebras f_{0} : B_{0} → C_{0} be given as above.

(i) There exists an A-extension C̃ and a morphism f̃: B̃ → C̃ completing diagram (3.2.4) if and only if ω(B̃, f_{0}, u) = 0. (i.e. ω(B̃, f_{0}, u) is the obstruction to the lifting of B̃ over f_{0}.)

(ii) Assume that the obstruction ω(B̃, f_{0}, u) vanishes. Then the set of isomorphism classes

of A-extensions C̃ as in (i) forms a torsor under the group:

Exal_{B_{0}}(C_{0}, J) ≃ 𝔼xt_{C_{0}}^{1}(𝕃_{C_{0}/B_{0}}^{a}, J).

(iii) The group of automorphisms of an A-extension C̃ as in (i) is naturally isomorphic to Der_{B_{0}}(C_{0}, J) (≃ 𝔼xt_{C_{0}}^{0}(𝕃_{C_{0}/B_{0}}^{a}, J)). □

3.2.10. The obstruction ω(B̃, f_{0}, u) depends functorially on u. More exactly, if we denote by

ω(B̃, f_{0}) ∈ 𝔼xt_{C_{0}}^{2}(𝕃_{C_{0}/B_{0}}^{a}, C_{0} ⊗_{B_{0}} I)

the obstruction corresponding to the natural morphism I → C_{0} ⊗_{B_{0}} I, then for any other morphism u : I → J we have

ω(B̃, f_{0}, u) = v_{!} ∘ ω(B̃, f_{0})

where v is the morphism (3.2.5). Taking lemma 3.2.6 into account we deduce Corollary 3.2.11. Suppose that B_{0} → C_{0} is flat. Then

(i) The class ω(B̃, f_{0}) is the obstruction to the existence of a flat deformation of C_{0} over B, i.e. of a B-extension C̃ as in (3.2.4)such that C is flat over B and C ⊗_{B} B_{0} → C_{0} is an isomorphism.

(ii) If the obstruction ω(B̃, f_{0}) vanishes, then the set of isomorphism classes of flat defor-

mations of C_{0} over B forms a torsor under the group Exal_{B_{0}}(C_{0}, C_{0} ⊗_{B_{0}} I). (iii) The group of automorphisms of a given flat deformation of C_{0} over B is naturally

isomorphic to Der_{B_{0}}(C_{0}, C_{0} ⊗_{B_{0}} I). □

3.2.12. Now, suppose we are given two A-extensions C̃^{1}, C̃^{2} with morphisms of A-extensions

\xymatrix{B̃ : \ar[d]^{f̃^{i}} & 0 \ar[r] & I \ar[r] \ar[d]^{u^{i}} & B \ar[r] \ar[d]^{f^{i}} & B_{0} \ar[r] \ar[d]^{f_{0}^{i}} & 0 \\ C̃^{i} : & 0 \ar[r] & J^{i} \ar[r] & C^{i} \ar[r] & C_{0}^{i} \ar[r] & 0}

and morphisms v : J^{1} → J^{2}, g_{0} : C_{0}^{1} → C_{0}^{2} such that

(3.2.13) u^{2} = v ∘ u^{1} and f_{0}^{2} = g_{0} ∘ f_{0}^{1}.

We consider the problem of finding a morphism of A-extensions

(3.2.14)  \xymatrix{C̃^{1} : \ar[d]^{g̃} & 0 \ar[r] & J^{1} \ar[r] \ar[d]^{v} & C^{1} \ar[r] \ar[d]^{g} & C_{0}^{1} \ar[r] \ar[d]^{g_{0}} & 0 \\ C̃^{2} : & 0 \ar[r] & J^{2} \ar[r] & C^{2} \ar[r] & C_{0}^{2} \ar[r] & 0}

such that f̃^{2} = g̃ ∘ f̃^{1}. Let us denote by e(C̃^{i}) ∈ 𝔼xt_{C_{0}^{i}}^{1}(𝕃_{C_{0}^{i}/B}^{a}, J^{i}) the classes defined by the B-extensions C̃^{1}, C̃^{2} via the isomorphisms (2.5.31)and (3.2.8), and by

v∗ : 𝔼xt_{C_{0}^{1}}^{1}(𝕃_{C_{0}^{1}/B}^{a}, J^{1}) → 𝔼xt_{C_{0}^{1}}^{1}(𝕃_{C_{0}^{1}/B}^{a}, J^{2})
∗g_{0} : 𝔼xt_{C_{0}^{2}}^{1}(𝕃_{C_{0}^{2}/B}^{a}, J^{2}) → 𝔼xt_{C_{0}^{2}}^{1}(C_{0}^{2} ⊗_{C_{0}^{1}} 𝕃_{C_{0}^{1}/B}^{a}, J^{2})

the canonical morphisms defined by v and g_{0}. Using the natural isomorphism 𝔼xt_{C_{0}^{1}}^{1}(𝕃_{C_{0}^{1}/B}^{a}, J^{2}) ≃ 𝔼xt_{C_{0}^{2}}^{1}(C_{0}^{2} ⊗_{C_{0}^{1}} 𝕃_{C_{0}^{1}/B}, J^{2})

we can identify the target of both v∗ and ∗g with 𝔼xt_{C_{0}^{1}}^{1}(𝕃_{C_{0}^{1}/B}^{a}, J^{2}). It is clear that the problem admits a solution if and only if the A-extensions v ∗ C̃^{1} and C̃^{2} ∗ g_{0} coincide, i.e. if and only if v ∗ e(C̃^{1}) − e(C̃^{2}) ∗ g_{0} = 0. By applying transitivity to the sequence of morphisms B → B_{0} → C_{0}^{1} we obtain an exact sequence

𝔼xt_{C_{0}^{1}}^{1}(𝕃_{C_{0}^{1}/B_{0}}^{a}, J^{2}) ↪ 𝔼xt_{C_{0}^{1}}^{1}(𝕃_{C_{0}^{1}/B}^{a}, J^{2}) → Hom_{C_{0}^{1}}(C_{0}^{1} ⊗_{B_{0}} I, J^{2}).

It follows from (3.2.13)that the image of v∗e(C̃^{1})−e(C̃^{2})∗g_{0} in the group Hom_{C_{0}^{1}}(C_{0}^{1}⊗_{B_{0}}I, J^{2}) vanishes, therefore

(3.2.15)  v ∗ e(C̃^{1}) − e(C̃^{2}) ∗ g_{0} ∈ 𝔼xt_{C_{0}^{1}}^{1}(𝕃_{C_{0}^{1}/B_{0}}^{a}, J^{2}).

In conclusion, we derive the following result as in [46, III.2.2.2].

Proposition 3.2.16. With the above notations, the class (3.2.15) is the obstruction to the existence of a morphism of A-extensions g̃: C̃^{1} → C̃^{2} as in (3.2.14)such that f̃^{2} = g̃∘f̃^{1}. When the obstruction vanishes, the set of such morphisms forms a torsor under the group Der_{B_{0}}(C_{0}^{1}, J^{2}) (the latter being identified with 𝔼xt_{C_{0}^{2}}^{0}(C_{0}^{2} ⊗_{C_{0}^{1}} 𝕃_{C_{0}^{1}/B_{0}}^{a}, J^{2})). □

3.2.17. For a given almost V -algebra A, we define the category A-w.Ét (resp. A-Ét) as the full subcategory of A-Alg consisting of all weakly étale (resp. étale) A-algebras. Notice that, by lemma 3.1.2(iv) all morphisms in A-w.Ét are weakly étale.

Theorem 3.2.18. Let A be a V^{a}-algebra.

(i) Let B be a weakly étale A-algebra, C any A-algebra and I ⊂ C a nilpotent ideal. Then the natural morphism

Hom_{A-Alg}(B, C) → Hom_{A-Alg}(B, C/I)

is bijective. (ii) Let I ⊂ A a nilpotent ideal and A′ := A/I. Then the natural functor

A-w.Ét → A′-w.Ét (ϕ : A → B) ↦ (1_{A′} ⊗_{A} ϕ : A′ → A′ ⊗_{A} B)

is an equivalence of categories. (iii) The equivalence of (ii) restricts to an equivalence A-Ét → A′-Ét.

Proof. By induction we can assume I^{2} = 0. Then (i) follows directly from proposition 3.2.16 and theorem 2.5.36. We show (ii) : by corollary 3.2.11 (and again theorem 2.5.36) a given weakly étale morphism ϕ′ : A′ → B′ can be lifted to a unique flat morphism ϕ : A → B. We need to prove that ϕ is weakly étale, i.e. that B is B ⊗_{A} B-flat. However, it is clear that μ_{B′/A′} : B′ ⊗_{A′} B′ → B′ is weakly étale, hence it has a flat lifting μ̃ : B ⊗_{A} B → C. Then the composition A → B ⊗_{A} B → C is flat and it is a lifting of ϕ′. We deduce that there is an isomorphism of A-algebras α : B → C lifting 1_{B′} and moreover the morphisms b ↦ μ̃(b ⊗ \underline{1}) and b ↦ μ̃(\underline{1} ⊗ b) coincide with α. Claim (ii) follows. To show (iii), suppose that A′ → B′ is étale and let I_{B′/A′} denote as usual the kernel of μ_{B′/A′}. By corollary 3.1.9 there is a natural morphism of almost algebras B′ ⊗_{A′} B′ → I_{B′/A′} which is clearly étale. Hence I_{B′/A′} lifts to a weakly étale B ⊗_{A} B-algebra C, and the isomorphism B′ ⊗_{A′} B′ ≃ I_{B′/A′} ⊕ B′ lifts to an isomorphism B ⊗_{A} B ≃ C ⊕ B of B ⊗_{A} B-algebras. It follows that B is an almost projective B ⊗_{A} B-module, i.e. A → B is étale, as claimed. □

We conclude with some results on deformations of almost modules. These can be established independently of the theory of the cotangent complex, along the lines of [46, IV.3.1.12].

3.2.19. We begin by recalling some notation from loc. cit. Let R be a ring and J ⊂ R an ideal with J^{2} = 0. Set R′ := R/J; an extension of R-modules \underline{M} := (0 → K → M →^{p} M′ → 0) where K and M′ are killed by J, defines a natural morphism of R′-modules u(\underline{M}) : J⊗_{R′}M′ → K such that u(\underline{M})(x ⊗ m′) = xm for x ∈ J, m ∈ M and p(m) = m′. By the local flatness criterion ([54, Th. 22.3]) M is flat over R if and only if M′ is flat over R′ and u(\underline{M}) is an isomorphism. One can then show the following. Proposition 3.2.20. (cp. [46, IV.3.1.5]) With the notation of (3.2.19)we have:

(i) Given R′-modules M′ and K and a morphism u′ : J ⊗_{R′} M′ → K there exists an obstruction ω(R, u′) ∈ Ext_{R′}^{2}(M′, K) whose vanishing is necessary and sufficient for the existence of an extension of R-modules \underline{M} of M′ by K such that u(\underline{M}) = u′. (ii) When ω(R, u′) = 0, the set of isomorphism classes of such extensions \underline{M} forms a torsor under Ext_{R′}^{1}(M′, K); the group of automorphisms of such an extension is isomorphic to Hom_{R′}(M′, K). □

Lemma 3.2.21. Let R be a ring, M a finitely generated R-module such that Ann_{R}M is a nilpotent ideal. Then R admits a filtration 0 = J_{m} ⊂ ... ⊂ J_{1} ⊂ J_{0} = R such that each J_{i}/J_{i+1} is a quotient of a direct sum of copies of M.

Proof. This is [41, 1.1.5]; for the convenience of the reader we reproduce the proof. Let I := F_{0}(M) ⊂ R; if M is generated by k elements, we have (Ann_{R}M)^{k} ⊂ I ⊂ Ann_{R}M, especially I is nilpotent.

Claim 3.2.22. It suffices to show that \overline{R} := R/I admits a filtration as above.

Proof of the claim: Indeed, if 0 = J′_{0} ⊂ J′_{1} ⊂ ... ⊂ J′_{n−1} ⊂ J′_{n} = R/I is such a filtration, we deduce filtrations 0 ⊂ J′_{1}(I^{t}/I^{t+1}) ⊂ ... ⊂ J′_{n−1}(I^{t}/I^{t+1}) ⊂ (I^{t}/I^{t+1}) for every t ∈ ℕ; the graded module associated to this filtration is a direct sum of quotients of modules of the form (J′_{k}/J′_{k+1}) ⊗_{R} (I^{t}/I^{t+1}), so the claim follows easily.

Let F_{0}(M/IM) be the 0-th Fitting ideal of the \overline{R}-module M/IM; we have F_{0}(M/IM) = F_{0}(M) · \overline{R} = 0, and (Ann\overline{R}M/IM)^{k} ⊂ F_{0}(M/IM), i.e. Ann\overline{R}M/IM is a nilpotent ideal. Thus, thanks to claim 3.2.22 we can replace R and M by \overline{R} and M/IM, and thereby reduce to the case where F_{0}(M) = 0. We claim that the filtration 0 = F_{0}(M) ⊂ F_{1}(M) ⊂ ... ⊂ F_{k+1}(M) = R will do in this case. Indeed, let L_{1} →^{ϕ} L_{0} → M → 0 be a presentation of M, where L_{0} and L_{1} are free R-modules and the rank of L_{0} equals k; by definition F_{j}(M) is the image of the map Λ_{R}^{k−j}L_{1} ⊗_{R} Λ_{R}^{j}L_{0} → Λ_{R}^{k}L_{0} ≃ R defined by the rule x ⊗ y ↦ Λ_{R}^{k−j}ϕ(x) ∧ y. We deduce easily that the induced surjection Λ_{R}^{k−j}L_{1} ⊗_{R} Λ_{R}^{j}L_{0} → F_{j}(M)/F_{j−1}(M) factors through the module Λ_{R}^{k−j}L_{1} ⊗_{R} Λ_{R}^{j}M; however the latter is a quotient of sums of M, at least when j ≥ 1, so the claim follows. □

Lemma 3.2.23. Let A → B be a finite morphism of almost algebras with nilpotent kernel. There exists m ≥ 0 such that the following holds. For every A-linear morphism ϕ : M → N, set ϕ_{B} := ϕ ⊗_{A} 1_{B} : M ⊗_{A} B → N ⊗_{A} B; then :

(i) Ann_{A}(Coker ϕ_{B})^{m} ⊂ Ann_{A}(Coker ϕ).

(ii) (Ann_{V} (Ker ϕ_{B}) · Ann_{V} (Tor_{1}^{A}(B, N)) · Ann_{V} (Coker ϕ))^{m} ⊂ Ann_{A}(Ker ϕ).

If B = A/I for some nilpotent ideal I, and I^{n} = 0, then we can take m = n in (i) and (ii).

Proof. Under the assumptions, we can find a finitely generated A_{∗}-module Q such that 𝔪·B_{∗} ⊂ Q ⊂ B_{∗}. By lemma 3.2.21 there exists a finite filtration 0 = J_{m} ⊂ ... ⊂ J_{1} ⊂ J_{0} = A_{∗} such that each J_{i}/J_{i+1} is a quotient of a direct sum of copies of Q. This implies that

(3.2.24)  Ann_{A}(M ⊗_{A} B)^{m} ⊂ Ann_{A}(M)

for every A-module M; (i) follows easily. Notice that if B = A/I and I^{n} = 0, then we

can take m = n in (3.2.24). For (ii) let C^{•} := Cone ϕ. We estimate H := H^{−1}(C^{•} ⊗^{L}_{A} B) in two ways. By the first spectral sequence of hyperhomology we have an exact sequence Tor_{1}^{A}(N, B) → H → Ker ϕ_{B}. By the second spectral sequence for hyperhomology we have an exact sequence Tor_{2}^{A}(Coker ϕ, B) → Ker(ϕ) ⊗_{A} B → H. Hence Ker(ϕ) ⊗_{A} B is annihilated by the product of the three annihilators in (ii) and the result follows by applying (3.2.24) with M := Ker ϕ. □

Lemma 3.2.25. Keep the assumptions of lemma 3.2.23 and let M be an A-module.

(i) If A → B is an epimorphism, M is flat and M_{B} := B ⊗_{A} M is almost projective over B, then M is almost projective over A.

(ii) If M_{B} is an almost finitely generated B-module then M is an almost finitely generated

A-module. (iii) If Tor_{1}^{A}(B, M) = 0 and M_{B} is almost finitely presented over B, then M is almost

finitely presented over A.

Proof. (i) : we have to show that Ext_{A}^{1}(M, N) is almost zero for every A-module N. Let I := Ker(A → B); by assumption I is nilpotent, so by the usual devissage we may assume that IN = 0. If χ ∈ Ext_{A}^{1}(M, N) is represented by an extension 0 → N → Q → M → 0 then after tensoring by B and using the flatness of M we get an exact sequence of B-modules 0 → N → B ⊗_{A} Q → M_{B} → 0. Thus χ comes from an element of Ext_{B}^{1}(M_{B}, N) which is almost zero by assumption.

(ii) : for a given finitely generated subideal 𝔪_{0} ⊂ 𝔪, let N ⊂ M_{B} be a finitely generated B-submodule such that 𝔪_{0}M_{B} ⊂ N. Since the induced map M_{∗}⊗_{A_{∗}}B_{∗} → (M_{B})_{∗} is almost surjective, we can find a finitely generated A-submodule N_{0} ⊂ M such that 𝔪_{0}N ⊂ Im((N_{0})_{B} → M_{B}); by lemma 3.2.23(i) it follows that 𝔪_{0}^{2n}(M/N_{0}) = 0 for some n ≥ 0 depending only on B, whence the claim.

(iii) : Let 𝔪_{0} be as above. By (ii), M is almost finitely generated over A, so we can choose a morphism ϕ : A^{r} → M such that 𝔪_{0} · Coker ϕ = 0. Consider ϕ_{B} := ϕ ⊗_{A} 1_{B} : B^{r} → M_{B}. By claim 2.3.12, there is a finitely generated submodule N of Ker ϕ_{B} containing 𝔪_{0}^{2} · Ker ϕ_{B}. Notice that Ker(ϕ)⊗_{A} B maps onto Ker(B^{r} → Im(ϕ)⊗_{A} B) and Ker(Im(ϕ)⊗_{A} B → M_{B}) ≃ Tor_{1}^{A}(B, Coker ϕ) is annihilated by 𝔪_{0}. Hence 𝔪_{0} · Ker ϕ_{B} is contained in the image of Ker ϕ and therefore we can lift a finite generating set {x′_{1}, ..., x′_{n}} for 𝔪_{0}^{2}N to almost elements {x_{1}, ..., x_{n}} of Ker ϕ. If we quotient A^{r} by the span of these x_{i}, we get a finitely presented A-module F with a morphism \overline{ϕ} : F → M such that Ker(\overline{ϕ} ⊗_{A} B) is annihilated by 𝔪_{0}^{4} and Coker \overline{ϕ} is annihilated by 𝔪_{0}. By lemma 3.2.23(ii) we derive 𝔪_{0}^{5m} · Ker \overline{ϕ} = 0 for some m ≥ 0. Since 𝔪_{0} is arbitrary, this proves the result. □

Remark 3.2.26. (i) Inspecting the proof, one sees that parts (ii) and (iii) of lemma 3.2.25 hold whenever (3.2.24) holds. For instance, if A → B is any faithfully flat morphism, then (3.2.24) holds with m := 1.

(ii) Consequently, if A → B is faithfully flat and M is an A-module such that M_{B} is flat (resp. almost finitely generated, resp. almost finitely presented) over B, then M is flat (resp. almost finitely generated, resp. almost finitely presented) over A.

(iii) On the other hand, we do not know whether a general faithfully flat morphism A → B descends almost projectivity. However, using (ii) and proposition 2.4.18 we see that if the Bmodule M_{B} is almost finitely generated projective, then M has the same property.

(iv) Furthermore, if B is faithfully flat and almost finitely presented as an A-module, then A → B does descend almost projectivity, as can be easily deduced from lemma 2.4.31(i) and proposition 2.4.18(ii).

3.2.27. We denote by A-Ét_{afp} the full subcategory of A-Ét consisting of all étale A-algebras B such that B is almost finitely presented as an A-module. Theorem 3.2.28. Let I ⊂ A be a nilpotent ideal, and set A′ := A/I.

(i) Suppose that 𝔪̃ is a (flat) V -module of homological dimension ≤ 1. Let P′ be an almost projective A′-module.

(a) There is an almost projective A-module P with A′ ⊗_{A} P ≃ P′.

(b) If P′ is almost finitely presented, then P is almost finitely presented. (ii) The equivalence of theorem 3.2.18(ii) restricts to an equivalence A-Ét_{afp} → A′-Ét_{afp}.

Proof. (i.a): as usual we reduce to I^{2} = 0. Then proposition 3.2.20(i) applies with R := A_{∗}, J := I_{∗}, R′ := A_{∗}/I_{∗}, M′ := P′_{!}, K := I_{∗} ⊗_{R′} P′_{!} and u′ := 1_{K}. We obtain a class ω(A_{∗}, u′) ∈ Ext_{R′}^{2}(P′_{!}, I_{∗} ⊗_{R′} P′_{!}) which gives the obstruction to the existence of a flat A_{∗}-module F lifting P′_{!}. Since P′_{!} is almost projective, lemma 2.4.14(ii) says that Ext_{R′}^{2}(P′_{!}, I_{∗} ⊗_{R′} P′_{!}) = 0, so such F can always be found, and then the A-module P = F^{a} is a flat lifting of P′; by lemma 3.2.25(i) we see that P is almost projective. Now (i.b) follows from (i.a), lemma 3.2.25(ii) and proposition 2.4.18(i).

(ii): in view of theorem 3.2.18(iii), we only have to show that an étale A-algebra B is almost finitely presented as an A-module whenever B ⊗_{A} A′ is almost finitely presented as an A′- module. However, the assertion is a direct consequence of lemma 3.2.25(iii). □

Remark 3.2.29. (i) According to proposition 2.1.12(ii.b), theorem 3.2.28(i) applies especially when 𝔪 is countably generated as a V -module.

(ii) For P and P′ as in theorem 3.2.28(i.b) let σ_{P} : P → P′ be the projection. It is natural to ask whether the pair (P, σ_{P}) is uniquely determined up to isomorphism, i.e. whether, for any other pair (Q, σ_{Q} : Q → P′) for which theorem 3.2.28(i.b) holds, there exists an Alinear isomorphism ϕ : P → Q such that σ_{Q} ∘ ϕ = σ_{P}. The answer is negative in general. Consider the case P′ := A′. Take P := Q := A and let σ_{P} be the natural projection, while σ_{Q} := (u′ · 1_{A′}) ∘ σ_{P}, where u′ is a unit in A′_{∗}. Then the uniqueness question amounts to whether every unit in A′_{∗} lifts to a unit of A_{∗}. The following counterexample is related to the fact that the completion of the algebraic closure \overline{ℚ}_{p} of ℚ_{p} is not maximally complete. Let V := \overline{ℤ}_{p}, the integral closure of ℤ_{p} in \overline{ℚ}_{p}. Then V is a non-discrete valuation ring of rank one, and we take for 𝔪 the maximal ideal of V, A := (V/p^{2}V)^{a} and A′ := A/pA. Choose a compatible system of roots of p. An almost element of A′ is just a V -linear morphism ϕ : colim_{n>0} p^{1/n!}V → V/pV. Such a ϕ can be represented (in a non-unique way) by an infinite series

of the form ∑_{n=1}^{∞} a_{n}p^{1−1/n!} (a_{n} ∈ V). The meaning of this expression is as follows. For every m > 0, scalar multiplication by the element ∑_{n=1}^{m} a_{n}p^{1−1/n!} ∈ V defines a morphism ϕ_{m} : p^{1/m!}V → V/pV. For m′ > m, let j_{m,m′} : p^{1/m!}V → p^{1/m′!}V be the imbedding. Then we have ϕ_{m′} ∘j_{m,m′} = ϕ_{m}, so that we can define ϕ := colim_{m>0} ϕ_{m}. Similarly, every almost element of

A can be represented by an expression of the form a_{0} + ∑_{n=1}^{∞} a_{n}p^{2−1/n!}. Now, if σ : A → A′ is the natural projection, the induced map σ_{∗} : A_{∗} → A′_{∗} is given by: a_{0} + ∑_{n=1}^{∞} a_{n}p^{2−1/n!} ↦ a_{0}. In particular, its image is the subring V/p ⊂ (V/p)_{∗} = A′_{∗}. For instance, the unit ∑_{n=1}^{∞} p^{1−1/n!} of A′_{∗} does not lie in the image of this map.

In the light of remark 3.2.29, the best one can achieve in general is the following result. Proposition 3.2.30. Assume (A) (see (2.1.6)) and keep the notation of theorem 3.2.28(i). Suppose that (Q, σ_{Q} : Q → P′) and (P, σ_{P} : P → P′) are two pairs as in remark 3.2.29(ii). Then for every ε ∈ 𝔪 there exist A-linear morphisms t_{ε} : P → Q and s_{ε} : Q → P such that

PQ(ε) \begin{matrix}σ_{Q} ∘ t_{ε} = & ε · σ_{P} & σ_{P} ∘ s_{ε} = & ε · σ_{Q} \\ s_{ε} ∘ t_{ε} = & ε^{2} · 1_{P} & t_{ε} ∘ s_{ε} = & ε^{2} · 1_{Q}. \end{matrix}

Proof. Since both Q and P are almost projective and σ_{P} , σ_{Q} are epimorphisms, there exist morphisms \overline{t}_{ε} : P → Q and \overline{s}_{ε} : Q → P such that σ_{Q} ∘ \overline{t}_{ε} = ε · σ_{P} and σ_{P} ∘ \overline{s}_{ε} = ε · σ_{Q}. Then we have σ_{P} ∘ (\overline{s}_{ε} ∘ \overline{t}_{ε} − ε^{2} · 1_{P}) = 0 and σ_{Q} ∘ (\overline{t}_{ε} ∘ \overline{s}_{ε} − ε^{2} ∘ 1_{Q}) = 0, i.e. the morphism u_{ε} := ε^{2} ·1_{P} −\overline{s}_{ε} ∘\overline{t}_{ε} (resp. v_{ε} := ε^{2} ·1_{Q} −\overline{t}_{ε} ∘\overline{s}_{ε}) has image contained in the almost submodule IP (resp. IQ). Since I^{m} = 0 this implies u_{ε}^{m} = 0 and v_{ε}^{m} = 0. Hence

ε^{2m} · 1_{P} = (ε^{2}1_{P})^{m} − u_{ε}^{m} = (∑_{a=0}^{m−1} ε^{2a}u_{ε}^{m−1−a}) ∘ \overline{s}_{ε} ∘ \overline{t}_{ε}.

Define \overline{s}_{(2m−1)ε} := (∑_{a=0}^{m−1} ε^{2a}u_{ε}^{m−1−a}) ∘ \overline{s}_{ε}. Notice that \overline{s}_{(2m−1)ε} = \overline{s}_{ε} ∘ (∑_{a=0}^{m−1} ε^{2a}v_{ε}^{m−1−a}). This implies the equalities \overline{s}_{(2m−1)ε} ∘ \overline{t}_{ε} = ε^{2m} · 1_{P} and \overline{t}_{ε} ∘ \overline{s}_{(2m−1)ε} = ε^{2m} · 1_{Q}. Then the pair (\overline{s}_{(2m−1)ε}, ε^{2(m−1)} · \overline{t}_{ε}) satisfies PQ(ε^{2m−1}). Under (A), every element of 𝔪 is a multiple of an element of the form ε^{2m−1}, therefore the claim follows for arbitrary ε ∈ 𝔪. □

3.3. Nilpotent deformations of torsors. For the considerations that follow it will be convenient to extend yet further our basic setup. Namely, suppose that T is any topos; we can define a basic setup relative to T as a pair (V, 𝔪) consisting of a T -ring V and an ideal 𝔪 ⊂ V satisfying the usual assumptions (2.1.1). Then most of the discussion of chapter 2 extends verbatim to this relative setting. Accordingly, we generally continue to use the same notation as in loc.cit.; however, if it is desirable for clarity’s sake, we may sometimes stress the dependance on T by mentioning it explicitly. For instance, an almost T -ring is an object of the category V^{a}-Alg of associative, commutative and unitary monoids of the abelian tensor category of V^{a}-modules, and sometimes the same category is denoted (T, V, 𝔪)^{a}-Alg.

3.3.1. Let T be a topos and (V, 𝔪) a basic setup relative to T. For every object U of T, we can consider the restriction (V_{/U}, 𝔪_{/U}) of (V, 𝔪) to T_{/U}, which is a basic setup relative to the latter topos. As usual, for every object X → U of T_{/U}, one defines V_{/U}(X) := Hom_{T_{/U}} (X, V) and similarly for 𝔪_{/U}(X). The restriction functor V -Mod → V_{/U}-Mod clearly preserves almost isomorphisms, whence a restriction functor

(T, V, 𝔪)^{a}-Mod → (T_{/U}, V_{/U}, 𝔪_{/U})^{a}-Mod M ↦ M_{/U}.

Similar functors exist for the categories of V^{a}-algebras, and more generally, for A-algebras, where A is any (T, V, 𝔪)^{a}-algebra.

3.3.2. For every almost T -module M, we define a functor M_{∗} : T^{o} → ℤ-Mod by the rule: U ↦ Hom_{V_{/U}^{a}} (V_{/U}^{a}, M_{/U}). Let N be a V -module; using the natural isomorphism

Hom_{V_{/U}} (𝔪̃_{/U}, N_{/U}) →^{∼} N_{∗}^{a}(U)

(cp. (2.2.4)) one sees that M_{∗} is a sheaf for the canonical topology of T, hence it representable by an abelian group object of T, which we denote by the same name. It is then easy to check that M_{∗} is a V -module, and that the functor M ↦ M_{∗} is right adjoint to the localization functor V -Mod → V^{a}-Mod. We can then generalize to the case of the localization functors A-Mod → A^{a}-Mod and A-Alg → A^{a}-Alg, for an arbitrary V -algebra A. Likewise, the left adjoint functors to localization M ↦ M_{!} and B ↦ B_{!!} are obtained in the same way as in the earlier treatment of the one-point topos.

3.3.3. Let T and (V, 𝔪) be as in (3.3.1) and let R be any V -algebra. An affine R-scheme is an object of the category R-Alg^{o}. An affine almost R-scheme (or an affine R^{a}-scheme) is an object of the category R^{a}-Alg^{o}. If X is an affine R^{a}-scheme, then we may write 𝒪_{X} in place of the R^{a}-algebra X^{o}, and an X^{o}-module will also be called a quasi-coherent 𝒪_{X}-module. A morphism ϕ : X → Y of affine almost schemes is the same as the morphism of almost algebras ϕ^{♯} := ϕ^{o} : 𝒪_{Y} → 𝒪_{X}; moreover, ϕ induces pullback (i.e. tensor product − ⊗_{𝒪_{Y}} 𝒪_{X}) and direct image functors (i.e. restriction of scalars), which we will denote

(3.3.4)  ϕ^{∗} : 𝒪_{Y} -Mod → 𝒪_{X}-Mod  and ϕ_{∗} : 𝒪_{X}-Mod → 𝒪_{Y} -Mod

respectively. In the same vein, if A is any R^{a}-algebra, we may write Spec A to denote the object A^{o} represented by A in the opposite category R^{a}-Alg^{o}. We say that X is flat over R^{a} if 𝒪_{X} is a flat R^{a}-algebra. Clearly the category of affine R^{a}-schemes admits arbitrary products. Hence, we can define an affine R^{a}-group scheme as a group object in the category of affine R^{a}-schemes (and likewise for the notion of affine R-group scheme).

3.3.5. The functors B ↦ B_{∗} and B ↦ B_{!!} induce functors on almost schemes, which we denote in the same way. Notice that the functor X ↦ X_{∗} (resp. X ↦ X_{!!}) from affine R^{a}- schemes to R-schemes (resp. to R_{!!}^{a}-schemes) commutes with all colimits (resp. with all limits). Especially, if G is an affine R^{a}-group scheme, then G_{!!} is an affine R_{!!}^{a}-group scheme.

3.3.6. Throughout the rest of this section we fix a V^{a}-algebra A and let S := SpecA.

Let X an affine A-scheme, G an affine A-group scheme. A right action of G on X is a morphism of S-schemes ρ : G ×_{S} X → X fulfilling the usual conditions. To the datum (X, G, ρ) one assigns its nerve G_{X}^{•} which is a simplicial affine S-scheme whose component in degree n is G_{X}^{n} := G^{n} ×_{S} X, and whose face and degeneracy morphisms

∂_{i} : G_{X}^{n+1} → G_{X}^{n}  σ_{j} : G_{X}^{n} → G_{X}^{n+1}  i = 0, ..., n + 1; j = 0, ..., n

are defined for every n ∈ ℕ as in [47, Ch.VI, §2.5].

3.3.7. Let (X, G, ρ) be as in (3.3.6), and let M be a quasi-coherent 𝒪_{X}-module. A G-action on M is a morphism of quasi-coherent 𝒪_{G_{X}^{1}}-modules

(3.3.8)  β : ∂_{0}^{∗}M → ∂_{1}^{∗}M

such that the following diagram commutes:

(3.3.9)  [figure: ∂_{0}^{∗}∂_{0}^{∗}M; ∂_{0}^{∗}β; ∂_{0}^{∗}∂_{1}^{∗}M; ∂_{2}^{∗}∂_{0}^{∗}M; ∂_{2}^{∗}β; ∂_{1}^{∗}∂_{0}^{∗}M; ∂_{1}^{∗}β; ∂_{1}^{∗}∂_{1}^{∗}M; ∂_{2}^{∗}∂_{1}^{∗}M]

and such that

(3.3.10)  σ_{0}^{∗}β = 1_{M}.

One also says that (M, β) is a G-equivariant 𝒪_{X}-module. One defines in the obvious way the morphisms of G-equivariant 𝒪_{X}-modules, and we denote by 𝒪_{X}-Mod_{G} the category of G-equivariant 𝒪_{X}-modules. Lemma 3.3.11. For every G-equivariant 𝒪_{X}-module M, the morphism (3.3.8) is an isomorphism.

Proof. We let τ : G_{X}^{1} → G_{X}^{2} be the morphism given on T -points by the rule: (g, x) ↦ (g, g^{−1}, x). Working out the identifications, one checks easily that τ^{∗}(3.3.9) boils down to the diagram

\xymatrix{∂_{0}^{∗}M \ar[rr]^{β} &  & ∂_{1}^{∗}M \\  & ∂_{1}^{∗}M. \ar[ul]^{τ^{∗}∂_{0}^{∗}β} \ar[ur]_{∂_{1}^{∗}σ_{0}^{∗}β}}

However, ∂_{1}^{∗}σ_{0}^{∗}β = 1_{∂_{1}^{∗}M}, in view of (3.3.10), hence β is an epimorphism and τ^{∗}∂_{0}^{∗}β is an monomorphism. Since ∂_{0} ∘ τ is an isomorphism, we deduce that β is already a monomorphism, whence the claim. □

Lemma 3.3.12. If G is a flat affine S-group scheme acting on an affine S-scheme X, then the category of G-equivariant quasi-coherent 𝒪_{X}-modules is abelian.

Proof. We need to verify that every (G-equivariant) morphism ϕ : M → N of G-equivariant 𝒪_{X}-modules admits a kernel and a cokernel (and then it will be clear that the kernel of N → Coker ϕ equals the cokernel of Ker ϕ → M, since the same holds for 𝒪_{X}-modules). The obvious candidates are the kernel K and cokernel C taken in the category of 𝒪_{X}-modules, and one has only to show that the G-actions of M and N induce G-actions on K and C. This is always the case for C (even when G is not flat). To deal with K, one remarks that both morphisms ∂_{0}, ∂_{1} : G_{X}^{1} → X are flat; indeed, this is clear for ∂_{1}. Then the same holds for ∂_{0} := ∂_{1} ∘ ω, where ω := (ρ, 1_{X}) : G_{X}^{1} → G_{X}^{1} is the isomorphism deduced from the action ρ : G_{X}^{1} → X. It follows that Ker (∂_{i}^{∗}ϕ) ≃ ∂_{i}^{∗}(Ker ϕ) for i = 0, 1, whence the claim. □

3.3.13. Let ϕ : X → Y be a morphism of affine S-schemes. A square zero deformation of X over Y is a datum of the form (j : X → X′, ℐ , β), consisting of:

(a) a morphism of Y -schemes j : X → X′ such that the induced morphism of 𝒪_{Y} -algebras j^{♯} : 𝒪_{X′} → 𝒪_{X} is an epimorphism and 𝒥 := Ker j^{♯} is a square zero ideal, and

(b) a quasi-coherent 𝒪_{X}-module ℐ with an isomorphism of 𝒪_{X}-modules β : j^{∗}𝒥 →^{∼} ℐ. The square zero deformations form a category Exal_{Y} (X, ℐ), with morphisms defined in the obvious way. As in the case of the one-point topos, we can compute the isomorphism classes of square zero deformations of X in terms of an appropriate cotangent complex. And, just as in the earlier treatment, we have to make sure that we are dealing with exact algebras, hence the right definition of the cotangent complex of ϕ is:

𝕃_{X/Y} := ι^{∗}𝕃_{(X⨿SpecV^{a})_{!!}/(Y ⨿SpecV^{a})_{!!}}

where ι : X_{!!} → (X ⨿ SpecV^{a})_{!!} is the natural morphism of schemes. This is an object of D(𝒪_{X_{!!}}-Mod).

3.3.14. Next, let G be a flat affine S-group scheme acting on X and Y, in such a way that ϕ is G-equivariant; it follows from lemma 3.3.12 that 𝒥 has a natural G-action. A G-equivariant square zero deformations of X over Y, is a datum (j : X → X′, ℐ , β) as above, such that, additionally, X′ and ℐ are endowed with a G-action and both j and β are G-equivariant. Let us denote by Exal_{Y} (X/G, ℐ) the category of such G-equivariant deformations. We aim to classify the isomorphism classes of objects of Exal_{Y} (X/G, ℐ), and more generally, study the G-equivariant deformation theory of X by means of an appropriate cotangent complex cohomology. This is achieved by the following device.

3.3.15. Let I be a (small) category; recall ([47, Ch.VI, §5.1]) that a fibred topos over I is a pseudo-functor X of I in the 2-category of of topoi, i.e. the datum consisting of:

(a) for every object i of I, a topos X_{i}

(b) for every arrow f : i → j in I, a morphism of topoi X_{f} : X_{i} → X_{j} (sometimes denoted f)

(c) for every composition i →^{f} j →^{g} k, a transitivity isomorphism X_{f,g} : X_{g} ∘ X_{f} →^{∼} X_{gf}, submitted to certain compatibility conditions (in practice, we will omit from the notation the transitivity isomorphisms).

Given a fibred topos X over I, one denotes by Top(X) the following category, which is easily seen to be a topos ([47, Ch.VI, §5.2]). An object of E of Top(X) is the datum of

(a) for every i ∈ I, an object E_{i} of X_{i}

(b) for every arrow f : i → j, a morphism E_{f} : f^{∗}E_{j} → E_{i} such that, for every composition i →^{f} j →^{g} k, one has E_{gf} = E_{f} ∘ f^{∗}E_{j}, provided one identifies (gf)^{∗}E_{k} with

f^{∗}g^{∗}E_{k} via X_{f,g}.

As an example we have the topos s.T whose objects are the cosimplicial objects of T; indeed this is the topos Top(F) associated to a fibred topos F, whose indexing category is the category Δ^{o} defined in (2.2) : to every object [n] of Δ^{o} one assigns F_{[n]} := T and for every morphism f : [n] → [m] of Δ^{o}, one takes F_{f} := 1_{T}, the identity functor of the topos T.

There is an obvious functor T → s.T that assigns to any object Z of T the constant cosimplicial T -object s.Z associated to Z. Especially, if (T, V, 𝔪) is a basic setup for T, then (s.T, s.V, s.𝔪) is a basic setup relative to s.T.

3.3.16. Suppose now that M_{•} is a cosimplicial V^{a}-module. By applying termwise the functor N ↦ N_{∗} we deduce a cosimplicial V -module (M_{•})_{∗}, whence an object of s.T which we denote by the same name. Clearly (M_{•})_{∗} is a s.V -module and we can therefore take its image in the localized category (s.V)^{a}-Mod. This defines a functor

(3.3.17)  s.(V^{a}-Mod) → (s.V)^{a}-Mod

and it is not difficult to see that (3.3.17) is an equivalence of categories. Similar equivalences then follow for categories of cosimplicial V^{a}-algebras (a.k.a. simplicial V^{a}-schemes) and the like. For instance, for (X, G, ρ) as in (3.3.6) we can regard the nerve G_{X}^{•} as an affine s.Sscheme.

3.3.18. More generally, let X_{•} be any simplicial S-scheme. A quasi-coherent 𝒪_{X_{•}}-module is the same as a cosimplicial 𝒪_{S}-module M_{•}, such that M_{n} is an 𝒪_{X_{n}}-module for every n ∈ ℕ, and the coface (resp. codegeneracy) morphisms ∂^{i} : M_{n} → M_{n+1} (resp. σ^{j} : M_{n} → M_{n−1}) are ∂_{i}-linear (resp. σ_{j}-linear), i.e. they induce M_{n}-linear morphisms M_{n} → ∂_{i∗}M_{n+1} (resp. M_{n} → σ_{j∗}M_{n−1}) (notation of (3.3.4)). It is convenient to introduce the following notation: for every i, n ∈ ℕ we let

\overline{∂}^{i} : ∂_{i}^{∗}M_{n} → M_{n+1}

the M_{n+1}-linear morphism deduced from ∂^{i} by extension of scalars (and likewise for \overline{σ}^{j}).

3.3.19. For every S-scheme X on which G acts, and for every n ∈ ℕ, set π_{X,n} := ∂_{1} ∘ ∂_{2} ∘ ... ∘ ∂_{n} : G_{X}^{n} → X. For any G-equivariant quasi-coherent 𝒪_{X}-module M, we define a quasicoherent 𝒪_{G_{X}^{•}}-module π_{X}^{∗}M as follows. According to (3.3.16), this is the same as defining a module π_{X}^{∗}M over the cosimplicial almost algebra 𝒪_{G_{X}^{•}}; then we set π_{X}^{∗}M_{n} := π_{X,n}^{∗}M for every object [n] of Δ^{o}. Next, we remark that π_{X,n−1} ∘ ∂_{i} = π_{X,n} for every i, n > 0, hence we have natural isomorphisms π_{X}^{∗}M_{n} →^{∼} ∂_{i}^{∗}π_{X}^{∗}M_{n−1}, from which we deduce the coface morphisms ∂^{i} : π_{X}^{∗}M_{n−1} → π_{X}^{∗}M_{n} for every i, n > 0. Finally we use the morphism (3.3.8) and the cartesian diagram:

\xymatrix{G_{X}^{n} \ar[r]^{∂_{0}} \ar[d]_{τ:=∂_{2}∘...∘∂_{n}} & G_{X}^{n−1} \ar[d]^{π_{X,n−1}} \\ G_{X}^{1} \ar[r]^{∂_{0}} & X}

to define ∂^{0} as the composition:

\xymatrix{M_{n−1} \ar[r] & ∂_{0}^{∗}M_{n−1} = ∂_{0}^{∗}π_{X,n−1}^{∗}M = τ^{∗}∂_{0}^{∗}M \ar[r]^{τ^{∗}β} & π_{X,n}^{∗}M = M_{n}.}

We leave to the reader the task of defining the codegeneracy morphisms; using the cocycle relation encoded in (3.3.9) one can then verify the required cosimplicial identities. In this way we obtain a functor

(3.3.20)  𝒪_{X}-Mod_{G} → 𝒪_{G_{X}^{•}}-Mod  M ↦ π_{X}^{∗}M.

Proposition 3.3.21. The functor (3.3.20) is fully faithful, and its essential image is the full subcategory of all quasi-coherent 𝒪_{G_{X}^{•}}-modules (M_{n} | n ∈ ℕ) such that \overline{∂}^{n} : ∂_{n}^{∗}M_{n−1} → M_{n} is an isomorphism for every n ∈ ℕ (notation of (3.3.18)).

Proof. Let ϕ_{•} : π_{X}^{∗}M → ϕ_{X}^{∗}N be a morphism of 𝒪_{G_{X}^{•}}-modules. Since M_{n} = π_{X,n}^{∗}M for every n ∈ ℕ, we see that ϕ_{•} is already determined by its component ϕ_{0} : M → N, whence the full faithfulness of (3.3.20). On the other hand, let M_{•} be a quasi-coherent 𝒪_{G_{X}^{•}}-module satisfying the condition of the lemma, and set M := M_{0}. We define a morphism β : ∂_{0}^{∗}M → ∂_{1}^{∗}M as the

composition ∂_{0}^{∗}M →^{\overline{∂}^{0}} M_{1} →^{(\overline{∂}^{1})^{−1}} ∂_{1}^{∗}M.

Claim 3.3.22. β defines an action of G on M. Proof of the claim: The identity σ_{0}^{∗}β = 1_{M} is immediate, hence it suffices to show that (3.3.9) commutes. This is the same as showing the commutativity of the following diagram:

[figure: ∂_{0}^{∗}∂_{1}^{∗}M = ∂_{2}^{∗}∂_{0}^{∗}M; ∂_{0}^{∗}(\overline{∂}^{1}); ∂_{2}^{∗}(\overline{∂}^{0}); ∂_{0}^{∗}M_{1}; _{\overline{∂}_{0}}; ∂_{2}^{∗}M_{1}; _{\overline{∂}_{2}}; ∂_{0}^{∗}∂_{0}^{∗}M; ∂_{0}^{∗}(\overline{∂}^{0}); M_{2}; ∂_{2}^{∗}∂_{1}^{∗}M; ∂_{2}^{∗}(\overline{∂}^{1}); ∂_{1}^{∗}∂_{0}^{∗}M; ∂_{1}^{∗}(\overline{∂}^{0}); ∂_{1}^{∗}M_{1}; _{\overline{∂}_{1}}; ∂_{1}^{∗}∂_{1}^{∗}M; ∂_{1}^{∗}(\overline{∂}^{1})]

which can be checked separately on each of its three quadrangular subdiagrams. Let us verify for instance the commutativity of the bottom left diagram. By linearity, we can simplify down to the diagram:

\xymatrix{M_{1} \ar[r]^{∂^{0}} & M_{2} \\ M \ar[u]^{∂^{0}} \ar[r]^{∂_{0}} & M_{1} \ar[u]_{∂^{1}}}

whose commutativity expresses one of the identities defining the cosimplicial module M_{•}.

To conclude the proof, it suffices to exhibit an isomorphism γ_{•} : π_{X}^{∗}M →^{∼} M_{•}. For every n ∈ ℕ, let γ_{n} : π_{X,n}^{∗}M → M_{n} be the morphism induced by the composition ∂^{n} ∘ ∂^{n−1} ∘ ... ∘ ∂^{1}; under our assumption, γ_{n} is an isomorphism, and we leave to the reader the verification that (γ_{n} | n ∈ ℕ) defines a morphism of cosimplicial modules. □

3.3.23. In case G is flat over S, ℐ is G-equivariant, and ϕ : X → Y is an equivariant morphism of affine S-schemes on which G acts, we deduce a natural functor:

(3.3.24)  Exal_{Y} (X/G, ℐ) → Exal_{G_{Y}^{•}} (G_{X}^{•}, π_{X}^{∗}ℐ).

Namely, to any G-equivariant square zero deformation D := (j : X → X′, ℐ , β) one assigns the datum G_{D} := (G_{j} : G_{X}^{•} → G_{X′}^{•}, π_{X}^{∗}ℐ , π_{X}^{∗}β). The flatness of G ensures that Ker(𝒪_{G_{X′}^{n}} → 𝒪_{G_{X}^{n}}) ≃ π_{X,n}^{∗}ℐ for every n ∈ ℕ, which means that G_{D} is indeed a deformation of G_{X}^{•} over G_{Y}^{•}.

The basic observation is contained in the following:

Lemma 3.3.25. Under the assumptions of (3.3.23), the functor (3.3.24) is an equivalence of categories.

Proof. Let (j : G_{X}^{•} → X′_{•}, π_{X}^{∗}ℐ , β) be a deformation of G_{X}^{•}. For every n ∈ ℕ we have a commutative diagram of affine S-schemes:

\xymatrix{X′_{n} \ar[r]^{∂} \ar[d] & X′_{0} \ar[d] \\ G_{Y}^{n} \ar[r]^{π_{Y}} & Y}

where ∂ := ∂_{1} ∘ ∂_{2} ∘ ... ∘ ∂_{n} and π_{Y,n} is the natural projection as in (3.3.19). We deduce a unique morphism α_{n} : X′_{n} → G_{Y}^{n} ×_{Y} X′_{0} ≃ G_{X′_{0}}^{n}. By construction, α_{n}^{♯} fits into a commutative diagram

[figure: 0; 𝒪_{G^{n}} ⊗_{𝒪_{S}} ℐ; 𝒪_{G^{n}} ⊗_{𝒪_{S}} 𝒪_{X′_{0}}; α_{n}^{♯}; 𝒪_{G^{n}} ⊗_{𝒪_{S}} 𝒪_{X}; 0; 0; 𝒪_{G^{n}} ⊗_{𝒪_{S}} ℐ; 𝒪_{X′_{n}}; 𝒪_{G^{n}} ⊗_{𝒪_{S}} 𝒪_{X}; 0]

hence α_{n} is an isomorphism for every n ∈ ℕ. To conclude, it suffices to verify that the system of morphisms (α_{n} | n ∈ ℕ) defines a morphism of simplicial S-schemes: α_{•} : X′_{•} → G_{X′_{0}}^{•}. This amounts to showing that the α_{n} commute with the face and degeneracy morphisms, which however is easily checked from the definition. □

3.3.26. Combining lemmata 3.3.25 and 2.5.17 (which holds verbatim in the present context) we derive a natural equivalence of categories:

Exal_{Y} (X/G, ℐ) → Exal_{G_{Y !!}^{•}}(G_{X!!}^{•}, (π_{X}^{∗}ℐ)_{∗}).

This enables us to use the usual theory of the cotangent complex to classify the G-equivariant deformations of X.

3.3.27. According to [47, Ch.VI, §5.3], for every n ∈ ℕ we have a morphism of topoi

[n]_{T} : T → s.T

called restriction to the n-th level. It is given by a pair of adjoint functors ([n]_{T}^{∗} , [n]_{T ∗}) such that [n]_{T}^{∗} : s.T → T is the functor that assigns to any cosimplicial object (X_{[k]} | k ∈ ℕ) the object X_{[n]} of T. For every k ∈ ℕ set N_{X,k} := G_{X}^{k} ⨿ Spec V^{a}; the system (N_{X,k} | k ∈ ℕ) defines a simplicial Spec V^{a}-scheme N_{X} (and likewise we define N_{Y}). In view of [46, Ch.II, (1.2.3.5)], we deduce natural isomorphisms of simplicial complexes of flat 𝒪_{N_{X,n!!}}-modules:

[n]_{T}^{∗} 𝕃_{N_{X!!}/N_{Y !!}} →^{∼} 𝕃_{N_{X,n!!}/N_{Y,n!!}} for every n ∈ ℕ

whence natural isomorphisms of simplicial complexes of flat 𝒪_{G_{X!!}^{n}}-modules:

[n]_{T}^{∗} 𝕃_{G_{X}^{•}/G_{Y}^{•}} := [n]_{T}^{∗} ι_{•}^{∗}𝕃_{N_{X!!}/N_{Y !!}} →^{∼} ι_{[n]}^{∗}[n]_{T}^{∗} 𝕃_{N_{X!!}/N_{Y !!}} →^{∼} ι_{[n]}^{∗}𝕃_{N_{X,n!!}/N_{Y,n!!}} =: 𝕃_{G_{X}^{n}/G_{Y}^{n}}

where ι_{•} : G_{X!!}^{•} → N_{X!!} is the morphism of simplicial schemes defined as in (3.3.13). In other words, 𝕃_{G_{X}^{•}/G_{Y}^{•}} is a mixed simplicial-cosimplicial module 𝕃_{••} whose rows 𝕃_{•n} are the cotangent complexes of the morphisms G_{ϕ} : G_{X}^{n} → G_{Y}^{n}. Furthermore, for every n ∈ ℕ and every i ≤ n + 1 we have a cartesian diagram

\xymatrix{G_{X}^{n+1} \ar[r]^{∂_{i}} \ar[d]_{G_{ϕ}^{n+1}} & G_{X}^{n} \ar[d]^{G_{ϕ}^{n}} \\ G_{Y}^{n+1} \ar[r]^{∂_{i}} & G_{Y}^{n}}

whose horizontal arrows are flat morphisms (since G is S-flat by assumption). Whence, taking into account theorem 2.5.35, a natural isomorphism in D(𝒪_{N_{X,n+1!!}}-Mod) :

\overline{∂}_{𝕃}^{i} : ∂_{i}^{∗}𝕃_{G_{X}^{n}/G_{Y}^{n}} →^{∼} 𝕃_{G_{X}^{n+1}/G_{Y}^{n+1}}

and, by unwinding the definitions, one sees that \overline{∂}_{𝕃}^{i} is induced by the i-th coface morphism 𝕃_{•n} → 𝕃_{•n+1} of the double complex 𝕃_{••}. Definition 3.3.28. Let G be an affine S-group scheme, ϕ : X → Y a G-equivariant morphism of affine S-schemes on which G acts. We say that ϕ is a G-torsor over Y if the action of G on Y is trivial (i.e. ρ : G ×_{S} Y → Y is the natural projection) and there exists a cartesian diagram of affine S-schemes

(3.3.29)  \xymatrix{G ×_{S} Z \ar[r]^{g} \ar[d]_{p_{Z}} & X \ar[d]^{ϕ} \\ Z \ar[r]^{f} & Y}

such that f is faithfully flat, g is G-equivariant for the action on G ×_{S} Z deduced from G, and p_{Z} is the natural projection.

3.3.30. Suppose that G is a flat affine S-group scheme and X → Y is a G-torsor over Y. Then the groupoid \xymatrix{G ×_{S} X \ar[r] \ar[r] & X} defined by the G-action on X is an effective equivalence relation (cp. (4.5.1)) and Y ≃ X/G. Furthermore, let X^{×_{Y} n} := X ×_{Y} X ×_{Y} .... ×_{Y} X (the n-th fold cartesian power of X over Y); there are natural identifications G_{X}^{n} ≃ X^{×_{Y} n+1}, amounting to an isomorphism of (semi-)simplicial S-schemes augmented over Y:

G_{X}^{•} →^{∼} [Y |X].

(cp. the notation of [5, Exp.V^{bis}, (1.2.7)]). We can regard the cosimplicial T -ring 𝒪_{G_{X}^{•}} as a ring of the topos Γ(Δ × T) deduced from the simplicial topos Δ × T (notation of [5, Exp.V^{bis}, (1.2.8)]), whence an augmentation of fibred topoi

(3.3.31)  θ : (Δ × T, 𝒪_{G_{X}^{•}}) → (Δ × T, 𝒪_{Y})

and it follows from the foregoing and from remark 3.1.3 that (3.3.31) is an augmentation of 2-cohomological descent (see [5, Exp.V^{bis}, Déf.2.2.6]). Denote by

\overline{θ}^{∗} : (T, 𝒪_{Y})-Mod → (Γ(Δ × T), 𝒪_{G_{X}^{•}})-Mod

the morphism obtained as the composition of the constant functor ε^{∗} : (T, 𝒪_{Y})-Mod → (Γ(Δ × T), 𝒪_{Y})-Mod and the functor Γ(θ^{∗}) : (Γ(Δ × T), 𝒪_{Y})-Mod → 𝒪_{G_{X}^{•}}-Mod. Since the morphisms \overline{∂}_{𝕃}^{i} are isomorphisms, it then follows from general cohomological descent ([5, Exp.V^{bis}, Prop.2.2.7]) that, for every k ∈ ℕ, the truncated system (τ_{[−k}𝕃_{G_{X}^{n}/G_{Y}^{n}} | n ∈ ℕ) is in the essential image of the functor L^{+}\overline{θ}^{∗}. In the following we will be only interested in the case where the cotangent complex is concentrated in degree zero, in which case one can avoid the recourse to cohomological descent, and rather appeal to more down-to-earth faithfully flat descent. In any case, the foregoing shows that there exists a uniquely determined pro-object (𝕃_{X/Y,k}^{G} | k ∈ ℕ) of D(𝒪_{Y} -Mod) such that

L^{+}\overline{θ}^{∗}(𝕃_{X/Y,k}^{G}) ≃ τ_{[−k}𝕃_{G_{X}^{•}/G_{Y}^{•}}^{a} for every k ∈ ℕ.

By trivial duality there follow natural isomorphisms for every k ∈ ℕ and every complex K^{•} ∈ D^{+}(𝒪_{G_{X}^{•}}-Mod) such that K = τ_{[−k}K :

(3.3.32)  𝔼xt_{𝒪_{G^{•}_{X}}}^{k} (𝕃_{G_{X}^{•}/G_{Y}^{•}}^{a} , K^{•}) ≃ 𝔼xt_{𝒪_{Y}}^{k} (𝕃_{X/Y,k}^{G}, R^{+}\overline{θ}_{∗}K^{•}).

Definition 3.3.33. Let G be an affine S-scheme and e : S → G its unit section. The colie complex of G is the complex of 𝒪_{S}-modules ℓ_{G/S} := e^{∗}𝕃_{G/S}^{a}. Proposition 3.3.34. Let G be a flat S-group scheme, ϕ : X → Y be a G-torsor over Y and π_{Y} : Y → S the structure morphism. Then, for every k ∈ ℕ, the complex 𝕃_{X/Y,k}^{G} is locally isomorphic to π_{Y}^{∗} τ_{[−k}ℓ_{G/S} in the fpqc topology of Y.

Proof. We will exhibit more precisely a faithfully flat morphism f : Z → Y such that, for every k ∈ ℕ there exist isomorphisms R^{+}f^{∗}𝕃_{X/Y,k}^{G} ≃ R^{+}f^{∗}π_{Y}^{∗} τ_{[−k}ℓ_{G/S} in D(𝒪_{Z}-Mod). Let π_{G} : G → S be the structure morphism; first of all we remark: Claim 3.3.35. 𝕃_{G/S}^{a} ≃ π_{G}^{∗}ℓ_{G/S}.

Proof of the claim: Indeed, π_{G} is trivially a G-torsor over S, hence we have a compatible system of isomorphisms τ_{[−k}𝕃_{G/S}^{a} ≃ π_{Y}^{∗} 𝕃_{G/S,k}^{G}. If e : S → G is the unit section, we deduce: τ_{[−k}ℓ_{G/S} ≃ e^{∗}τ_{[−k}𝕃_{G/S}^{a} ≃ 𝕃_{G/S,k}^{G} for every k ∈ ℕ. After taking π_{G}^{∗} of the two sides, the claim follows.

Let now f : Z → Y be a faithfully flat morphism that trivializes the given G-torsor ϕ : X → Y, so we have a cartesian diagram (3.3.29). Denoting by p_{G} : G ×_{S} Z → G the natural projection, we deduce (since G is S-flat) an isomorphism g^{∗}𝕃_{X/Y} ≃ p_{G}^{∗}𝕃_{G/S}. Whence, in view of claim 3.3.35 :

(3.3.36)  p_{Z}^{∗}f^{∗}𝕃_{X/Y,k}^{G} ≃ p_{G}^{∗}π_{G}^{∗}τ_{[−k}ℓ_{G/S} ≃ p_{Z}^{∗}π_{Z}^{∗}τ_{[−k}ℓ_{G/S} ≃ p_{Z}^{∗}f^{∗}π_{Y}^{∗} τ_{[−k}ℓ_{G/S}.

Let e_{Z} := e ×_{S} 1_{Z} : Z → G ×_{S} Z; the claim follows after applying e_{Z}^{∗} to (3.3.36). □

3.3.37. Finally we can wrap up this section with a discussion of equivariant deformations of torsors. Hence, let ϕ : X → Y be a G-torsor over Y, and j_{Y} : Y → Y′ a morphism of affine S-schemes such that ℐ := Ker(j_{Y}^{♯} : 𝒪_{Y′} → 𝒪_{Y}) is a square zero ideal of 𝒪_{Y′}. We wish to classify the square zero deformations of the torsor ϕ over Y′, that is, the isomorphism classes of cartesian diagrams

\xymatrix{X \ar[d]_{ϕ} \ar[r]^{j_{X}} & X′ \ar[d]^{ϕ′} \\ Y \ar[r]^{j_{Y}} & Y′}

such that ϕ′ is a G-torsor over Y′ and j_{X} is G-equivariant. Theorem 3.3.38. Suppose that G is flat over S, that H_{i}(ℓ_{G/S}) = 0 for i > 0 and that H_{0}(ℓ_{G/S}) is an almost finitely generated projective 𝒪_{S}-module. Furthermore, suppose that the homological dimension of 𝔪̃ is ≤ 1. Then, in the situation of (3.3.37), we have:

(i) The pro-object (𝕃_{X/Y,k}^{G} | k ∈ ℕ) is constant, isomorphic to a complex of D(𝒪_{Y} -Mod) concentrated in degree zero, that we shall denote by 𝕃_{X/Y}^{G}.

(ii) The set of isomorphism classes of square zero deformations of the torsor ϕ : X → Y over Y′ is a torsor under the group Ext_{𝒪_{Y}}^{1} (𝕃_{X/Y}^{G} , ℐ).

(iii) The group of automorphisms of a square zero deformation ϕ′ : X′ → Y′ as in (3.3.37) is naturally isomorphic to Ext_{𝒪_{Y}}^{0} (𝕃_{X/Y}^{G} , ℐ).

Proof. (i) follows easily from proposition 3.3.34; moreover it follows from remark 3.2.26(iii) that H_{0}(𝕃_{X/Y}^{G})^{a} is an almost finitely generated projective 𝒪_{Y} -module. Let 𝒥 := Ker(j^{♯} : 𝒪_{X′} → 𝒪_{X}); by flatness, the natural morphism ϕ^{∗}ℐ → 𝒥 is a G-equivariant isomorphism. We notice that, for every quasi-coherent 𝒪_{Y} -module M, there is a natural isomorphism

\overline{θ}^{∗}M_{!} →^{∼} (π_{X}^{∗}ϕ^{∗}M)_{!}.

By cohomological descent (or else, by plain old-fashioned faithfully flat descent) it follows that the counit of the adjunction:

ℐ_{!} → R^{+}\overline{θ}_{∗}π_{X}^{∗}𝒥_{!}

is an isomorphism, whence, in light of (3.3.32), natural isomorphisms:

(3.3.39)  𝔼xt_{𝒪_{G^{•}_{X}}}^{k} (𝕃_{G_{X}^{•}/G_{Y}^{•}}^{a} , π_{X}^{∗}𝒥) ≃ Ext_{𝒪_{Y}}^{k} (H_{0}(𝕃_{X/Y}^{G}), ℐ)  for every k ∈ ℕ.

Claim 3.3.40. The diagram of functors:

\xymatrix{𝒪_{Y} -Mod \ar[r]^{(−)_{!}} \ar[d]_{ϕ^{∗}} & 𝒪_{Y!!}-Mod \ar[d]^{ϕ_{!!}^{∗}} \\ 𝒪_{X}-Mod \ar[r]^{(−)_{!}} & 𝒪_{X!!}-Mod}

is essentially commutative. Proof of the claim: Indeed, both of the two possible composition of arrows represent a functor 𝒪_{Y} -Mod → 𝒪_{X!!}-Mod which is left adjoint to the functor M ↦ ϕ_{∗}M^{a}.

Claim 3.3.41. The counit of the adjunction: ε : H_{0}(𝕃_{X/Y}^{G}) → H_{0}(𝕃_{X/Y}^{G})_{!}^{a} is an isomorphism.

Proof of the claim: It follows easily from claim 3.3.40 and lemma 2.5.28 that ϕ^{∗}(ε) is an isomorphism. Since ϕ is faithfully flat, the assertion follows.

Combining (3.3.32), claim 3.3.41 and lemma 2.4.14(i),(ii), we deduce that

𝔼xt_{𝒪_{G^{•}_{X}}}^{2} (𝕃_{G_{X}^{•}/G_{Y}^{•}} , π_{X}^{∗}𝒥) = 0.

However, by (3.3.26)and the usual arguments (cp. section 3.2) one knows that the obstruction to deforming the torsor ϕ is a class in the latter group; since the obstruction vanishes, one deduces (ii). (iii) follows in the same way. □

3.4. Descent. Faithfully flat descent in the almost setting presents no particular surprises: since the functor A ↦ A_{!!} preserves faithful flatness of morphisms (see remark 3.1.3) many wellknown results for usual rings and modules extend verbatim to almost algebras.

3.4.1. So for instance, faithfully flat morphisms are of universal effective descent for the fibred categories F : V^{a}-Alg.Mod^{o} → V^{a}-Alg^{o} and G : V^{a}-Alg.Morph^{o} → V^{a}-Alg^{o} (see definition 2.5.22: for an almost V -algebra B, the fibre F_{B} (resp. G_{B}) is the opposite of the category of B-modules (resp. B-algebras)). Then, using remark 3.2.26, we deduce also universal effective descent for the fibred subcategories of flat (resp. almost finitely generated, resp. almost finitely presented, resp. almost finitely generated projective) modules. Likewise, a faithfully flat morphism is of universal effective descent for the fibred subcategories Ét^{o} → V^{a}-Alg^{o} of étale (resp. w.Ét^{o} → V^{a}-Alg^{o} of weakly étale) algebras.

3.4.2. More generally, since the functor A ↦ A_{!!} preserves pure morphisms in the sense of [58], and since, by a theorem of Olivier (loc. cit.), pure morphisms are of universal effective descent for modules, the same holds for pure morphisms of almost algebras.

3.4.3. Non-flat descent is more delicate. Our results are not as complete here as it could be wished, but nevertheless, they suffice for the further study of étale and unramified morphisms that shall be taken up in chapter 5. (and they also cover the cases needed in [34]). Our first statement is the almost version of a theorem of Gruson and Raynaud ([42, Part II, Th. 1.2.4]). Proposition 3.4.4. A finite monomorphism of almost algebras descends flatness.

Proof. Let ϕ : A → B be such a morphism. Under the assumption, we can find a finite A_{∗}- module Q such that 𝔪B_{∗} ⊂ Q ⊂ B_{∗}. One sees easily that Q is a faithful A_{∗}-module, so by [42, Part II, Th. 1.2.4 and lemma 1.2.2], Q satisfies the following condition :

(3.4.5)

If (0 → N → L → P → 0) is an exact sequence of A_{∗}-modules with L flat, such that Im(N ⊗_{A_{∗}} Q) is a pure submodule of L ⊗_{A_{∗}} Q, then P is flat.

Now let M be an A-module such that M ⊗_{A} B is flat. Pick an epimorphism p : F → M with F free over A. Then \underline{Y} := (0 → Ker(p ⊗_{A} 1_{B}) → F ⊗_{A} B → M ⊗_{A} B → 0) is universally exact over B, hence over A. Consider the sequence \underline{X} := (0 → Im(Ker(p)_{!} ⊗_{A_{∗}} Q) → F_{!} ⊗_{A_{∗}} Q → M_{!} ⊗_{A_{∗}} Q → 0). Clearly \underline{X}^{a} ≃ \underline{Y}. However, it is easy to check that a sequence \underline{E} of A-modules is universally exact if and only if the sequence \underline{E}_{!} is universally exact over A_{∗}. We conclude that \underline{X} = (\underline{X}^{a})_{!} is a universally exact sequence of A_{∗}-modules, hence, by condition (3.4.5), M_{!} is flat over A_{∗}, i.e. M is flat over A as required. □

Corollary 3.4.6. Let A → B be a finite morphism of almost algebras, with nilpotent kernel. If C is a flat A-algebras such that C ⊗_{A} B is weakly étale (resp. étale) over B, then C is weakly étale (resp. étale) over A.

Proof. In the weakly étale case, we have to show that the multiplication morphism μ : C ⊗_{A} C → C is flat. As N := Ker(A → B) is nilpotent, the local flatness criterion reduces the question to the situation over A/N. So we may assume that A → B is a monomorphism. Then C ⊗_{A} C → (C ⊗_{A} C)⊗_{A} B is a monomorphism, but μ⊗_{C⊗_{A}C} 1_{(C⊗_{A}C)⊗_{A}B} is the multiplication morphism of C ⊗_{A} B, which is flat by assumption. Therefore, by proposition 3.4.4, μ is flat.

For theétale case, we have to show that C is almost finitely presented as a C⊗_{A}C-module. By hypothesis C ⊗_{A} B is almost finitely presented as a C ⊗_{A} C ⊗_{A} B-module and we know already that C is flat as a C ⊗_{A} C-module, so by lemma 3.2.25(iii) (applied to the finite morphism C ⊗_{A} C → C ⊗_{A} C ⊗_{A} B) the claim follows. □

3.4.7. Suppose now that we are given a cartesian diagram 𝒟 of almost algebras

(3.4.8)  \xymatrix{A_{0} \ar[r]^{f_{2}} \ar[d]_{f_{1}} & A_{2} \ar[d]^{g_{2}} \\ A_{1} \ar[r]^{g_{1}} & A_{3}}

such that one of the morphisms A_{i} → A_{3} (i = 1, 2) is an epimorphism. Diagram 𝒟 induces an essentially commutative diagram for the corresponding categories A_{i}-Mod, where the arrows are given by the “extension of scalars” functors. We define the category of 𝒟-modules as the 2-fibre product 𝒟-Mod := A_{1}-Mod ×_{A_{3}-Mod} A_{2}-Mod. Recall (see [8, Ch.VII §3] or [43, Ch.I] for generalities on 2-categories and 2-fibre products) that 𝒟-Mod is the category whose objects are the triples (M_{1}, M_{2}, ξ), where M_{i} is an A_{i}-module (i = 1, 2) and ξ : A_{3} ⊗_{A_{1}} M_{1} →^{∼} A_{3} ⊗_{A_{2}} M_{2} is an A_{3}-linear isomorphism. There follows a natural functor

π : A_{0}-Mod → 𝒟-Mod.

Given an object (M_{1}, M_{2}, ξ) of 𝒟-Mod, let us denote M_{3} := A_{3} ⊗_{A_{2}} M_{2}; we have a natural morphism M_{2} → M_{3}, and ξ gives a morphism M_{1} → M_{3}, so we can form the fibre product T (M_{1}, M_{2}, ξ) := M_{1} ×_{M_{3}} M_{2}. In this way we obtain a functor T : 𝒟-Mod → A_{0}-Mod, and we leave to the reader the verification that T is right adjoint to π. Let us denote by ε : 1_{ℳ_{0}} → T ∘ π and η : π ∘ T → 1_{𝒟-Mod} the unit and counit of the adjunction. Lemma 3.4.9. The functor π induces an equivalence of full subcategories :

{X ∈ Ob(A_{0}-Mod) | ε_{X} is an isomorphism} →^{π} {Y ∈ Ob(𝒟-Mod) | η_{Y} is an isomorphism}

having T as quasi-inverse.

Proof. General nonsense. □

Lemma 3.4.10. Let M be any A_{0}-module. Then ε_{M} is an epimorphism. If Tor_{1}^{A_{0}}(M, A_{3}) = 0, then ε_{M} is an isomorphism.

Proof. Indeed, ε_{M} : M → (A_{1} ⊗_{A_{0}} M) ×_{A_{3}⊗_{A_{0}}M} (A_{2} ⊗_{A_{0}} M) is the natural morphism. So, the assertions follow by applying − ⊗_{A_{0}} M to the short exact sequence of A_{0}-modules

(3.4.11)  0 → A_{0} →^{f} A_{1} ⊕ A_{2} →^{g} A_{3} → 0

where f(a) := (f_{1}(a), f_{2}(a)) and g(a, b) := g_{1}(a) − g_{2}(b). □

3.4.12. There is another case of interest, in which ε_{M} is an isomorphism. Namely, suppose that one of the morphisms A_{i} → A_{3} (i = 1, 2), say A_{1} → A_{3}, has a section. Then also the morphism A_{0} → A_{2} gains a section s : A_{2} → A_{0} and we have the following : Lemma 3.4.13. In the situation of (3.4.12), suppose that the A_{0}-module M arises by extension of scalars from an A_{2}-module M′, via the section s : A_{2} → A_{0}. Then ε_{M} is an isomorphism.

Proof. Indeed, in this case (3.4.11) is split exact as a sequence of A_{2}-modules, and it remains such after tensoring by M′. □

Lemma 3.4.14. η_{(M_{1},M_{2},ξ)} is an isomorphism for all objects (M_{1}, M_{2}, ξ).

Proof. To fix ideas, suppose that A_{1} → A_{3} is an epimorphism. Consider any 𝒟-module (M_{1}, M_{2}, ξ). Let M := T (M_{1}, M_{2}, ξ); we deduce a natural morphism

ϕ : (M ⊗_{A_{0}} A_{1}) ×_{M⊗_{A_{0}}A_{3}} (M ⊗_{A_{0}} A_{2}) → M_{1} ×_{M_{3}} M_{2}

such that ϕ ∘ ε_{M} = 1_{M}. It follows that ε_{M} is injective, hence it is an isomorphism, by lemma 3.4.10. We derive a commutative diagram with exact rows :

[figure: 0; M; (M ⊗_{A_{0}} A_{1}) ⊕ (M ⊗_{A_{0}} A_{2}); ϕ_{1}⊕ϕ_{2}; M ⊗_{A_{0}} A_{3}; ϕ_{3}; 0; 0; M; M_{1} ⊕ M_{2}; M_{3}; 0.]

From the snake lemma we deduce

(∗)  Ker(ϕ_{1}) ⊕ Ker(ϕ_{2}) ≃ Ker(ϕ_{3})
(∗∗)  Coker(ϕ_{1}) ⊕ Coker(ϕ_{2}) ≃ Coker(ϕ_{3}).

Since M_{3} ≃ M_{1} ⊗_{A_{1}} A_{3} we have A_{3} ⊗_{A_{1}} Coker ϕ_{1} ≃ Coker ϕ_{3}. But by assumption A_{1} → A_{3} is an epimorphism, so also Coker ϕ_{1} → Coker ϕ_{3} is an epimorphism. Then (∗∗) implies that Coker ϕ_{2} = 0. But ϕ_{3} = 1_{A_{3}} ⊗_{A_{2}} ϕ_{2}, thus Coker ϕ_{3} = 0 as well. We look at the exact sequence 0 → Ker ϕ_{1} → M ⊗_{A_{0}} A_{1} →^{ϕ_{1}} M_{1} → 0 : applying A_{3} ⊗_{A_{1}} − we obtain an epimorphism A_{3} ⊗_{A_{1}} Ker ϕ_{1} → Ker ϕ_{3}. From (∗) it follows that Ker ϕ_{2} = 0. In conclusion, ϕ_{2} is an isomorphism. Hence the same is true for ϕ_{3} = 1_{A_{3}} ⊗_{A_{2}} ϕ_{2}, and again (∗), (∗∗) show that ϕ_{1} is an isomorphism as well, which implies the claim. □

Lemma 3.4.15. In the situation of (3.4.7), let M be any A_{0}-module and n ≥ 1 an integer. The following conditions are equivalent:

(a) Tor_{j}^{A_{0}}(M, A_{i}) = 0 for every j ≤ n and i = 1, 2, 3.
(b) Tor_{j}^{A_{i}}(A_{i} ⊗_{A_{0}} M, A_{3}) = 0 for every j ≤ n and i = 1, 2.

Proof. There is a base change spectral sequence

(3.4.16)  E_{pq}^{2} := Tor_{p}^{A_{i}}(Tor_{q}^{A_{0}}(M, A_{i}), A_{3}) ⇒ Tor_{p+q}^{A_{0}}(M, A_{3}).

If now (a) holds, we deduce that Tor_{p}(A_{i} ⊗_{A_{0}} M, A_{3}) ≃ Tor_{p}(M, A_{3}) whenever p ≤ n, and then (b) follows. Conversely, suppose that (b) holds; we show (a) by induction on n. Say that the morphism A_{1} → A_{3} is an epimorphism, so that the same holds for the morphism A_{0} → A_{2}, and denote by I the common kernel of these morphisms. For n = 1 and i = 1, the assumption is equivalent to saying that the natural morphism I ⊗_{A_{1}} (A_{1} ⊗_{A_{0}} M) → (A_{1} ⊗_{A_{0}} M) is injective. It follows that the same holds for the morphism I ⊗_{A_{0}} M → M, which already shows that Tor_{1}^{A_{0}}(M, A_{2}) = 0. Next, the assumption for i = 2 means that the term E_{10}^{2} of the spectral sequence (3.4.16)vanishes, whence an isomorphism Tor_{1}^{A_{0}}(M, A_{3}) ≃ Tor_{1}^{A_{0}}(M, A_{2})⊗_{A_{2}} A_{3} ≃ 0. Finally, we use the long exact Tor sequence arising from the short exact sequence (3.4.11) to deduce that also Tor_{1}^{A_{0}}(M, A_{1}) vanishes. Let now n > 1 and suppose that assertion (a) is already known for all j < n. We choose a presentation

(3.4.17)  0 → R → F → M → 0

with F flat over A_{0}; using the long exact Tor sequence we deduce that Tor_{j}^{A_{0}}(M, A_{i}) ≃ Tor_{j−1}^{A_{0}}(R, A_{i}) for every j > 1 and every i ≤ 3. Moreover, assertion (a) taken with j = 1 shows that the sequence A_{i}⊗_{A_{0}}(3.4.17) is again exact for i ≤ 3, therefore Tor_{1}^{A_{0}}(R, A_{i}) vanishes as well for i ≤ 3. In other words, the A_{0}-module R fulfills condition (b) for every j < n, hence the inductive assumption shows that Tor_{j}^{A_{0}}(R, A_{i}) = 0 for every j < n and i = 1, 2, 3; in turns this implies the sought vanishing for M. □

The following lemma 3.4.18(iii) generalizes a result of D.Ferrand ([35, lemma]). Lemma 3.4.18. Let M be any A_{0}-module. We have:

(i) Ann_{A_{0}}(M ⊗_{A_{0}} A_{1}) · Ann_{A_{0}}(M ⊗_{A_{0}} A_{2}) ⊂ Ann_{A_{0}}(M).

(ii) M admits a three-step filtration 0 ⊂ Fil_{0}M ⊂ Fil_{1}M ⊂ Fil_{2}M = M such that Fil_{0}M

and gr_{2}M are A_{2}-modules and gr_{1}M is an A_{1}-module. (iii) If (A_{1} × A_{2}) ⊗_{A_{0}} M is flat over A_{1} × A_{2}, then M is flat over A_{0}.

Proof. To fix ideas, suppose that A_{1} → A_{3} is an epimorphism, and let I be its kernel; let also M_{i} := A_{i} ⊗_{A_{0}} M for i = 1, 2, 3.

(i): clearly I ≃ Ker(A_{0} → A_{2}), therefore aM ⊂ IM for every a ∈ Ann_{A_{0}}(M_{2}). On the other hand, the natural morphism I ⊗_{A_{1}} M_{1} →^{∼} I ⊗_{A_{0}} M → IM is obviously an epimorphism, whence the assertion.

(ii): we set Fil_{0}M := Ker(ε_{M} : M → M_{1} ×_{M_{3}} M_{2}); using the short exact sequence (3.4.16) we see that Fil_{0}M ≃ Tor_{1}^{A_{0}}(M, A_{3})/Tor_{1}^{A_{0}}(A_{1}⊕A_{2}, M). Obviously I annihilates Fil_{0}M (since it annihilates already Tor_{1}^{A_{0}}(M, A_{3})), hence the latter is an A_{2}-module. By lemma 3.4.10, we have M′ := M/Fil_{0}M ≃ M_{1} ×_{M_{3}} M_{2}. We can then filter the latter module by defining Fil_{0}M′ := 0, Fil_{1}M′ := Ker(M′ → M_{2}) ≃ Ker(M_{1} → M_{3}), which is a A_{1}-module, and Fil_{2}M′ := M′. Since gr_{2}M′ ≃ M_{2}, the assertion follows.

(iii): in view of lemma 3.4.15, it suffices to show the following: Claim 3.4.19. M is flat over A_{0} if and only if M_{i} is A_{i}-flat and Tor_{1}^{A_{0}}(M, A_{i}) = 0 for i ≤ 2. Proof of the claim: It suffices to prove the non-obvious implication, and in view of (ii) we are reduced to showing that Tor_{1}^{A_{0}}(M, L) = 0 whenever L is an A_{i}-module, for i = 1, 2. However, for any A_{i}-module L we have a base change spectral sequence E_{pq}^{2} := Tor_{p}^{A_{i}}(Tor_{q}^{A_{0}}(M, A_{i}), L) ⇒ Tor_{p+q}^{A_{0}}(M, L). If Tor_{1}^{A_{0}}(M, A_{i}) = 0, this yields Tor_{1}^{A_{0}}(M, L) ≃ Tor_{1}^{A_{i}}(M_{i}, L), which vanishes when M_{i} is A_{i}-flat. □

3.4.20. For any V^{a}-algebra A, let A-Mod_{fl} (resp. A-Mod_{proj}, resp. A-Mod_{afpr}) denote the full subcategory of A-Mod consisting of all flat (resp. almost projective, resp. almost finitely generated projective) A-modules. For any integer n ≥ 1, let A_{0}-Mod_{n} be the full subcategory of all A_{0}-modules satisfying condition (a) of lemma 3.4.15; let also A_{i}-Mod_{n} (for i = 1, 2) be the full subcategory of A_{i}-Mod consisting of all A_{i}-modules M such that Tor_{j}^{A_{i}}(M, A_{3}) = 0 for every j ≤ n. Finally, Let A-Alg_{fl} be the full subcategory of A-Alg consisting of all flat A-algebras. Proposition 3.4.21. In the situation of (3.4.7), the natural essentially commutative diagram:

\xymatrix{A_{0}-Mod_{?} \ar[r] \ar[d] & A_{2}-Mod_{?} \ar[d] \\ A_{1}-Mod_{?} \ar[r] & A_{3}-Mod_{?}}

is 2-cartesian (i.e. cartesian in the 2-category of categories, cp. [43, Ch.I]) whenever ? = “fl” or ? = “proj” or ? = “afpr”, or ? = n, for any integer n ≥ 1.

Proof. The assertion for flat almost modules follows directly from lemmata 3.4.9, 3.4.10, 3.4.14 and 3.4.18(iii). Similarly the assertion for the categories A_{i}-Mod_{n} follows from the same lemmata and from lemma 3.4.15. Set B := A_{1} × A_{2}. To establish the assertion for projective modules, it suffices to show that, if P is an A_{0}-module such that B ⊗_{A_{0}} P is almost projective over B, then P is almost projective over A_{0}, or which is the same, that alExt_{A_{0}}^{i}(P, N) ≃ 0 for all i > 0 and any A_{0}-module N. We know already that P is flat. Let M be any A_{0}-module and

N any B-module. The standard isomorphism RHom_{B}(B⊗^{L}_{A_{0}}M, N) ≃ RHom_{A_{0}}(M, N) yields a natural isomorphism alExt_{B}^{i}(B ⊗_{A_{0}} M, N) ≃ alExt_{A_{0}}^{i}(M, N), whenever Tor_{j}^{A_{0}}(B, M) = 0 for every j > 0. In particular, we have alExt_{A_{0}}^{i}(P, N) ≃ 0 whenever N comes from either an A_{1}-module, or an A_{2}-module. In view of lemma 3.4.18(ii), we deduce that the sought vanishing holds in fact for every A_{0}-module N. Finally, suppose that P ⊗_{A} B is almost finitely generated projective over B; we have to show that P is almost finitely generated. To this aim, notice that Ann_{A_{0}}(P ⊗_{A} B)^{2} ⊂ Ann_{A_{0}}(P), in view of lemma 3.4.18(i); then the claim follows from remark 3.2.26(i). □

Corollary 3.4.22. In the situation of (3.4.7), the natural essentially commutative diagram:

\xymatrix{A_{0}-𝒞 \ar[r] \ar[d] & A_{2}-𝒞 \ar[d] \\ A_{1}-𝒞 \ar[r] & A_{3}-𝒞}

is 2-cartesian whenever 𝒞 is one of the categories Alg_{fl}, Ét, w.Ét, Ét_{afp} (notation of (3.2.27)). □

3.4.23. Next we want to reinterpret the equivalences of proposition 3.4.21 in terms of descent data. If F : 𝒞 → V^{a}-Alg^{o} is a fibred category over the opposite of the category of almost algebras, and if X → Y is a given morphism of almost algebras, we shall denote by Desc(𝒞 , Y/X) the category of objects of the fibre category F_{Y}, endowed with a descent datum relative to the morphism X → Y (cp. [39, Ch.II §1]). In the arguments hereafter, we consider morphisms of almost algebras and modules, and one has to reverse the direction of the arrows to pass to morphisms in the relevant fibred category. Denote by p_{i} : Y → Y ⊗_{X} Y (i = 1, 2), resp. p_{ij} : Y ⊗_{X} Y → Y ⊗_{X} Y ⊗_{X} Y (1 ≤ i < j ≤ 3) the usual morphisms.

3.4.24. As an example, Desc(V^{a}-Alg.Mod^{o}, Y/X) consists of the pairs (M, β) where M is a Y -module and β is a Y ⊗_{X} Y -linear isomorphism β : p_{2}^{∗}(M) →^{∼} p_{1}^{∗}(M) such that

(3.4.25)  p_{12}^{∗}(β) ∘ p_{23}^{∗}(β) = p_{13}^{∗}(β).

3.4.26. Let now I ⊂ X be an ideal, and set \overline{X} := X/I, \overline{Y} := Y/IY. For any F : 𝒞 → V^{a}-Alg^{o} as in (3.4.23), one has an essentially commutative diagram:

(3.4.27)  \xymatrix{Desc(𝒞 , Y/X) \ar[r] \ar[d] & Desc(𝒞 , \overline{Y} /\overline{X}) \ar[d] \\ F_{Y} \ar[r] & F\overline{Y} .}

This induces a functor :

(3.4.28)  Desc(𝒞 , Y/X) → Desc(𝒞 , \overline{Y} /\overline{X}) ×_{F\overline{Y}} F_{Y} .

Lemma 3.4.29. In the situation of (3.4.26), suppose moreover that the natural morphism I → IY is an isomorphism. Then diagram (3.4.27) is 2-cartesian whenever 𝒞 is one of the fibred categories V^{a}-Alg.Mod^{o}, V^{a}-Alg.Morph^{o}, Ét^{o}, w.Ét^{o}.

Proof. For any n > 0, denote by Y^{⊗n} (resp. \overline{Y}^{⊗n}) the n-fold tensor product of Y (resp. \overline{Y}) with itself over X (resp. \overline{X}), and by ρ_{n} : Y^{⊗n} → \overline{Y}^{⊗n} the natural morphism. First of all we claim that, for every n > 0, the natural diagram of almost algebras

(3.4.30)  \xymatrix{Y^{⊗n} \ar[r]^{ρ_{n}} \ar[d]_{μ_{n}} & \overline{Y}_{⊗n} \ar[d]^{_{\overline{μ}_{n}}} \\ Y \ar[r]^{ρ_{1}} & \overline{Y}}

is cartesian (where μ_{n} and \overline{μ}_{n} are n-fold multiplication morphisms). For this, we need to verify that, for every n > 0, the induced morphism Ker ρ_{n} → Ker ρ_{1} (defined by multiplication of the first two factors) is an isomorphism. It then suffices to check that the natural morphism Ker ρ_{n} → Ker ρ_{n−1} is an isomorphism for all n > 1. Indeed, consider the commutative diagram

[figure: I ⊗_{X} Y^{⊗n−1}; p; IY^{⊗n−1}; i; ψ; Y^{⊗n−1}; ϕ⊗1_{Y^{⊗n−1}}; 1_{Y^{⊗n−1}}; I ⊗_{X} Y^{⊗n−1}; p′; Ker ρ_{n}; i′; Y^{⊗n}; μ_{Y/X}⊗1_{Y^{⊗n−2}}; Y^{⊗n−1}.]

From IY = ϕ(Y), it follows that p′ is an epimorphism. Hence also ψ is an epimorphism. Since i is a monomorphism, it follows that ψ is also a monomorphism, hence ψ is an isomorphism and the claim follows easily.

We consider first the case 𝒞 := V^{a}-Alg.Mod^{o}; we see that (3.4.30)is a diagram of the kind considered in (3.4.8), hence, for every n > 0, we have the associated functor π_{n} : Y^{⊗n}-Mod → \overline{Y}^{⊗n}-Mod ×_{\overline{Y} -Mod} Y -Mod and also its right adjoint T_{n}. Denote by \overline{p}_{i} : \overline{Y} → \overline{Y}^{⊗2} (i = 1, 2) the usual morphisms, and similarly define \overline{p}_{ij} : \overline{Y}^{⊗2} → \overline{Y}^{⊗3}. Suppose there is given a descent datum (\overline{M}, \overline{β}) for \overline{M}, relative to \overline{X} → \overline{Y}. The cocycle condition (3.4.25) implies easily that \overline{μ}_{2}^{∗}(\overline{β}) is the identity on \overline{μ}_{2}^{∗}(\overline{p}_{i}^{∗}\overline{M}) = \overline{M}. It follows that the pair (\overline{β}, 1_{M}) defines an isomorphism π_{2}(p_{1}^{∗}M) →^{∼} π_{2}(p_{2}^{∗}M) in the category \overline{Y}^{⊗2}-Mod ×_{\overline{Y} -Mod} Y -Mod. Hence T_{2}(\overline{β}, 1_{M}) : T_{2} ∘ π_{2}(p_{1}^{∗}M) → T_{2} ∘ π_{2}(p_{2}^{∗}M) is an isomorphism. However, we remark that either morphism \overline{p}_{i} yields a section for μ_{2}, hence we are in the situation contemplated in lemma 3.4.13, and we derive an isomorphism β : p_{2}^{∗}(M) →^{∼} p_{1}^{∗}(M). We claim that (M, β) is an object of Desc(𝒞 , Y/X), i.e. that β verifies the cocycle condition (3.4.25). Indeed, we can compute: π_{3}(p_{ij}^{∗}β) = (ρ_{3}^{∗}(p_{ij}^{∗}β), μ_{3}^{∗}(p_{ij}^{∗}β)) and by construction we have ρ_{3}^{∗}(p_{ij}^{∗}β) = \overline{p}_{ij}^{∗}(\overline{β}) and μ_{3}^{∗}(p_{ij}^{∗}β) = μ_{2}^{∗}(β) = 1_{M}. Therefore, the cocycle identity for \overline{β} implies the equality π_{3}(p_{12}^{∗}β) ∘ π_{3}(p_{23}^{∗}β) = π_{3}(p_{13}^{∗}β). If we now apply the functor T_{3} to this equality, and then invoke again lemma 3.4.13, the required cocycle identity for β will ensue. Clearly β is the only descent datum on M lifting \overline{β}. This proves that (3.4.28)is essentially surjective. The same sort of argument also shows that the functor (3.4.28) induces bijections on morphisms, so the lemma follows in this case. Next, the case 𝒞 := V^{a}-Alg.Morph^{o} can be deduced formally from the previous case, by applying repeatedly natural isomorphisms of the kind p_{i}^{∗}(M ⊗_{Y} N) ≃ p_{i}^{∗}(M) ⊗_{Y ⊗_{X}Y} p_{i}^{∗}(N) (i = 1, 2). Finally, the “étaleness” of an object of Desc(V^{a}-Alg.Morph^{o}, Y/X) can be checked on its projection onto Y -Alg^{o}, hence also the cases 𝒞 := w.Ét^{o} and 𝒞 := Ét^{o} follow directly. □

3.4.31. Now, let B := A_{1}×A_{2}; to an object (M, β) in Desc(V^{a}-Alg.Mod^{o}, B/A) we assign a 𝒟-module (M_{1}, M_{2}, ξ) (notation of (3.4.7)) as follows. Set M_{i} := A_{i} ⊗_{B} M (i = 1, 2) and A_{ij} := A_{i} ⊗_{A_{0}} A_{j}. We can write B ⊗_{A_{0}} B = ∏_{i,j=1}^{2} A_{ij} and β gives rise to the A_{ij}-linear isomorphisms β_{ij} : A_{ij} ⊗_{B⊗_{A_{0}}B} p_{2}^{∗}(M) →^{∼} A_{ij} ⊗_{B⊗_{A_{0}}B} p_{1}^{∗}(M). In other words, we obtain isomorphisms β_{ij} : A_{i} ⊗_{A_{0}} M_{j} → M_{i} ⊗_{A_{0}} A_{j}. However, we have a natural isomorphism A_{12} ≃ A_{3} (indeed, suppose that A_{1} → A_{3} is an epimorphism with kernel I; then I is also an ideal of A_{0} and A_{0}/I ≃ A_{2}; now the claim follows by remarking that IA_{1} = I). Hence we can choose ξ = β_{12}. In this way we obtain a functor :

(3.4.32)  Desc(V^{a}-Alg.Mod^{o}, B/A_{0}) → 𝒟-Mod^{o}.

Proposition 3.4.33. The functor (3.4.32)is an equivalence of categories.

Proof. Let us say that A_{1} → A_{3} is an epimorphism with kernel I. Then I is also an ideal of B and we have B/I ≃ A_{3} × A_{2} and A_{0}/I ≃ A_{2}. We intend to apply lemma 3.4.29 to the morphism A_{0} → B. However, the induced morphism \overline{B} := B/I → \overline{A}_{0} := A_{0}/I in V^{a}-Alg^{o} has a section, and hence it is of universal effective descent for every fibred category. Thus, we can replace in (3.4.28)the category Desc(V^{a}-Alg.Mod^{o}, \overline{B}/\overline{A}_{0}) by \overline{A}_{0}-Mod^{o}, and thereby, identify (up to equivalence) the target of (3.4.28)with the 2-fibred product (A_{1}-Mod× A_{2}-Mod)^{o}×_{(A_{3}-Mod×A_{2}-Mod)^{o}} A_{2}-Mod^{o}. The latter is equivalent to the category 𝒟-Mod^{o} and the resulting functor Desc(V^{a}-Alg.Mod^{o}, B/A_{0}) → 𝒟-Mod^{o} is canonically isomorphic to (3.4.32), whence the claim. □

Putting together propositions 3.4.21 and 3.4.33 we obtain the following : Corollary 3.4.34. In the situation of (3.4.8), the morphism A_{0} → A_{1}×A_{2} is of effective descent for the fibred categories of flat modules and of almost projective modules. □

3.4.35. Next we would like to give sufficient conditions to ensure that a morphism of almost algebras is of effective descent for the fibred category w.Ét^{o} → V^{a}-Alg^{o} of weakly étale algebras (resp. for étale algebras). To this aim we are led to the following : Definition 3.4.36. A morphism ϕ : A → B of almost algebras is said to be strictly finite if Ker ϕ is nilpotent and B ≃ R^{a}, where R is a finite A_{∗}-algebra. Theorem 3.4.37. Let ϕ : A → B be a strictly finite morphism of almost algebras. Then :

(i) For every A-algebra C, the induced morphism C → C ⊗_{A} B is again strictly finite. (ii) If M is a flat A-module and B ⊗_{A} M is almost projective over B, then M is almost

projective over A. (iii) A → B is of universal effective descent for the fibred categories of weakly étale (resp.

étale) almost algebras.

Proof. (i): suppose that B = R^{a} for a finite A_{∗}-algebra R; then S := C_{∗} ⊗_{A_{∗}} R is a finite C_{∗}- algebra and S^{a} ≃ C. It remains to show that Ker(C → C ⊗_{A} B) is nilpotent. Suppose that R is generated by n elements as an A_{∗}-module and let F_{A_{∗}}(R) (resp. F_{C_{∗}}(S)) be the Fitting ideal of R (resp.of S); we have Ann_{C_{∗}}(S)^{n} ⊂ F_{C_{∗}}(S) ⊂ Ann_{C_{∗}}(S) (see [50, Ch.XIX Prop.2.5]); on the other hand F_{C_{∗}}(S) = F_{A_{∗}}(R) · C_{∗}, so the claim is clear.

(iii): we shall consider the fibred category F : w.Ét^{o} → V^{a}-Alg^{o}; the same argument applies also to étale almost algebras. We begin by establishing a very special case : Claim 3.4.38. Assertion (iii) holds when B = (A/I_{1}) × (A/I_{2}), where I_{1} and I_{2} are ideals in A such that I_{1} ∩ I_{2} is nilpotent. Proof of the claim: First of all we remark that the situation considered in the claim is stable under arbitrary base change, therefore it suffices to show that ϕ is of F -2-descent in this case. Then we factor ϕ as a composition A → A/Ker ϕ → B and we remark that A → A/Ker ϕ is of F -2-descent by theorem 3.2.18; since a composition of morphisms of F -2-descent is again of F -2-descent, we are reduced to show that A/Ker ϕ → B is of F -2-descent, i.e. we can assume that Ker ϕ ≃ 0. However, in this case the claim follows easily from corollary 3.4.34.

Claim 3.4.39. More generally, assertion (iii) holds when B = ∏_{i=1}^{n} A/I_{i}, where I_{1}, ..., I_{n} are ideals of A, such that ⋂_{i=1}^{n} I_{i} is nilpotent. Proof of the claim: We prove this by induction on n, the case n = 2 being covered by claim 3.4.38. Therefore, suppose that n > 2, and set B′ := A/(⋂_{i=1}^{n−1} I_{j}). By induction, the morphism B′ → ∏_{i=1}^{n−1} A/I_{i} is of universal F -2-descent. However, according to [39, Ch.II Prop.1.1.3], the sieves of universal F -2-descent form a topology on V^{a}-Alg^{o}; for this topology, {A, B} is a covering family of A × B and (A → B′ × (A/I_{n}))^{o} is a covering morphism, hence {B′, A/I_{n}} is a covering family of A, and then, by composition of covering families, {∏_{i=1}^{n−1} A/I_{i}, A/I_{n}} is a covering family of A, which is equivalent to the claim.

Now, let A → B be a general strictly finite morphism, so that B = R^{a} for some finite A_{∗}-algebra R. Pick generators f_{1}, ..., f_{m} of the A_{∗}-module R, and monic polynomials p_{1}(X), ..., p_{m}(X) such that p_{i}(f_{i}) = 0 for i = 1, ..., m. Claim 3.4.40. There exists a finite and faithfully flat extension C of A_{∗} such that the images in C[X] of p_{1}(X),...,p_{m}(X) split as products of monic linear factors. Proof of the claim: This extension C can be obtained as follows. It suffices to find, for each i = 1, ..., m, an extension C_{i} that splits p_{i}(X), because then C := C_{1} ⊗_{A_{∗}} ... ⊗_{A_{∗}} C_{m} will split them all, so we can assume that m = 1 and p_{1}(X) = p(X); moreover, by induction on the degree of p(X), it suffices to find an extension C′ such that p(X) factors in C′[X] as a product of the form p(X) = (X − α) · q(X), where q(X) is a monic polynomial of degree deg(p) − 1. Clearly we can take C′ := A_{∗}[T]/(p(T)).

Given a C as in claim 3.4.40, we remark that the morphism A → C^{a} is of universal F - 2-descent. Considering again the topology of universal F -2-descent, it follows that A → B is of universal F -2-descent if and only if the same holds for the induced morphism C^{a} → C^{a} ⊗_{A} B. Therefore, in proving assertion (iii) we can replace ϕ by 1_{C} ⊗_{A} ϕ and assume from start that the polynomials p_{i}(X) factor in A_{∗}[X] as product of linear factors. Now, let d_{i} := deg(p_{i}) and p_{i}(X) := ∏_{j}^{d_{i}}(X−α_{ij}) (for i = 1, ..., m). We get a surjective homomorphism of A_{∗}-algebras D := A_{∗}[X_{1}, ..., X_{m}]/(p_{1}(X_{1}), ..., p_{m}(X_{m})) → R by the rule X_{i} ↦ f_{i} (i = 1, ..., m). Moreover, any sequence \underline{α} := (α_{1,j_{1}}, α_{2,j_{2}}, ..., α_{m,j_{m}}) yields a homomorphism ψ\underline{α} : D → A_{∗}, determined by the assignment X_{i} ↦ α_{i,j_{i}}. A simple combinatorial argument shows that ∏\underline{α} Ker ψ\underline{α} = 0, where \underline{α} runs over all the sequences as above. Hence the product map ∏\underline{α} ψ\underline{α} : D → ∏\underline{α} A_{∗} has nilpotent kernel. We notice that the A_{∗}-algebra (∏\underline{α} A_{∗}) ⊗_{D} R is a quotient of ∏\underline{α} A_{∗}, hence it can be written as a product of rings of the form A_{∗}/I\underline{α}, for various ideals I\underline{α}. By (i), the kernel of the induced homomorphism R → ∏\underline{α} A_{∗}/I\underline{α} is nilpotent, hence the same holds for the kernel of the composition A → ∏\underline{α} A/Ia ; \underline{α}, which is therefore of the kind considered in claim 3.4.39. Hence A → ∏\underline{α} A/Ia ; \underline{α} is of universal F -2-descent. Since such morphisms form a topology, it follows that also A → B is of universal F -2-descent, which concludes the proof of (iii).

Finally, let M be as in (ii) and pick again C as in the proof of claim 3.4.40. By remark 3.2.26(iv), M is almost projective over A if and only if C^{a} ⊗_{A} M is almost projective over C^{a}; hence we can replace ϕ by 1_{C^{a}} ⊗_{A} ϕ, and by arguing as in the proof of (iii), we can assume from start that B = ∏_{j=1}^{n} A/I_{j} for ideals I_{j} ⊂ A, j = 1, ..., n such that I := ⋂_{j=1}^{n} I_{j} is nilpotent. By an easy induction, we can furthermore reduce to the case n = 2. We factor ϕ as A → A/I → B; by proposition 3.4.21 it follows that (A/I) ⊗_{A} M is almost projective over A/I, and then lemma 3.2.25(i) says that M itself is almost projective. □

Remark 3.4.41. It is natural to ask whether theorem 3.4.37 holds if we replace everywhere “strictly finite” by “finite with nilpotent kernel” (or even by “almost finite with nilpotent kernel”). We do not know the answer to this question.

3.4.42. On the category V^{a}-Alg (taken in some universe) consider the topologies τ_{e} (resp. τ_{w}) of universal effective descent for the fibred category Ét^{o} (resp. w.Ét^{o}). For a ring R denote by Idemp(R) the set of idempotents of R. Proposition 3.4.43. With the notation of (3.4.42)we have:

(i) The presheaf A ↦ Idemp(A_{∗}) is a sheaf for both τ_{e} and τ_{w}.

(ii) If f : A → B is an étale (resp. weakly étale) morphism of almost V -algebras and there is a covering family {(A → A_{α})^{o}} for τ_{e} (resp. τ_{w}) such that A_{α} → A_{α} ⊗_{A} B is an almost projective epimorphism for all α, then f is an almost projective epimorphism.

(iii) τ_{e} is finer than τ_{w}.

Proof. (i): use descent of morphisms and the bijection

Hom_{A-Alg}(A × A, A) →^{∼} Idemp(A_{∗}) ϕ ↦ ϕ_{∗}(1, 0).

(ii): by remark 3.1.8, Ker(A_{α} → A_{α} ⊗_{A} B) is generated by e_{α} ∈ Idemp(A_{α∗}). e_{α} and e_{β} agree in (A_{α} ⊗_{A} A_{β})_{∗}, so by (i) there is an idempotent e ∈ A_{∗} that restricts to e_{α} in Idemp(A_{α∗}), for each α. The A-algebras B and A/eA become isomorphic after applying − ⊗_{A} A_{α}; these isomorphisms are unique and are compatible on A_{α} ⊗_{A} A_{β}, hence they patch to an isomorphism B ≃ A/eA.

(iii): we have to show that if A is an almost V -algebra, R a sieve of (V^{a}-Alg)^{o}/A and R is of universal w.Ét^{o}-2-descent, then R is of universal Ét^{o}-2-descent. Since the assumption is stable under base change, it suffices to show that R is of Ét^{o}-2-descent. Descent of morphisms is clear. Let R be the sieve generated by a family of morphisms {(A → A_{α})^{o}}. Any descent datum consisting of étale A_{α}-algebras B_{α} and isomorphisms A_{α} ⊗_{A} B_{β} ≃ B_{α} ⊗_{A} A_{β} satisfying the cocycle condition, becomes effective when we pass to w.Ét^{o}. So one has to verify that if B is a weakly étale A-algebra such that B ⊗_{A} A_{α} is étale over A_{α} for all α, then B is étale over A. Indeed, an application of (ii) gives that B ⊗_{A} B → B is almost projective. □

3.4.44. We conclude with a digression to explain the relationship between our results and known facts that can be extracted from the literature. So, we now place ourselves in the “classical limit” 𝔪 := V (cp. example 2.1.2(ii)). In this case, weakly étale morphisms had already been considered in some earlier work, and they were called “absolutely flat” morphisms. A ring homomorphism A → B is étale in the usual sense of [40] if and only if it is absolutely flat and of finite presentation. Let us denote by u.Ét^{o} the fibred category over V -Alg^{o}, whose fibre over a V -algebra A is the opposite of the category of étale A-algebras in the usual sense. We claim that, if a morphism A → B of V -algebras is of universal effective descent for the fibred category w.Ét^{o} (resp. Ét^{o}), then it is a morphism of universal effective descent for u.Ét^{o}. Indeed, let C be an étale A-algebra (in the sense of definition 3.1.1) and such that C ⊗_{A} B is étale over B in the usual sense. We have to show that C is étale in the usual sense, i.e. that it is of finite presentation over A. This amounts to showing that, for every filtered inductive system (A_{λ})_{λ∈Λ} of A-algebras, we have colim_{λ∈Λ} Hom_{A-Alg}(C, A_{λ}) ≃ Hom_{A-Alg}(C, colim_{λ∈Λ} A_{λ}). Since, by assumption, this is known after extending scalars to B and to B ⊗_{A} B, it suffices to show that, for any A-algebra D, the natural sequence

\xymatrix{Hom_{A-Alg}(C, D) \ar[r] & Hom_{B-Alg}(C_{B}, D_{B}) \ar[r] \ar[r] & Hom_{B⊗_{A}B-Alg}(C_{B⊗_{A}B}, D_{B⊗_{A}B})}

is exact. For this, note that Hom_{A-Alg}(C, D) = Hom_{D-Alg}(C_{D}, D) (and similarly for the other terms) and by hypothesis (D → D ⊗_{A} B)^{o} is a morphism of 1-descent for the fibred category w.Ét^{o} (resp. Ét^{o}).

As a consequence of these observations and of theorem 3.4.37, we see that any finite ring homomorphism ϕ : A → B with nilpotent kernel is of universal effective descent for the fibred category of étale algebras. This fact was known as follows. By [40, Exp.IX, 4.7], Spec(ϕ) is of universal effective descent for the fibred category of separated étale morphisms of finite type. One has to show that if X is such a scheme over A, such that X ⊗_{A} B is affine, then X is affine. This follows by reduction to the noetherian case and [27, Ch.II, 6.7.1].

3.5. Behaviour of étale morphisms under Frobenius. We consider the following category ℬ of basic setups. The objects of ℬ are the pairs (V, 𝔪), where V is a ring and 𝔪 is an ideal of V with 𝔪 = 𝔪^{2} and 𝔪̃ is flat. The morphisms (V, 𝔪_{V}) → (W, 𝔪_{W}) between two objects of ℬ are the ring homomorphisms f : V → W such that 𝔪_{W} = f(𝔪_{V}) · W.

3.5.1. We have a fibred and cofibred category ℬ-Mod → ℬ (see [40, Exp.VI §5,6,10] for generalities on fibred categories). An object of ℬ-Mod (which we may call a “ℬ-module”) consists of a pair ((V, 𝔪), M), where (V, 𝔪) is an object of ℬ and M is a V -module. Given two objects X := ((V, 𝔪_{V}), M) and Y := ((W, 𝔪_{W}), N), the morphisms X → Y are the pairs (f, g), where f : (V, 𝔪_{V}) → (W, 𝔪_{W}) is a morphism in ℬ and g : M → N is an f-linear map.

3.5.2. Similarly one has a fibred and cofibred category ℬ-Alg → ℬ of ℬ-algebras. We will also need to consider the fibred and cofibred category ℬ-Mon → ℬ of non-unitary commutative ℬ-monoids: an object of ℬ-Mon is a pair ((V, 𝔪), A) where A is a V -module endowed with a morphism A ⊗_{V} A → A subject to associativity and commutativity conditions, as discussed in section 2.2. The fibre over an object (V, 𝔪) of ℬ, is the category of V -monoids denoted (V, 𝔪)-Mon or simply V -Mon.

3.5.3. The almost isomorphisms in the fibres of ℬ-Mod → ℬ give a multiplicative system Σ in ℬ-Mod, admitting a calculus of both left and right fractions. The “locally small” conditions are satisfied (see [67, p.381]), so that one can form the localised category ℬ^{a}-Mod := Σ^{−1}(ℬ-Mod). The fibres of the localised category over the objects of ℬ are the previously considered categories of almost modules. Similar considerations hold for ℬ-Alg and ℬ-Mon, and we get the fibred and cofibred categories ℬ^{a}-Mod → ℬ, ℬ^{a}-Alg → ℬ and ℬ^{a}-Mon → ℬ. In particular, for every object (V, 𝔪) of ℬ, we have an obvious notion of almost V -monoid and the category consisting of these is denoted V^{a}-Mon.

3.5.4. The localisation functors

ℬ-Mod → ℬ^{a}-Mod : M ↦ M^{a} ℬ-Alg → ℬ^{a}-Alg : B ↦ B^{a}

have left and right adjoints. These adjoints can be chosen as functors of categories over ℬ such that the adjunction units and counits are morphisms over identity arrows in ℬ. On the fibres these induce the previously considered left and right adjoints M ↦ M_{!}, M ↦ M_{∗}, B ↦ B_{!!}, B ↦ B_{∗}. We will use the same notation for the corresponding functors on the larger categories. Then it is easy to check that the functor M ↦ M_{!} is cartesian and cocartesian (i.e. it sends cartesian arrows to cartesian arrows and cocartesian arrows to cocartesian arrows), M ↦ M_{∗} and B ↦ B_{∗} are cartesian, and B ↦ B_{!!} is cocartesian.

3.5.5. Let ℬ/𝔽_{p} be the full subcategory of ℬ consisting of all objects (V, 𝔪) where V is an 𝔽_{p}-algebra. Define similarly ℬ-Alg/𝔽_{p}, ℬ-Mon/𝔽_{p} and ℬ^{a}-Alg/𝔽_{p}, ℬ^{a}-Mon/𝔽_{p}, so that we have again fibred and cofibred categories ℬ^{a}-Alg/𝔽_{p} → ℬ/𝔽_{p} and ℬ^{a}-Alg/𝔽_{p} → ℬ/𝔽_{p} (resp. the same for non-unitary monoids). We remark that the categories ℬ^{a}-Alg/𝔽_{p} and ℬ^{a}-Mon/𝔽_{p} have small limits and colimits, and these are preserved by the projection to ℬ/𝔽_{p}. Especially, if A → B and A → C are two morphisms in ℬ^{a}-Alg/𝔽_{p} or ℬ^{a}-Mon/𝔽_{p}, we can define B ⊗_{A} C as such a colimit.

3.5.6. If A is a (unitary or non-unitary) ℬ-monoid over 𝔽_{p}, we denote by Φ_{A} : A → A the Frobenius endomorphism: x ↦ x^{p}. If (V, 𝔪) is an object of ℬ/𝔽_{p}, it follows from proposition 2.1.7(ii) that Φ_{V} : (V, 𝔪) → (V, 𝔪) is a morphism in ℬ. If B is an object of ℬ-Alg/𝔽_{p} (resp. ℬ-Mon/𝔽_{p}) over V, then the Frobenius map induces a morphism Φ_{B} : B → B in ℬ-Alg/𝔽_{p} (resp. ℬ-Mon/𝔽_{p}) over Φ_{V}. In this way we get a natural transformation from the identity functor of ℬ-Alg/𝔽_{p} (resp. ℬ-Mon/𝔽_{p}) to itself that induces a natural transformation on the identity functor of ℬ^{a}-Alg/𝔽_{p} (resp. ℬ^{a}-Mon/F_{p}).

3.5.7. Using the pull-back functors, any object B of ℬ-Alg over V defines new objects B_{(m)} of ℬ-Alg (m ∈ ℕ) over V, where B_{(m)} := (Φ_{V}^{m})^{∗}(B), which is just B considered as a V - algebra via the homomorphism V ⟶^{Φ^{m}} V → B. These operations also induce functors B ↦ B_{(m)} on almost ℬ-algebras. Definition 3.5.8. Let (V, 𝔪) be an object of ℬ/𝔽_{p}.

(i) We say that a morphism f : A → B of almost V -algebras (resp. almost V -monoids) is invertible up to Φ^{m} if there exists a morphism f′ : B → A in ℬ^{a}-Alg (resp. ℬ^{a}-Mon) over Φ_{V}^{m}, such that f′ ∘ f = Φ_{A}^{m} and f ∘ f′ = Φ_{B}^{m}.

(ii) We say that an almost V -monoid I (e.g. an ideal in a V^{a}-algebra) is Frobenius nilpotent if Φ_{I} is nilpotent.

3.5.9. Notice that a morphism f of V^{a}-Alg (or V^{a}-Mon) is invertible up to Φ^{m} if and only if f_{∗} : A_{∗} → B_{∗} is so as a morphism of 𝔽_{p}-algebras. Lemma 3.5.10. Let (V, 𝔪) be an object of ℬ/𝔽_{p} and let f : A → B, g : B → C be morphisms of almost V -algebras or almost V -monoids.

(i) If f (resp. g) is invertible up to Φ^{n} (resp. Φ^{m}), then g ∘ f is invertible up to Φ^{m+n}. (ii) If f (resp. g ∘ f) is invertible up to Φ^{n} (resp. Φ^{m}), then g is invertible up to Φ^{m+n}. (iii) If g (resp.g ∘ f) is invertible up to Φ^{n} (resp. Φ^{m}), then f is invertible up to Φ^{m+n}. (iv) The Frobenius morphisms induce Φ_{V} -linear morphisms (i.e. morphisms in ℬ^{a}-Mod

over Φ_{V}) Φ′ : Ker f → Ker f and Φ′′ : Coker f → Coker f, and f is invertible up to some power of Φ if and only if both Φ′ and Φ′′ are nilpotent.

(v) Consider a map of short exact sequences of almost V -monoids :

\xymatrix{0 \ar[r] & A′ \ar[r] \ar[d]_{f′} & A \ar[r] \ar[d]_{f} & A′′ \ar[r] \ar[d]_{f′′} & 0 \\ 0 \ar[r] & B′ \ar[r] & B \ar[r] & B′′ \ar[r] & 0}

and suppose that two of the morphisms f′, f, f′′ are invertible up to a power of Φ. Then also the third morphism has this property.

Proof. (i): if f′ is an inverse of f up to Φ^{n} and g′ is an inverse of g up to Φ^{m}, then f′ ∘ g′ is an inverse of g ∘ f up to Φ^{m+n}. (ii): given an inverse f′ of f up to Φ^{n} and an inverse h′ of h := g ∘ f up to Φ^{m}, let g′ := Φ_{B}^{n} ∘ f ∘ h′. We compute :

g ∘ g′ = g ∘ Φ_{B}^{n} ∘ f ∘ h′ = Φ_{C}^{n} ∘ g ∘ f ∘ h = Φ_{C}^{n} ∘ Φ_{C}^{m}
g′ ∘ g = Φ_{B}^{n} ∘ f ∘ h′ ∘ g = f ∘ h′ ∘ g ∘ Φ_{B}^{n} = f ∘ h′ ∘ g ∘ f ∘ f′
= f ∘ Φ_{A}^{m} ∘ f′ = Φ_{B}^{m} ∘ f ∘ f′ = Φ_{B}^{m} ∘ Φ_{B}^{n}.

(iii) is similar and (iv) is an easy diagram chasing left to the reader. (v) follows from (iv) and the snake lemma. □

Lemma 3.5.11. Let (V, 𝔪) be an object of ℬ/𝔽_{p}.

(i) If f : A → B is a morphism of almost V -algebras, invertible up to Φ^{n}, then so is A′ → A′ ⊗_{A} B for every morphism A → A′ of almost algebras. (ii) If f : (V, 𝔪_{V}) → (W, 𝔪_{W}) is a morphism in ℬ/𝔽_{p}, the functors

f_{∗} : (V, 𝔪_{V})^{a}-Alg → (W, 𝔪_{W})^{a}-Alg and f^{∗} : (W, 𝔪_{W})^{a}-Alg → (V, 𝔪_{V})^{a}-Alg

preserve the class of morphisms invertible up to Φ^{n}.

Proof. (i): given f′ : B → A_{(m)}, construct a morphism A′ ⊗_{A} B → A′_{(m)} using the morphism A′ → A′_{(m)} coming from Φ_{A′}^{m} and f′. (ii): the assertion for f^{∗} is clear, and the assertion for f_{∗} follows from (i). □

Remark 3.5.12. Statements like those of lemma 3.5.11 hold for the classes of flat, (weakly) unramified, (weakly) étale morphisms. Theorem 3.5.13. Let (V, 𝔪) be an object of ℬ/𝔽_{p} and f : A → B a weakly étale morphism of almost V -algebras.

(i) If f is invertible up to Φ^{n} (n ≥ 0), then it is an isomorphism.

(ii) For every integer m ≥ 0 the natural square diagram

(3.5.14)  \xymatrix{A \ar[r]^{f} \ar[d]_{Φ_{A}^{m}} & B \ar[d]^{Φ_{B}^{m}} \\ A_{(m)} \ar[r]^{f_{(m)}} & B_{(m)}}

is cocartesian.

Proof. (i): we first show that f is faithfully flat. Since f is flat, it remains to show that if M is an A-module such that M ⊗_{A} B = 0, then M = 0. It suffice to do this for M := A/I, for an arbitrary ideal I of A. After base change by A → A/I, we reduce to show that B = 0 implies A = 0. However, A_{∗} → B_{∗} is invertible up to Φ^{n}, so Φ_{A_{∗}}^{n} = 0 which means A_{∗} = 0. In particular, f is a monomorphism, hence the proof is complete in case that f is an epimorphism. In general, consider the composition B ⟶^{1_{B}⊗f} B ⊗_{A} B ⟶^{μ_{B/A}} B. From lemma 3.5.11(i) it follows that 1_{B}⊗f is invertible up to Φ^{n}; then lemma 3.5.10(ii) says that μ_{B/A} is invertible up to Φ^{n}. The latter is also weakly étale; by the foregoing we derive that it is an isomorphism. Consequently 1_{B} ⊗ f is an isomorphism, and finally, by faithful flatness, f itself is an isomorphism.

(ii): the morphisms Φ_{A}^{m} and Φ_{B}^{m} are invertible up to Φ^{m}. By lemma 3.5.11(i) it follows that 1_{B} ⊗ Φ_{A}^{m} : B → B ⊗_{A} A_{(m)} is invertible up to Φ^{m}; hence, by lemma 3.5.10(ii), the morphism h : B ⊗_{A} A_{(m)} → B_{(m)} induced by Φ_{B}^{m} and f_{(m)} is invertible up to Φ^{2m} (in fact one verifies that it is invertible up to Φ^{m}). But h is a morphism of weakly étale A_{(m)}-algebras, so it is weakly étale, so it is an isomorphism by (i). □

Remark 3.5.15. Theorem 3.5.13(ii) extends a statement of Faltings ([34, p.10]) for his notion of almost étale extensions.

3.5.16. We recall (cp. [39, Ch.0, 3.5]) that a morphism f : X → Y of objects in a site is called bicovering if the induced map of associated sheaves of sets is an isomorphism; if f is squarable (“quarrable” in French), this is equivalent to the condition that both f and the diagonal morphism X → X ×_{Y} X are covering morphisms.

3.5.17. Let F → E be a fibered category and f : P → Q a squarable morphism of E. Consider the following condition:

(3.5.18) for every base change P ×_{Q}Q′ → Q′ of f, the inverse image functor F_{Q′} → F_{P ×_{Q}Q′} is an equivalence of categories.

Inspecting the arguments in [39, Ch.II, §1.1] one can show: Lemma 3.5.19. With the above notation, let τ be the topology of universal effective descent relative to F → E. Then we have :

(i) if (3.5.18)holds, then f is a covering morphism for the topology τ.

(ii) f is bicovering for τ if and only if (3.5.18) holds both for f and for the diagonal morphism P → P ×_{Q} P. □

Remark 3.5.20. In [39, Ch.II, 1.1.3(iv)] it is stated that “la réciproque est vraie si i = 2”, meaning that (3.5.18) is equivalent to the condition that f is bicovering for τ. (Actually the cited statement is given in terms of presheaves, but one can show that (3.5.18) is equivalent to the corresponding condition for the fibered category F^{+} → Ê_{U} considered in loc.cit.) However, this fails in general : as a counterexample we can give the following. Let E be the category of schemes of finite type over a field k; set P = 𝔸_{k}^{1}, Q = Spec k. Finally let F → E be the discretely fibered category defined by the presheaf X ↦ H^{0}(X, ℤ). Then it is easy to show that f satisfies (3.5.18) but the diagonal map does not, so f is not bicovering. The mistake in the proof is in [39, Ch.II, 1.1.3.5], where one knows that F^{+}(d) is an equivalence of categories (notation of loc.cit.) but one needs it also after base changes of d. Lemma 3.5.21. (i) Let f : A → B be a morphism of V^{a}-algebras.

(i) If f is invertible up to Φ^{m}, then the induced functors A-Ét → B-Ét and A-w.Ét → B-w.Ét are equivalences of categories.

(ii) If f is weakly étale and C → D is a morphism of A-algebras invertible up to Φ^{m}, then the induced map: Hom_{A-Alg}(B, C) → Hom_{A-Alg}(B, D) is bijective.

Proof. We first consider (i) for the special case where f := Φ_{A}^{m} : A → A_{(m)}. The functor (Φ_{V}^{m})^{∗} : V^{a}-Alg → V^{a}-Alg induces a functor (−)_{(m)} : A-Alg → A_{(m)}-Alg, and by restriction (see remark 3.5.11) we obtain a functor (−)_{(m)} : A-Ét → A_{(m)}-Ét; by theorem 3.5.13(ii), the latter is isomorphic to the functor (Φ^{m})_{∗} : A-Ét → A_{(m)}-Ét of the lemma. Furthermore, from remark 2.1.4(ii) and (2.2.4) we derive a natural ring isomorphism ω : A_{(m)∗} ≃ A_{∗}, hence an essentially commutative diagram

\xymatrix{A-Ét \ar[r] \ar[d]_{(Φ^{m})_{∗}} & A-Alg \ar[r]^{α} \ar[d]_{(−)_{(m)}} & (A_{∗}, 𝔪 · A_{∗})^{a}-Alg \ar[d]_{ω^{∗}} \\ A_{(m)}-Ét \ar[r] & A_{(m)}-Alg \ar[r]^{β} & (A_{(m)∗}, 𝔪 · A_{(m)∗})^{a}-Alg}

where α and β are the equivalences of remark 2.2.12. Clearly α and β restrict to equivalences on the corresponding categories of étale algebras, hence the lemma follows in this case.

For the general case of (i), let f′ : B → A_{(m)} be a morphism as in definition 3.5.8. Diagram (3.5.14) induces an essentially commutative diagram of the corresponding categories of algebras, so by the previous case, the functor (f′)_{∗} : B-Ét → A_{(m)}-Ét has both a left quasi-inverse and a right quasi-inverse; these quasi-inverses must be isomorphic, so f_{∗} has a quasi-inverse as desired. Finally, we remark that the map in (ii) is the same as the map Hom_{C-Alg}(B ⊗_{A} C, C) → Hom_{D-Alg}(B ⊗_{A} D, D), and the latter is a bijection in view of (i). □

Remark 3.5.22. Notice that lemma 3.5.21(ii) generalises the lifting theorem 3.2.18(i) (in case V is an 𝔽_{p}-algebra). Similarly, it follows from lemmata 3.5.21(i) and 3.5.10(iv) that, in case V is an 𝔽_{p}-algebra, one can replace “nilpotent” in theorem 3.2.18(ii),(iii) by “Frobenius nilpotent”.

3.5.23. In the following, τ will denote indifferently the topology of universal effective descent defined by either of the fibered categories w.Ét^{o} → V^{a}-Alg^{o} or Ét^{o} → V^{a}-Alg^{o}. Proposition 3.5.24. If f : A → B is a morphism of almost V -algebras which is invertible up to Φ^{m}, then f^{o} is bicovering for the topology τ.

Proof. In light of lemmata 3.5.19(ii) and 3.5.21(i), it suffices to show that μ_{B/A} is invertible up

to a power of Φ. For this, factor the identity morphism of B as B ⟶^{1_{B}⊗f} B ⊗_{A} B ⟶^{μ_{B/A}} B and argue as in the proof of theorem 3.5.13. □

Proposition 3.5.25. Let A → B be a morphism of almost V -algebras and I ⊂ A an ideal. Set \overline{A} := A/I and \overline{B} := B/IB. Suppose that either

(a) I → IB is an epimorphism with nilpotent kernel, or

(b) V is an 𝔽_{p}-algebra and I → IB is invertible up to a power of Φ. Then we have :

(i) conditions (a) and (b) are stable under any base change A → C.

(ii) (A → B)^{o} is covering (resp. bicovering) for τ if and only if (\overline{A} → \overline{B})^{o} is.

Proof. Suppose first that I → IB is an isomorphism; in this case we claim that IC → I(C ⊗_{A} B) is an epimorphism and Ker(IC → I(C ⊗_{A} B))^{2} = 0 for any A-algebra C. Indeed, since by assumption I ≃ IB, C ⊗_{A} B acts on C ⊗_{A} I, hence Ker(C → C ⊗_{A} B) annihilates C ⊗_{A} I, hence annihilates its image IC, whence the claim. If, moreover, V is an 𝔽_{p}-algebra, lemma 3.5.10(iv) implies that IC → I(C ⊗_{A} B) is invertible up to a power of Φ.

In the general case, consider the intermediate almost V -algebra A_{1} := \overline{A} ×\overline{B} B equipped with the ideal I_{1} := 0 ×\overline{B} (IB). In case (a), I_{1} = IA_{1} and A → A_{1} is an epimorphism with nilpotent kernel, hence it remains such after any base change A → C. To prove (i) in case (a), it suffices then to consider the morphism A_{1} → B, hence we can assume from start that I → IB is an isomorphism, which is the case already dealt with. To prove (i) in case (b), it suffices to consider the cases of (A, I) → (A_{1}, I_{1}) and (A_{1}, I_{1}) → (B, IB). The second case is treated above. In the first case, we do not necessarily have I_{1} = IA_{1} and the assertion to be checked is that, for every A-algebra C, the morphism IC → I_{1}(A_{1} ⊗_{A} C) is invertible up to a power of Φ. We apply lemma 3.5.10(v) to the commutative diagram with exact rows:

[figure: 0; I; A; A/I; 0; 0; IB; A_{1}; A/I; 0]

to deduce that A → A_{1} is invertible up to some power of Φ, hence so is C → A_{1} ⊗_{A} C, which implies the assertion.

As for (ii), we remark that the “only if” part is trivial; and we assume therefore that (\overline{A} → \overline{B})^{o} is τ-covering (resp. τ-bicovering). Consider first the assertion for “covering”. We need to show that (A → B)^{o} is of universal effective descent for F, where F is either one of our two fibered categories. In light of (i), this is reduced to the assertion that (A → B)^{o} is of effective descent for F. We notice that (A → A_{1})^{o} is bicovering for τ (in case (a) by theorem 3.2.18 and lemma 3.5.19(ii), in case (b) by proposition 3.5.24). As (\overline{A} → A_{1}/I_{1})^{o} is an isomorphism, the assertion is reduced to the case where I → IB is an isomorphism. In this case, by lemma 3.4.29, there is a natural equivalence: Desc(F, B/A) →^{∼} Desc(F, \overline{B}/\overline{A}) ×_{F\overline{B}} F_{B}. Then the assertion follows easily from corollary 3.4.22. Finally suppose that (\overline{A} → \overline{B})^{o} is bicovering. The foregoing already says that (A → B)^{o} is covering, so it remains to show that (B ⊗_{A} B → B)^{o} is also covering. The above argument again reduces to the case where I → IB is an isomorphism. Then, as in the proof of lemma 3.4.29, the induced morphism I(B ⊗_{A} B) → IB is an isomorphism as well. Thus the assertion for “bicovering” is reduced to the assertion for “covering”. □

We conclude this section with a result of a more special nature, which can be interpreted as an easy case of almost purity in positive characteristic.

3.5.26. We suppose now that the basic setup (V, 𝔪) consists of a perfect 𝔽_{p}-algebra V, i.e. such that the Frobenius endomorphism Φ_{V} : V → V is bijective; moreover we assume that there exists a non-zero-divisor ε ∈ 𝔪 such that 𝔪 = ⋃_{n>0} ε^{1/p^{m}}V. Let us denote by V^{a}-Ét_{uafp} (resp. V [ε^{−1}]-u.Ét_{fp}) the category of uniformly almost finite projective étale V^{a}-algebras (resp. of finite étale V [ε^{−1}]-algebra in the usual sense of [40]). We will be concerned with the natural functor:

(3.5.27)  V^{a}-Ét_{uafp} → V [ε^{−1}]-u.Ét_{fp} : A ↦ A_{∗}[ε^{−1}].

Theorem 3.5.28. Under the assumptions of (3.5.26), the functor (3.5.27) is an equivalence of categories.

Proof. Let R be a finite étale V [ε^{−1}]-algebra. Since V [ε^{−1}] is perfect, the same holds for R, in view of theorem 3.5.13(ii) (applied in the classical limit case of example 2.1.2(ii)). Let us choose a finite V -algebra R_{0} ⊂ R such that R_{0}[ε^{−1}] = R and define R_{1} := ⋃_{n∈ℕ} Φ_{R}^{−n}(R_{0}). Claim 3.5.29. The V^{a}-algebra R_{1}^{a} does not depend on the choice of R_{0}. Proof of the claim: Let R′_{0} ⊂ R be another finite V -algebra such that R′_{0}[ε^{−1}] = R; clearly we have ε^{m}R_{0} ⊂ R′_{0} ⊂ ε^{−m}R_{0} for m ∈ ℕ sufficiently large. It follows that ε^{m/p^{n}}Φ_{R}^{−n}(R_{0}) ⊂ Φ_{R}^{−n}(R′_{0}) ⊂ ε^{−m/p^{n}}Φ_{R}^{−n}(R_{0}) for every n ∈ ℕ. The claim readily follows.

Claim 3.5.30. R_{1}^{a} is an unramified V^{a}-algebra. Proof of the claim: Let e ∈ R ⊗_{V [ε^{−1}]} R be the idempotent provided by proposition 3.1.4; for m ∈ ℕ large enough, ε^{m} · e is contained in the subring R_{0} ⊗_{V} R_{0}. Hence, for every n ∈ ℕ, ε^{m/p^{n}} · e ∈ Φ_{R}^{−n}(R_{0}) ⊗_{V} Φ_{R}^{−n}(R_{0}), so e defines an almost element in (R_{1} ⊗_{V} R_{1})^{a} which fulfills the conditions (i)-(iii) of proposition 3.1.4, and the claim follows.

Claim 3.5.31. R_{1}^{a} is a uniformly almost finite V^{a}-algebra. Proof of the claim: For large enough m ∈ ℕ we have: R_{0} ⊂ Φ_{R}^{−1}(R_{0}) ⊂ ε^{−m}R_{0}, therefore Φ_{R}^{−n}(R_{0}) ⊂ Φ_{R}^{−(n+1)}(R_{0}) ⊂ ε^{−m/p^{n}}Φ_{R}^{−n}(R_{0}) for every n ∈ ℕ. By an easy induction we deduce: Φ_{R}^{−(n+k)}(R_{0}) ⊂ ∏_{j=0}^{k} ε^{−m/p^{n+j}} · Φ_{R}^{−n}(R_{0}) ⊂ ε^{−m/p^{n−1}}Φ_{R}^{−n}(R_{0}) for every n, k ∈ ℕ. Finally, this implies that R_{1} ⊂ ε^{−m/p^{n−1}}Φ_{R}^{−n}(R_{0}) for every n ∈ ℕ and the claim follows.

Claim 3.5.32. Let S be the integral closure of V in R; then R_{1}^{a} = S^{a}. Proof of the claim: Let us endow R with the unique ring topology τ such that the induced subspace topology on R_{0} is the ε-adic topology and R_{0} is open in R. It is easy to check that S consists of power-bounded elements of R relative to the topology τ. Since clearly R_{1} ⊂ S, it suffices therefore to show that (R_{1}^{a})_{∗} ⊂ R is the subring of all power-bounded elements of R. However, (R_{1}^{a})_{∗} can be characterized as the subring of all x ∈ R such that 𝔪 · x ⊂ R_{1}; this already implies that (R_{1}^{a})_{∗} consists of power-bounded elements. On the other hand, if x ∈ R is power-bounded, it follows that δ · x is topologically nilpotent for every δ ∈ 𝔪; since R_{0} is open in R, it follows that, for every δ ∈ 𝔪 there exists n_{0} ∈ ℕ such that (δ · x)^{n} ∈ R_{0} for every n > n_{0}. By taking n := p^{k} for sufficiently large k ∈ ℕ, we deduce that Φ_{R}^{k}(δ · x) ∈ R_{0}, that is δ · x ∈ R_{1}, and the claim follows.

Claim 3.5.33. R_{1}^{a} is an almost projective V^{a}-algebra. Proof of the claim: As a special case of claim 3.5.32, let W be the integral closure of V in V [ε^{−1}]; then:

(3.5.34)  W^{a} = V^{a}.

Next, let Tr_{R/V [ε^{−1}]} : R → V [ε^{−1}] be the trace map of the finite étale extension V [ε^{−1}] → R; recall that Tr_{R/V [ε^{−1}]} sends elements integral over V to elements integral over V (to see this, we can assume that R has constant rank n over V [ε^{−1}]; then the assertion can be checked after a faithfully flat base change V [ε^{−1}] → S, so we can further suppose that R ≃ V [ε^{−1}]^{n}, in which case everything is clear); it then follows from claim 3.5.32 and (3.5.34)that Tr_{R/V [ε^{−1}]}^{a} restricts to a morphism T : R_{1}^{a} → V^{a}. Furthermore, let e ∈ R ⊗_{V [ε^{−1}]} R be the idempotent defining the diagonal imbedding; by claim 3.5.30, for every δ ∈ 𝔪 we can write δ · e = ∑_{i}^{n} x_{i} ⊗ y_{i} for certain x_{i}, y_{i} ∈ R_{1}. By remark 4.1.17 (whose proof does not use theorem 3.5.28) we deduce the identity: δ · b = ∑_{i}^{n} x_{i} · T (b · y_{i}) for every b ∈ (R_{1}^{a})_{∗}. This allows us to define morphisms α : R_{1}^{a} → (V^{a})^{n}, β : (V^{a})^{n} → R_{1}^{a} with β ∘ α = δ · 1_{R_{1}^{a}}, namely α(b) = (T (b · y_{1}), ..., T (b · y_{n})) and β(v_{1}, ..., v_{n}) = ∑_{i}^{n} x_{i} · v_{i} for every b ∈ R_{1}^{a} and v_{1}, ..., v_{n} ∈ V_{∗}^{a}. By lemma 2.4.15, the claim follows.

Claim 3.5.35. The functor (3.5.27)is fully faithful. Proof of the claim: First of all, it is clear that, for every flat V^{a}-algebras A, B, the natural map

(3.5.36)  Hom_{V^{a}-Alg}(A, B) → Hom_{V [ε^{−1}]-Alg}(A_{∗}[ε^{−1}], B_{∗}[ε^{−1}])

is injective, since A_{∗} ⊂ A_{∗}[ε^{−1}] and similarly for B. Suppose now that A and B are étale and almost finite over V^{a}; then Φ_{A} and Φ_{B} are automorphisms, due to theorem 3.5.13(ii) and the assumption that V is perfect. Let ψ : A_{∗}[ε^{−1}] → B_{∗}[ε^{−1}] be any map of V [ε^{−1}]-algebras; since A is almost finite, we have ψ(A_{∗}) ⊂ ε^{−m}B_{∗} for m ∈ ℕ large enough. Since Frobenius commutes with every ring homomorphism, we deduce ψ(A_{∗}) = ψ(Φ_{A_{∗}}^{−n}(A_{∗})) ⊂ ε^{−m/p^{n}}Φ_{B_{∗}}^{−n}(B_{∗}) = ε^{−m/p^{n}}B_{∗} for every n ∈ ℕ, so ψ induces a morphism ψ^{a} : A → B, which shows that (3.5.36) is surjective.

It now follows from claims 3.5.29, 3.5.30, 3.5.31, 3.5.33 that the assignment R → R_{1}^{a} defines a quasi-inverse to 3.5.27; together with claim 3.5.35, this concludes the proof of the theorem.

□

4. Fine study of almost projective modules

An alternative title for this chapter could have been “Everything you can do with traces”. Right at the outset we find the definition of the trace map of an almost projective almost finitely generated A-module. The whole purpose of the chapter is to showcase the versatility of this construction, a real swiss-knife of almost linear algebra. For instance, we apply it to characterize étale morphisms (theorem 4.1.14); more generally, it is used to define the different ideal of an almost finite A-algebra. In section 4.3 it is employed in an essential way to study the important class of A-modules of finite rank, i.e. those almost projective A-modules P such that Λ_{A}^{i}P = 0 for sufficiently large i ∈ ℕ. A rather complete and satisfactory description is achieved for such A-modules (proposition 4.3.27). This is further generalized in theorem 4.3.28, to arbitrary A-modules so called of almost finite rank (see definition 4.3.9(ii)). The interest of the latter class is that it contains basically all the almost projective modules found in nature; indeed, we cannot produce a single example of an almost projective module that is almost finitely generated but has not almost finite rank (but we suspect that they do exist). In any case, almost finitely generated modules whose rank is not almost finite are certainly rather weird beasts : some clue about their looks can be gained by analyzing the structure of invertible modules : we do this at the end of section 4.4.

The other main construction of chapter 4 is the splitting algebra of an almost projective module, introduced in section 4.4 : with its aid we show that A-modules of finite rank are locally free in the flat topology of A. It should be clear that this is a very pleasant and useful culmination for our study of almost projective modules; we put it to use right away in the following section 4.5, where we show that an étale groupoid of finite rank over the category of affine almost schemes (more prosaically, the opposite of the category of almost algebras) is universally effective, that is, it admits a good quotient, as in the classical algebro-geometric setting.

4.1. Almost traces. Let A be a V^{a}-algebra. Definition 4.1.1. Let P be an almost finitely generated projective A-module. Then ω_{P/A} is an isomorphism by lemma 2.4.29(b). The trace morphism of P is the A-linear morphism

tr_{P/A} := ev_{P/A} ∘ ω_{P/A}^{−1} : End_{A}(P)^{a} → A.

We let ζ_{P} be the unique almost element of P ⊗_{A} P^{∗} such that ω_{P/A}(ζ_{P}) = 1_{P}. Lemma 4.1.2. Let M, N be almost finitely generated projective A-modules, and ϕ : M → N, ψ : N → M two A-linear morphisms. Then :

(i) tr_{M/A}(ψ ∘ ϕ) = tr_{N/A}(ϕ ∘ ψ).

(ii) If ψ ∘ ϕ = a · 1_{M} and ϕ ∘ ψ = a · 1_{N} for some a ∈ A_{∗}, and if, furthermore, there exist u ∈ End_{A}(M), v ∈ End_{A}(N) such that v∘ϕ = ϕ∘u, then a·(tr_{M/A}(u)−tr_{N/A}(v)) = 0.

Proof. (i) : by lemma 2.4.29(i), the natural morphism N ⊗_{A}alHom_{A}(M, A) → alHom_{A}(M, N) is an isomorphism (and similarly when we exchange the roles of M and N). By A-linearity, we can therefore assume that ϕ (resp. ψ) is of the form x ↦ n · α(x) for some n ∈ N_{∗}, α : M → A (resp. of the form x ↦ m · β(x) for some m ∈ M_{∗}, β : N → A). Then a simple computation yields:

ϕ ∘ ψ = ω_{N/A}(n · α(m) ⊗ β) ψ ∘ ψ = ω_{M/A}(m · β(n) ⊗ α)

and the claim follows directly from the definition of the trace morphism. For (ii) we compute using (i) : a·tr_{M/A}(u) = tr_{M/A}(ψ∘ϕ∘u) = tr_{M/A}(ψ∘v∘ϕ) = tr_{N/A}(v∘ϕ∘ψ) = a·tr_{N/A}(v). □

Lemma 4.1.3. Let M, N be two almost finitely generated projective A-modules, ϕ ∈ End_{A}(M) and ψ ∈ End_{A}(N). Then tr_{M⊗_{A}N/A}(ϕ ⊗ ψ) = tr_{M/A}(ϕ) · tr_{N/A}(ψ).

Proof. As usual we can suppose that ϕ = ω_{M/A}(m ⊗ α), ψ = ω_{N/A}(n ⊗ β) for some α ∈ M^{∗}, β ∈ N^{∗}. Then ϕ ⊗ ψ = ω_{M⊗_{A}N/A}((m ⊗ n) ⊗ (α ⊗ β)) and the sought identity follows by explicit calculation. □

Proposition 4.1.4. Let \underline{M} = (0 → M_{1} →^{i} M_{2} →^{p} M_{3} → 0) be an exact sequence of almost finitely generated projective A-modules, and let \underline{u} = (u_{1}, u_{2}, u_{3}) : \underline{M} → \underline{M} be an endomorphism of \underline{M}, given by endomorphisms u_{i} : M_{i} → M_{i} (i = 1, 2, 3). Then we have tr_{M_{2}/A}(u_{2}) = tr_{M_{1}/A}(u_{1}) + tr_{M_{3}/A}(u_{3}).

Proof. Suppose first that there exists a splitting s : M_{3} → M_{2} for p, so that we can view u_{2} as a

matrix ( \begin{matrix}u_{1} & v \\ 0 & u_{3} \end{matrix} ), where v ∈ Hom_{A}(M_{3}, M_{1}). By additivity of the trace, we are then reduced

to show that tr_{M_{2}/A}(i ∘ v ∘ p) = 0. By lemma 4.1.2(i), this is the same as tr_{M_{3}/A}(p ∘ i ∘ v), which obviously vanishes. In general, for any a ∈ 𝔪 we consider the morphism μ_{a} = a · 1_{M_{3}} and the pull back morphism \underline{M} ∗ μ_{a} → \underline{M} :

[figure: 0; M_{1}; i; M_{2}; p; M_{3}; 0; 0; M_{1}; P; p′; ϕ; M_{3}; μ_{a}; 0.]

Pick a morphism j : M_{3} → M_{2} such that p ∘ j = a · 1_{M_{3}}; the pair (j, 1_{M_{3}}) determines a morphism σ : M_{3} → P such that σ ∘ p′ = 1_{M_{3}}, i.e. the sequence \underline{M} ∗ μ_{a} is split exact; this sequence also inherits the endomorphism \underline{u} ∗ μ_{a} = (u_{1}, v, u_{3}), for a certain v ∈ End_{A}(P). The pair of morphisms (a · 1_{M_{2}}, p) determines a morphism ψ : M_{2} → P, and it is easy to check that ϕ ∘ ψ = a · 1_{M_{2}} and ψ ∘ ϕ = a · 1_{P}. We can therefore apply lemma 4.1.2 to deduce that a · (tr_{P/A}(v) − tr_{M/A}(u_{2})) = 0. By the foregoing we know that tr_{P/A}(v) = tr_{M_{1}/A}(u_{1}) + tr_{M_{3}/A}(u_{3}), so the claim follows. □

Lemma 4.1.5. Let A be a V^{a}-algebra.

(i) If P := M ⊗_{A} N is an almost projective and faithful (resp. and almost finitely generated) A-module, then so are M and N.

(ii) If M ⊗_{A} N ≃ A, then the evaluation map ev_{M} : M ⊗_{A} M^{∗} → A is an isomorphism. (iii) An invertible A-module is faithful and almost finitely generated projective. (iv) An epimorphism ϕ : M → N of invertible A-modules is an isomorphism.

Proof. Clearly (iii) is just a special case of (i). We show (i): by proposition 2.4.28(iv) we know that ℰ_{P/A} = A; however, one checks easily that ℰ_{P/A} ⊂ ℰ_{N/A}, whence

(4.1.6)  ℰ_{N/A} = A.

Therefore N will be faithful, as soon as it is shown to be almost projective, again by virtue of proposition 2.4.28(iv). In any case, (4.1.6) means that, for every ε ∈ 𝔪, we can find an almost element of the form ∑_{i=1}^{n} x_{i} ⊗ ϕ_{i} ∈ N ⊗_{A} N^{∗}, such that ∑_{i=1}^{n} ϕ_{i}(x_{i}) = ε. We use such an element to define morphisms A → N^{n} → A whose composition equals ε · 1_{N}. After tensoring by M, we obtain morphisms M → P → M whose composition is ε·1_{M}. Then, since P is almost projective, it follows easily that so must be M; similarly, if P is almost finitely generated, the same follows for M. By symmetry, the same holds for N.

(ii): notice that, by (i), we know already that M and N are almost finitely generated projective. By lemma 4.1.3 we deduce that tr_{M/A}(1_{M}) · tr_{N/A}(1_{N}) = 1, so both factors are invertible in A_{∗}. It follows that the morphism A → End_{A}(M) given by a ↦ a · 1_{M} provides a splitting for the trace morphism (and similarly for N in place of M). Thus we can write End_{A}(M) ≃ A ⊕ X, End_{A}(N) ≃ A ⊕ Y for some A-modules X, Y. However, on one hand we have a natural isomorphism End_{A}(M) ⊗_{A} End_{A}(N) ≃ A; on the other hand, we have a decomposition End_{A}(M) ⊗_{A} End_{A}(N) ≃ A ⊕ X ⊕ Y ⊕ (X ⊗_{A} Y); working out the identifications, one sees that the induced isomorphism A ⊕ X ⊕ Y ⊕ (X ⊗_{A} Y) ≃ A restricts to the identity morphism on the direct summand A; it follows that X = Y = 0, which readily implies the claim.

(iv): in view of (ii) we can replace ϕ by ϕ ⊗_{A} 1_{M^{∗}}, and thereby assume that M = A. Then N ≃ A/Ker(ϕ); it is clear that such a module is faithful if and only if Ker(ϕ) = 0. By (iii), the claim follows. □

Lemma 4.1.5 explains why we do not insist, in the definition of an invertible A-module, that it should be almost projective or almost finitely generated: both conditions can be deduced.

4.1.7. Suppose now that B is an almost finite projective A-algebra. For any b ∈ B_{∗}, denote by μ_{b} : B → B the B-linear morphism b′ ↦ b · b′. The map b ↦ μ_{b} defines a B-linear monomorphism μ : B → End_{A}(B)^{a}. The composition

Tr_{B/A} := tr_{B/A} ∘ μ : B → A

will also be called the almost trace morphism of the A-algebra B. Proposition 4.1.8. Let A and B be as in (4.1.7).

(i) If ϕ : A → B is an isomorphism, then Tr_{B/A} = ϕ^{−1}. (ii) If C any other A-algebra, then Tr_{C⊗_{A}B/C} = 1_{C} ⊗_{A} Tr_{B/A}. (iii) If C is an almost finite projective B-algebra, then Tr_{C/A} = Tr_{B/A} ∘ Tr_{C/B}.

Proof. (i) and (ii) are left as exercises for the reader. We verify (iii). It comes down to checking that the following diagram commutes:

\xymatrix{C ⊗_{B} alHom_{B}(C, B) \ar[r] \ar[d]^{_{ev_{C/B}}} & C ⊗_{B} alHom_{A}(C, B) \ar[r]^{∼} & C ⊗_{A} alHom_{A}(C, A) \ar[d]^{_{ev_{C/A}}} \\ B \ar[rr]^{_{Tr_{B/A}}} &  & A.}

Therefore, pick c ∈ C_{∗} and ϕ ∈ Hom_{B}(C, B). For every ε ∈ 𝔪 we can find elements b_{1}, ..., b_{k} ∈ B_{∗} and ϕ_{1}, ..., ϕ_{k} ∈ Hom_{A}(C, A) such that ε · ϕ(x) = ∑_{i} b_{i} · ϕ_{i}(x) for every x ∈ C_{∗}. The Blinearity of ϕ translates into the identity:

(4.1.9)  ∑_{i} b_{i} · ϕ_{i}(b · x) = ∑_{i} b · b_{i} · ϕ_{i}(x)  for all b ∈ B_{∗}, x ∈ C_{∗}.

Then ε · ev_{C/B}(c ⊗ ϕ) = ∑_{i} b_{i} · ϕ_{i}(c) and we need to show that

(4.1.10)  Tr_{B/A}(∑_{i} b_{i} · ϕ_{i}(c)) = ∑_{i} ϕ_{i}(c · b_{i}).

For every i ≤ k, let μ_{i} : A → B be the morphism a ↦ b_{i} · a (for all a ∈ A_{∗}); furthermore, let μ_{c} : B → C be the morphism b ↦ c · b (for all b ∈ B_{∗}). In view of (4.1.9), the left-hand side of (4.1.10) is equal to tr_{B/A}(∑_{i} μ_{i} ∘ ϕ_{i} ∘ μ_{c}). By lemma 4.1.2(i), we have tr_{B/A}(μ_{i} ∘ ϕ_{i} ∘ μ_{c}) = tr_{A/A}(ϕ_{i} ∘ μ_{c} ∘ μ_{i}) = ϕ_{i}(c · b_{i}) for every i ≤ k. The claim follows. □

Corollary 4.1.11. Let A → B be a faithfully flat almost finitely presented and étale morphism of almost V -algebras. Then Tr_{B/A} : B → A is an epimorphism.

Proof. Under the stated hypotheses, B is an almost projective A-module (by proposition 2.4.18). Let C = Coker(Tr_{B/A}) and Tr_{B/B⊗_{A}B} the trace morphism for the morphism of almost V - algebras μ_{B/A}. By faithful flatness, the natural morphism C → C ⊗_{A} B = Coker(Tr_{B⊗_{A}B/B}) is a monomorphism, hence it suffices to show that Tr_{B⊗_{A}B/B} is an epimorphism (here B ⊗_{A} B is considered as a B-algebra via the second factor). However, from proposition 4.1.8(i) and (iii) we see that Tr_{B/B⊗_{A}B} is a right inverse for Tr_{B⊗_{A}B/B}. The claim follows. □

4.1.12. It is useful to introduce the A-linear morphism

t_{B/A} := Tr_{B/A} ∘ μ_{B/A} : B ⊗_{A} B → A.

We can view t_{B/A} as a bilinear form; it induces an A-linear morphism

τ_{B/A} : B → B^{∗} = alHom_{A}(B, A)

characterized by the equality t_{B/A}(b_{1} ⊗ b_{2}) = τ_{B/A}(b_{1})(b_{2}) for all b_{1}, b_{2} ∈ B_{∗}. We say that t_{B/A} is a perfect pairing if τ_{B/A} is an isomorphism. Lemma 4.1.13. Let A → B be an almost finite projective morphism of V^{a}-algebras and C any A-algebra. Denote by η_{B,C} : C ⊗_{A} alHom_{A}(B, A) → alHom_{C}(C ⊗_{A} B, C) the natural isomorphism provided by lemma 2.4.31(i). Then :

(i) τ_{B/A} is B-linear (for the natural B-module structure of B^{∗} defined in remark 2.4.25); (ii) η_{B,C} is C ⊗_{A} B-linear; (iii) η_{B,C} ∘ (1_{C} ⊗ τ_{B/A}) = τ_{C⊗_{A}B/C}.

Proof. For any b ∈ B_{∗}, let ξ_{b} : B → A the A-linear morphism defined by the rule b′ ↦ Tr_{B/A}(b′ · b) for all b′ ∈ B_{∗}. Then, directly from the definition we can compute: (η_{B,C} ∘ (1_{C} ⊗ τ_{B/A}))(c ⊗ b)(c′ ⊗ b′) = η_{B,C}(c ⊗ ξ_{b})(c′ ⊗ b′) = c · c′ · Tr_{B/A}(b′ · b) for all b, b′ ∈ B_{∗}, c, c′ ∈ C_{∗}. But by proposition 4.1.8(ii), the latter expression can be rewritten as τ_{C⊗_{A}B/C}(c ⊗ b)(c′ ⊗ b′), which shows (iii). The proofs of (i) and (ii) are similar direct verifications: we show (i) and leave (ii) to the reader. Let us pick any b, b′, b′′ ∈ B_{∗}; then (b · τ_{B/A}(b′))(b′′) = τ_{B/A}(b′)(bb′′) = Tr_{B/A}(bb′b′′) = (τ_{B/A}(bb′))(b′′). □

Theorem 4.1.14. An almost finite projective morphism ϕ : A → B of almost V -algebras is étale if and only if the trace form t_{B/A} is a perfect pairing.

Proof. By lemma 4.1.13, we have a commutative diagram:

(4.1.15)  \xymatrix{(B ⊗_{A} B) ⊗_{B} B \ar[r]^{∼} \ar[d]_{1_{B⊗_{A}B}⊗_{B}τ_{B}} & B ⊗_{A} B \ar[d]_{1_{B}⊗_{A}τ_{B}} \ar[dr]^{τ_{B⊗_{A}B/B}} \\ (B ⊗_{A} B) ⊗_{B} B^{∗} \ar[r]^{∼} & B ⊗_{A} B^{∗} \ar[r]^{η_{B,B}} & alHom_{B}(B ⊗_{A} B, B)}

in which all the morphisms are B ⊗_{A} B-linear (here we take the B-module structure on B ⊗_{A} B given by multiplication on the right factor). Suppose now that ϕ is étale; then, by corollary 3.1.9, there is an isomorphism of B-algebras: B ⊗_{A} B ≃ I_{B/A} ⊕ B. It follows that τ_{B⊗_{A}B/B} = τ_{B/B} ⊕ τ_{I_{B/A}/B}. Especially, 1_{B} ⊗_{B⊗_{A}B} τ_{B⊗_{A}B/B} is the identity morphism of B (by proposition 4.1.8(i)). This means that in the diagram B ⊗_{B⊗_{A}B} (4.1.15)all the arrows are isomorphisms. In particular, τ_{B/A} is an isomorphism, as claimed.

To prove the converse, we consider the almost element ζ_{B} of the B ⊗_{A} B-module B ⊗_{A} B^{∗}. Viewing B^{∗} as a B-module in the natural way (cp. remark 2.4.25), we also get a scalar multiplication morphism σ_{B^{∗}/B} : B ⊗_{A} B^{∗} → B^{∗} (see (2.2.5)). Claim 4.1.16. With the above notation we have: I_{B/A} · ζ_{B} = 0 and σ_{B^{∗}/B}(ζ_{B}) = Tr_{B/A}. Proof of the claim: Notice that ω_{B/A} is also B ⊗_{A} B-linear for the B ⊗_{A} B-module structure on End_{A}(B) such that ((b ⊗ b′) · ϕ)(b′′) = b′ · ϕ(b · b”) for every b, b′, b” ∈ B_{∗} and every ϕ ∈ End_{A}(B). We compute ω_{B/A}((b ⊗ b′) · ζ_{B})(b”) = ((b ⊗ b′) · ω_{B/A}(ζ_{B}))(b”) = b · b′ · b”. Whence x · ζ_{B} = μ_{B/A}(x) · ζ_{B} for every x ∈ B ⊗_{A} B_{∗} which implies the first claimed identity. Next we compute: σ_{B^{∗}/B}(ζ_{B})(b) = ev_{B}((1 ⊗ b) · ζ_{B}) = (tr_{B/A} ∘ ω_{B/A})((1 ⊗ b) · ζ_{B}) = tr_{B/A}((1 ⊗ b) · ω_{B/A}(ζ_{B})) = tr_{B/A}((1 ⊗ b) · 1_{B}) = Tr_{B/A}(b) for every b ∈ B_{∗}. The claim follows.

Suppose now that τ_{B/A} is an isomorphism. Then we can define e := (1_{B} ⊗ τ_{B/A}^{−1})(ζ_{B}). From claim 4.1.16 and lemma 4.1.13(i) we derive that I_{B/A} · e = 0 and τ_{B/A}(σ_{B/B}(e)) = Tr_{B/A}.

The latter equality implies that σ_{B/B}(e) = 1, in other words μ_{B/A}(e) = 1. We see therefore that e satisfies conditions (ii) and (iii) of proposition 3.1.4 and therefore also condition (i), since the latter is an easy consequence of the other two. Thus A → B is an étale morphism, as claimed. □

Remark 4.1.17. By inspection of the proof of theorem 4.1.14, we see that the following has been shown. Let A → B be an étale morphism of V^{a}-algebras. Then (1_{B} ⊗ τ_{B/A})(e_{B/A}) = ζ_{B}. Definition 4.1.18. The nilradical of an almost algebra A is the ideal nil(A) = nil(A_{∗})^{a} (where, for a ring R, we denote by nil(R) the ideal of nilpotent elements in R). We say that A is reduced if nil(A) ≃ 0.

4.1.19. Notice that, if R is a V -algebra, then every nilpotent ideal in R^{a} is of the form I^{a}, where I is a nilpotent ideal in R (indeed, it is of the form I^{a} where I is an ideal, and 𝔪 · I is seen to be nilpotent). It follows easily that nil(A) is the colimit of the nilpotent ideals in A; moreover nil(R)^{a} = nil(R^{a}). Using this one sees that A/nil(A) is reduced. Proposition 4.1.20. Let A → B be an étale almost finitely presented morphism of almost algebras. If A is reduced then B is reduced as well.

Proof. Under the stated hypothesis, B is an almost projective A-module (by virtue of proposition 2.4.18(ii)). Hence, for given ε ∈ 𝔪, pick a sequence of morphisms B →^{u_{ε}} A^{n} →^{v_{ε}} B such that v_{ε} ∘ u_{ε} = ε · 1_{B}; let μ_{b} : B → B be multiplication by b ∈ B_{∗} and define ν_{b} : A^{n} → A^{n} by ν_{b} = v_{ε} ∘ μ_{b} ∘ u_{ε}. One verifies easily that ν_{b}^{m} = ε^{m−1} · ν_{b^{m}} for all integers m > 0. Now, suppose that b ∈ nil(B_{∗}). It follows that b^{m} = 0 for m sufficiently large, hence ν_{b}^{m} = 0 for m sufficiently large. Let 𝔭 be any prime ideal of A_{∗}; let π : A_{∗} → A_{∗}/𝔭 be the natural projection and F the fraction field of A_{∗}/𝔭. The F -linear morphism ν_{b∗} ⊗_{A_{∗}} 1_{F} is nilpotent on the vector space F^{n}, hence π ∘ tr_{A_{∗}^{n}/A_{∗}}(ν_{b∗}) = tr_{F^{n}/F} (ν_{b∗} ⊗_{A_{∗}} 1_{F}) = 0. This shows that tr_{A_{∗}^{n}/A_{∗}}(ν_{b∗}) lies in the intersection of all prime ideals of A_{∗}, hence it is nilpotent. Since by hypothesis A is reduced, we get tr_{A_{∗}^{n}/A_{∗}}(ν_{b∗}) = 0, whence tr_{A^{n}/A}(ν_{b}) = 0. Using lemma 4.1.2(i) we deduce ε · tr_{B/A}(μ_{b}) = 0, and finally, tr_{B/A}(b) = 0. Now, for any b′ ∈ B_{∗}, the almost element bb′ will be nilpotent as well, so the same conclusion applies to it. This shows that τ_{B/A}(b) = 0. But by hypothesis B is étale over A, hence theorem 4.1.14 yields b = 0, as required. □

Remark 4.1.21. Let M be an A-module. We say that an almost element a of A is M-regular if the multiplication morphism m ↦ am : M → M is a monomorphism. Assume (A) (see (2.1.6)) and suppose furthermore that 𝔪 is generated by a multiplicative system 𝒮 which is a cofiltered semigroup under the preorder structure (𝒮 , ≻) induced by the divisibility relation in V. We say that 𝒮 is archimedean if, for all s, t ∈ 𝒮 there exists n > 0 such that s^{n} ≻ t. Suppose that 𝒮 is archimedean and that A is a reduced almost algebra. Then 𝒮 consists of A-regular elements. Indeed, by hypothesis nil(A_{∗})^{a} = 0; since the annihilator of 𝒮 in A_{∗} is 0 we get nil(A_{∗}) = 0. Suppose that s · a = 0 for some s ∈ 𝒮 and a ∈ A_{∗}. Let t ∈ 𝒮 be arbitrary and pick n > 0 such that t^{n} ≻ s. Then (ta)^{n} = 0 hence ta = 0 for all t ∈ 𝒮, hence a = 0. Definition 4.1.22. Let ϕ : A → B be an almost finite projective morphism of V^{a}-algebras. By (4.1.12), we can assign to ϕ a B-linear trace morphism τ_{B/A} : B → B^{∗}. The different ideal of the morphism ϕ is the ideal 𝒟_{B/A} := Ann_{B}(Coker τ_{B/A}) ⊂ B.

Lemma 4.1.23. Let M_{1} →^{ϕ} M_{2} →^{ψ} M_{3} be two A-linear morphisms of invertible A-modules M_{i} (i ≤ 3) and C an A-algebra. Then:

(i) Ann_{A}(Coker(ψ ∘ ϕ)) = Ann_{A}(Coker ϕ) · Ann_{A}(Coker ψ).

(ii) Ann_{C}(Coker(1_{C} ⊗_{A} ψ)) = C · Ann_{A}(Coker ψ).

Proof. (i): Since, by lemma 4.1.5(iii), M_{3} is faithfully flat, lemma 2.4.6 yields

(4.1.24)  Ann_{A}(Coker ψ) = Ann_{A}(Coker(ψ ⊗_{A} 1_{M_{3}^{∗}}))

and likewise for ϕ; hence we can replace M_{i} by M_{i}⊗_{A}M_{3}^{∗} and suppose that M_{3} = A. Moreover, since M_{2} is invertible, ev_{M_{2}/A} is an isomorphism, by lemma 4.1.5(ii). Let ẽv_{M_{2}/A} : M_{2}^{∗} ⊗_{A} M_{2} → A be the map given by the rule: ϕ ⊗ x ↦ ev_{M_{2}/A}(x ⊗ ϕ), for every ϕ ∈ (M_{2}^{∗})_{∗} and x ∈ M_{2}. Set λ := ev_{M_{2}/A}∘(ϕ⊗_{A}1_{M_{2}^{∗}}) : M_{1}⊗_{A}M_{2}^{∗} → A; then ϕ∘(1_{M_{1}}⊗_{A}ẽv_{M_{2}/A}) = λ⊗_{A}1_{M_{2}}, so that Ann_{A}(ϕ) = Ann_{A}(λ ⊗_{A} 1_{M_{2}}) = Ann_{A}(λ). Thus, we can replace ϕ by λ ⊗_{A} 1_{M_{2}} and then we have to show that

Ann_{A}(Coker ψ ∘ (λ ⊗_{A} 1_{M_{2}})) = Ann_{A}(Coker ψ) · Ann_{A}(Coker λ).

However, quite generally we have:

(4.1.25)  Ann_{A}(Coker(M → A)) = Im(M → A)

for any A-linear morphism M → A. Hence we compute: Ann_{A}(Coker ψ ∘ (λ ⊗_{A} 1_{M_{2}})) = Im(ψ∘(λ⊗_{A}1_{M_{2}})) = ψ(Im(λ⊗_{A}1_{M_{2}})) = ψ(Im(λ)·M_{2}) = Im(λ)·Im(ψ) = Ann_{A}(Coker λ)· Ann_{A}(Coker ψ).

(ii): again, using (4.1.24)we reduce to the case where M_{3} = A; then the claim follows easily from (4.1.25). □

Lemma 4.1.26. Let ϕ : A → B be a morphism of V^{a}-algebras as in definition 4.1.22. Let C be an A-algebra. Suppose that either C is flat over A, or B^{∗} is an invertible B-module for its natural B-module structure. Then 𝒟_{C⊗_{A}B/C} = 𝒟_{B/A} · (C ⊗_{A} B).

Proof. Under the stated assumptions, alHom_{A}(B, A) is an almost finitely generated projective A-module. In particular, Coker τ_{B/A} is almost finitely generated; If C is flat over A, it follows by lemma 2.4.6 that Ann_{C⊗_{A}B}(C ⊗_{A} Coker τ_{B/A}) = 𝒟_{B/A} · (C ⊗_{A} B); if B^{∗} is an invertible Bmodule, the same holds by virtue of lemma 4.1.23(ii). However, by lemma 4.1.13(iii), the trace pairing is preserved under arbitrary base changes, so: C⊗_{A}Coker τ_{B/A} ≃ Coker(1_{C}⊗_{A}τ_{B/A}) ≃ Coker τ_{C⊗_{A}/B}, which shows the claim. □

Proposition 4.1.27. Let B → C be a morphism of A-algebras, and suppose that B (resp. C) is an almost finite projective A-algebra (resp. B-algebra). Suppose moreover that B^{∗} := alHom_{A}(B, A) (resp. C^{∗} := alHom_{B}(C, B)) is an invertible B-module (resp. C-module) for its natural B-module (resp. C-module) structure. Then

𝒟_{C/A} = 𝒟_{C/B} · 𝒟_{B/A}.

Proof. Let C_{/A}^{∗} := alHom_{A}(C, A) and define a C-linear morphism ξ : alHom_{B}(C, B^{∗}) → C_{/A}^{∗} by the rule: ϕ ↦ (c ↦ ϕ(c)(1)) for every ϕ ∈ Hom_{B}(C, B^{∗}) and c ∈ C_{∗}. Claim 4.1.28. C_{/A}^{∗} is an invertible C-module and ξ is an isomorphism.

Proof of the claim: By lemma 2.4.29(i), the natural morphism λ : C^{∗}⊗_{B} B^{∗} → alHom_{B}(C, B^{∗}) is a C-linear isomorphism. It suffices therefore to show that ξ ∘ λ^{−1} : C^{∗} ⊗_{B} B^{∗} → C_{/A}^{∗} is an isomorphism. One verifies easily that ξ ∘ λ^{−1} is defined by the rule: ϕ ⊗ ψ ↦ ψ ∘ ϕ, and then the claim follows from lemma 2.4.31(iii).

Unwinding the definitions, one verifies that the following diagram commutes:

\xymatrix{C \ar[r]^{τ_{C/A}} \ar[d]_{τ_{C/B}} & C_{/A}^{∗} \\ C^{∗} \ar[r]^{alHom_{B}(C,τ_{B/A})} & alHom_{B}(C, B^{∗}). \ar[u]_{ξ}}

Thus, taking into account claim 4.1.28, and lemma 4.1.23(i), we have Ann_{C}(Coker τ_{C/A}) = Ann_{C}(Coker τ_{C/B})·Ann_{C}(Coker(alHom_{B}(C, τ_{B/A}))). However, Coker(alHom_{B}(C, τ_{B/A})) ≃ C^{∗} ⊗_{B} Coker τ_{B/A} by lemma 2.4.29(b). By lemma 4.1.5(iii), C^{∗} is faithfully flat; consequently:

Ann_{C}(Coker(alHom_{B}(C, τ_{B/A}))) = Ann_{C}(Coker τ_{B/A})

which implies the assertion. □

Lemma 4.1.29. Let ϕ : A → B be a morphism of V^{a}-algebras as in definition 4.1.22. Suppose moreover that B^{∗} is an invertible B-module for its natural B-module structure. Then ϕ is étale if and only if 𝒟_{B/A} = B.

Proof. By theorem 4.1.14 it follows easily that 𝒟_{B/A} = B whenever ϕ is étale. Conversely, suppose that 𝒟_{B/A} = B; it then follows that τ_{B/A} is an epimorphism. Again by theorem 4.1.14, we need only show that τ_{B/A} is an isomorphism. This follows from lemma 4.1.5(iv). □

The following lemma will be useful when we will compute the different ideal in situations such as those contemplated in proposition 6.3.11. Lemma 4.1.30. Let A be a V^{a}-algebra, B an almost finite almost projective A-algebra, and let {B_{α} | α ∈ J} be a net of A-subalgebras of B, with B_{α} almost finite projective over A for every α ∈ J, such that lim_{α∈J} B_{α} = B in ℐ_{A}(B). Then lim_{α∈J} 𝒟_{B_{α}/A} = 𝒟_{B/A}.

Proof. For given α ∈ J, let ε ∈ V such that εB ⊂ B_{α}; lemma 4.1.2(ii) implies that ε · Tr_{B_{α}/A}(b) = ε · Tr_{B/A}(b) for every b ∈ B_{α∗}. Hence the diagrams:

\xymatrix{B \ar[r]^{μ_{ε}} \ar[d]_{ε·τ_{B/A}} & B_{α} \ar[d]^{τ_{Bα/A}} & B_{α} \ar[d]_{τ_{Bα/A}} \ar[r] & B \ar[d]^{ε·τ_{B/A}} \\ B^{∗} \ar[r] & B_{α}^{∗} & B_{α}^{∗} \ar[r]^{μ_{ε}^{∗}} & B^{∗}}

commute. The rightmost diagram implies that 𝒟_{B_{α}/A} · Im μ_{ε}^{∗} ⊂ Im(ε · τ_{B/A}) ⊂ Im τ_{B/A}. Hence ε · 𝒟_{B_{α}/A} ⊂ 𝒟_{B/A}, so finally Ann_{V} (B/B_{α}) · 𝒟_{B_{α}/A} ⊂ 𝒟_{B/A}. From the leftmost diagram we deduce that ε · 𝒟_{B/A} (which is an ideal in B_{α}) annihilates Coker(ε · τ_{B/A} : B → B^{∗}) and on the other hand Ann_{V} (B/B_{α}) obviously annihilates Coker(B^{∗} → B_{α}^{∗}); we deduce that Ann_{V} (B/B_{α})^{2} · 𝒟_{B/A} ⊂ 𝒟_{B_{α}/A}, whence the claim. □

4.2. Endomorphisms of 𝔾̂_{m}. This section is dedicated to a discussion of the universal ring that classifies endomorphisms of the formal group 𝔾̂_{m}. The results of this section will be used in sections 4.3 and 4.4.

4.2.1. For every ring R and every integer n ≥ 0 we introduce the ”n-truncated” version of 𝔾̂_{m,R}. This is the scheme 𝔾_{m,R}(n) := Spec R[T]/(T^{n+1}), endowed with the multiplication morphism which is associated to the co-multiplication map

R[T]/(T^{n+1}) → R[T, S]/(T, S)^{n+1} T ↦ T + S + T · S.

Then in the category of formal schemes we have a natural identification 𝔾̂_{m,R} ≃ colim_{n∈ℕ}𝔾_{m,R}(n).

4.2.2. In the terminology of [52, §II.4], 𝔾_{m}(n) is the n-bud of 𝔾̂_{m}. We will be mainly interested in the endomorphisms of 𝔾_{m}(n), but before we can get to that, we will need some complements on buds over artinian ring. Therefore, suppose we have a cartesian diagram of artinian rings

(4.2.3)  \xymatrix{R_{3} \ar[r] \ar[d] & R_{1} \ar[d] \\ R_{2} \ar[r] & R_{0}}

such that one of the two maps R_{i} → R_{0} (i = 1, 2) is surjective. For any ring R, we define the category Bud(n, d, R) of n-buds over R whose underlying R-algebra is isomorphic to R[T_{1}, ..., T_{d}]/(T_{1}, ..., T_{d})^{n+1}. Lemma 4.2.4. Let S be a finite flat augmented algebra over a local noetherian ring R; let I be the augmentation ideal, and suppose that I^{n+1} = 0. Let κ be the residue field of R, and suppose that S ⊗_{R} κ ≃ κ[t_{1}, ..., t_{d}]/(t_{1}, ..., t_{d})^{n+1}. Then S ≃ R[T_{1}, ..., T_{d}]/(T_{1}, ..., T_{d})^{n+1}.

Proof. Let ε : S → R be the augmentation map. For every i = 1, ..., d, pick a lifting T′_{i} ∈ S of t_{i}; set T_{i} = T′_{i} − ε(T′_{i}). By Nakayama’s lemma, the monomials T_{1}^{a_{1}} · ... · T_{d}^{a_{d}} with ∑_{i=1}^{d} a_{i} ≤ n generate the R-module S. Furthermore, under the stated hypothesis, S is a free R-module, and its rank is equal to dim_{κ} S ⊗_{R} κ; hence the above monomials form an R-basis of S. Clearly the elements T_{i} lie in the augmentation ideal of S, therefore every product of n + 1 of them equals zero; in other words, the natural morphism R[X_{1}, ..., X_{d}] → S given by X_{i} ↦ T_{i} is surjective, with kernel containing J := (X_{1}, ..., X_{d})^{n+1}; but by comparing the ranks over R we see that this kernel cannot be larger than J. The assertion follows. □

Proposition 4.2.5. In the situation of (4.2.3), the natural functor

Bud(n, d, R_{3}) → Bud(n, d, R_{1}) ×_{Bud(n,d,R_{0})} Bud(n, d, R_{2})

is an equivalence of categories.

Proof. Let ℳ_{i,proj} (i = 0, ..., 3) be the category of projective R_{i}-modules. By our previous discussion on descent, we already know that (4.2.3) induces a natural equivalence between ℳ_{3,proj} and the 2-fibered product ℳ_{1,proj} ×_{ℳ_{0,proj}} ℳ_{2,proj}. It is easy to see that this equivalence respects the rank of R_{i}-modules, hence induces a similar equivalence for the categories ℳ_{i,f.f.} of free R_{i}-modules of finite rank. Given two objects M := (M_{1}, M_{2}, α : M_{1}⊗_{R_{1}}R_{0} →^{∼} M_{2}⊗_{R_{2}}R_{0}) and N := (N_{1}, N_{2}, β : N_{1} ⊗_{R_{1}} R_{0} →^{∼} N_{2} ⊗_{R_{2}} R_{0}), define the tensor product M ⊗ N := (M_{1} ⊗_{R_{1}} N_{1}, M_{2} ⊗_{R_{2}} N_{2}, α ⊗_{R_{0}} β). Then one checks easily that the above equivalences respect tensor products. It follows formally that one has analogous equivalences for the categories of finite flat R_{i}-algebras. From there, one further obtains equivalences on the categories of such R_{i}-algebras that are augmented over R_{i}, and even on the subcategories R_{i}-Alg_{aug.fl.}^{(n)} of those augmented R_{i}-algebras such that the (n+1)-th power of the augmentation ideal vanishes. These categories admit finite coproducts, that are constructed as follows. For augmented R_{i}-algebras ε_{A} : A → R_{i} and ε_{B} : B → R_{i}, set (A → R_{i}) ⊗ (B → R_{i}) := (A ⊗_{R_{i}} B/Ker(ε_{A} ⊗_{R_{i}} ε_{B})^{n+1} → R_{i}); this is a coproduct of A and B. By formal reasons, the foregoing equivalences of categories respect these coproducts. Finally, an object of Bud(n, d, R_{i}) can be defined as a commutative group object in (R_{i}-Alg_{aug.fl.}^{(n)})^{o}, such that its underlying R_{i}-algebra is isomorphic to R_{i}[T_{1}, ..., T_{d}]/(T_{1}, ..., T_{d})^{n+1}. By formal categorical considerations we see that the foregoing equivalence induces equivalences on the commutative group objects in the respective categories. It remains to check that an R_{3}-algebra S such that S ⊗_{R_{3}} R_{i} ≃ R_{i}[T_{1}, ..., T_{d}]/(T_{1}, ..., T_{d})^{n+1}, (for i = 1, 2) is itself of the form R_{i}[T_{1}, ..., T_{d}]/(T_{1}, ..., T_{d})^{n+1}. However, this follows readily from lemma 4.2.4 and the fact that one of the maps R_{3} → R_{i} (i = 1, 2) is surjective. □

4.2.6. For a given ring R, the endomorphisms of 𝔾_{m,R}(n) are all the polynomials f(T) := a_{0}+a_{1}·T +...+a_{n}·T^{n} such that f(T)+f(S)+f(T)·f(S) ≡ f(T +S+T ·S) (mod (T, S)^{n+1}). This relationship translates into a finite set of polynomial identities for the coefficients a_{0}, ..., a_{n}, and using these identities we can therefore define a quotient 𝒢_{n} of the ring in n indeterminates ℤ[X_{1}, ..., X_{n}] which will be the “universal ring of endomorphisms” of 𝔾_{m}(R), i.e., such that X_{1} ·T +X_{2} ·T^{2} +...+X_{n} ·T^{n} is an endomorphism of 𝔾_{m,𝒢_{n}}(n) and such that, for every ring R, and every f(T) as above, the map ℤ[X_{1}, ..., X_{n}] → R given by X_{i} ↦ a_{i} (i = 1, ..., n) factors through a (necessarily unique) map 𝒢_{n} → R. One of the main results of this section will be a simple and explicit description of the ring 𝒢_{n}. Proposition 4.2.7. 𝒢_{n} is a smooth ℤ-algebra.

Proof. We know already that 𝒢_{n} is of finite type over ℤ, therefore it suffices to show that, for every prime ideal 𝔭 of 𝒢_{n}, the local ring 𝒢_{n,𝔭} is formally smooth for the 𝔭-adic topology (see [30, Ch.IV, Prop.17.5.3]). Therefore, let R_{1} → R_{0} be a surjective homomorphism of local artinian rings; we need to show that the natural map End(𝔾_{m,R_{1}}(n)) → End(𝔾_{m,R_{0}}(n)) is surjective. Let f ∈ End(𝔾_{m,R_{0}}(n)); we define an automorphism χ of 𝔾_{m,R_{0}} ×_{R_{0}} 𝔾_{m,R_{0}} := R_{0}[T, S]/(T, S)^{n+1}, the n-bud of 𝔾̂_{m} × 𝔾̂_{m}, by setting (T, S) ↦ (T, f(T) + S + f(T) · S). Then, thanks to proposition 4.2.5, we obtain an n-bud X_{n} over R_{2} := R_{1} ×_{R_{0}} R_{1}, by gluing two copies of 𝔾_{m,R_{1}} × 𝔾_{m,R_{1}} along the automorphism χ. Claim 4.2.8. The n-bud X_{n} is isomorphic to 𝔾_{m,R_{2}} × 𝔾_{m,R_{2}} if and only if χ lifts to an automorphism of 𝔾_{m,R_{1}} × 𝔾_{m,R_{1}}. Proof of the claim: Taking into account the decription of B(n, d, R_{2}) as 2-fibered product of categories, the proof amounts to a simple formal verification, which is best left to the reader.

Claim 4.2.9. There exists a compatible system of k-buds X_{k} over R_{2} for every k > n, such that X_{k} reduces to X_{k−1} over R_{2}, and specializes to 𝔾_{m,R_{1}}^{2}(k) over the quotient R_{1} of R_{2}. Proof of the claim: In case R_{2} is a torsion-free ℤ-algebra, this follows from [52, Ch.II, §4.10] and an easy induction. If R_{2} is a general artinian ring, choose a torsion-free ℤ-algebra R_{3} with a surjective homomorphism R_{3} → R_{2}. By loc. cit. (and an easy induction) we can find an n-bud Y_{n} over R_{3} such that Y_{n} specialises to X_{n} on the quotient R_{2}, and Y_{n} reduces to 𝔾_{m,R_{3}}^{2}(1) over R_{3}. Then, again by loc. cit, we can find a compatible system of k-buds Y_{k} on R_{3} for every k > n, such that Y_{k} reduces to Y_{k−1} over R_{3} and specializes to 𝔾_{m,R_{1}}^{2}(k) over the quotient R_{1} of R_{3}. The claim holds if we take X_{k} equal to the specialization of Y_{k} over R_{2}.

The direct limit (in the category of formal schemes) of the system (X_{k})_{k≥n} is a formal group X̂ over R_{2}, such that X̂ ⊗_{R_{2}} R_{1} ≃ 𝔾̂_{m,R_{1}} × 𝔾̂_{m,R_{1}}. This formal group gives rise to a p-divisible group (X̂(n))_{n≥0}, where X̂(n) is the kernel of multiplication by p^{n} in X̂. For every m ∈ ℕ, X̂(m) is a finite flat group scheme over R_{2}, such that X̂(m)×_{R_{2}} R_{1} ≃ μ_{p^{m},R_{1}} ×μ_{p^{m},R_{1}}. Denote by X̂(m)^{∗} the Cartier dual of X̂(m) (cp. [56, §III.14]). Then X̂(m)^{∗} ×_{R_{2}} R_{1} ≃ (ℤ/p^{m}ℤ)_{R_{1}}^{2}, in particular it has p^{2m} connected components. Since the pair (R_{2}, R_{1}) is henselian, it follows that X̂(m)^{∗} must have p^{2m} connected components as well, and consequently X̂(m)^{∗} ≃ (ℤ/p^{m}ℤ)_{R_{2}}^{2}. Finally, this shows that X̂(n) ≃ μ_{p^{m},R_{2}} × μ_{p^{m},R_{2}}, whence X̂ ≃ 𝔾_{m,R_{2}} × 𝔾_{m,R_{2}}. From claim 4.2.8, we deduce that χ lifts to an automorphism χ′ of 𝔾_{m,R_{1}}(n) × 𝔾_{m,R_{1}}(n). Let i : 𝔾_{m,R_{1}}(n) → 𝔾_{m,R_{1}}(n) × 𝔾_{m,R_{1}}(n), π : 𝔾_{m,R_{1}}(n) × 𝔾_{m,R_{1}}(n) → 𝔾_{m,R_{1}}(n) be respectively the imbedding of the first factor, and the projection onto the second factor; clearly π ∘ χ′ ∘ i yields a lifting of f(T), as required. □

4.2.10. Next, let us remark that, for every n ∈ ℕ, the polynomial (1 + T)^{X} − 1 := X · T + \binom{X}{2}·T^{2} +...+\binom{X}{n}·T^{n} ∈ ℚ[X, T] is an endomorphism of 𝔾_{m,ℚ[X]}(n). As a consequence, there is a unique ring homomorphism 𝒢_{n} → ℤ[X, \binom{X}{2}, ..., \binom{X}{n}] representing this endomorphism. The following theorem will show that this homomorphism is an isomorphism. Theorem 4.2.11. The functor

ℤ-Alg → Set R ↦ End_{R}(𝔾_{m,R}(n))

is represented by the ring ℤ[X, \binom{X}{2}, ..., \binom{X}{n}].

Proof. The above discussion has already furnished us with a natural surjective map ρ : 𝒢_{n} → ℤ[X, \binom{X}{2}, ..., \binom{X}{n}]. Therefore, it suffices to show that this map is injective.

Claim 4.2.12. ρ ⊗_{ℤ} 1_{ℚ} is an isomorphism.

Proof of the claim: First of all, the map ρ can be characterized in the following way. The identity map 𝒢_{n} → 𝒢_{n} determines an endomorphism f(T) := a_{0} + a_{1} · T + ... + a_{n} · T^{n} of 𝔾_{m,𝒢_{n}}(n); then ρ is the unique ring homomorphism such that ρ(f) := f(a_{0})+f(a_{1})·T +...+f(a_{n})·T^{n} = (1 + T)^{X} − 1. On the other hand, the ring 𝒢_{n} ⊗_{ℤ} ℚ represents endomorphisms of 𝔾_{m}(n) in the category of ℚ-algebras. However, for every n ∈ ℕ and for every ℚ-algebra R, there is an isomorphism

log : 𝔾_{m,R}(n) ⟶^{∼} 𝔾_{a,R}(n)

to the n-bud of the additive formal group 𝔾̂_{a,R}. The endomorphism group of 𝔾_{a,R}(n) is easily computed, and found to be isomorphic to R. In other words, the universal ring representing endomorphisms of 𝔾_{a}(n) over ℚ-algebras is just ℚ[X], and the bijection Hom_{ℚ-Alg}(ℚ[X], R) ≃ End(𝔾_{a,R}(n)) assigns to a homomorphism ϕ : ℚ[X] → R, the endomorphism g_{ϕ}(T) := ϕ(X)·T. It follows that, for any ℚ-algebra R there is a natural bijection Hom_{ℚ-Alg}(ℚ[X], R) ≃ End(𝔾_{m,R}(n)) given by: (ϕ : ℚ[X] → R) ↦ exp(ϕ(X) · log(1 + T)) − 1 = (1 + T)^{ϕ(X)} − 1. Especially, f(T) can be written in the form (1 + T)^{ψ(X)} − 1 for a unique ring homomorphism ψ : ℚ[X] → 𝒢_{n} ⊗_{ℤ} ℚ. Clearly ψ is inverse to ρ ⊗_{ℤ} 1_{ℚ}.

In view of claim 4.2.12, we are thus reduced to show that 𝒢_{n} is a flat ℤ-algebra, which follows readily from proposition 4.2.7. □

4.2.13. Furthermore, 𝒢_{n} is endowed with a co-addition, i.e. a ring homomomorphism 𝒢_{n} → 𝒢_{n}⊗_{ℤ}𝒢_{n} satisfying the usual co-associativity and co-commutativity conditions. The co-addition is given by the rule:

coadd : 𝒢_{n} → 𝒢_{n} ⊗_{ℤ} 𝒢_{n} \binom{X}{k} ↦ ∑_{i+j=k} \binom{X}{i} ⊗ \binom{X}{j}.

Moreover, for every k ∈ ℤ, we have a ring homomorphism π_{k} : 𝒢_{n} → ℤ, which corresponds to the endomorphism of 𝔾_{m,ℤ}(n) given by the rule: T ↦ (1 + T)^{k} − 1 (raising to the k-th power in 𝔾_{m,ℤ}(n)). Hence we derive, for every k ∈ ℤ, a ring homomorphism

(4.2.14)  \xymatrix{𝒢_{n} \ar[r]^{coadd} & 𝒢_{n} ⊗_{ℤ} 𝒢_{n} \ar[r]^{1_{𝒢n} ⊗π_{k}} & 𝒢_{n}.}

Remark 4.2.15. (i) On 𝒢_{n} ⊗_{ℤ} ℚ = ℚ[X], (4.2.14) is the unique map such that \binom{X}{i} ↦ \binom{X+k}{i} for all i ≤ n, therefore we see that \binom{X+k}{i} ∈ 𝒢_{n} for all k ∈ ℤ, n ≥ 0 and 0 ≤ i ≤ n. Moreover, (4.2.14)is clearly an automorphism for every k ∈ ℤ.

(ii) It is also interesting (though it will not be needed in this work) to remark that 𝒢_{n} is endowed additionally with a co-composition structure, so that 𝒢_{n} is actually a co-ring, and it represents the functor R ↦ End(𝔾_{m,R}(n)) from ℤ-algebras to unitary rings. One can check that the co-composition map is given by the rule:

\binom{X}{k} ↦ ∑_{ϕ} \binom{X}{ϕ} ⊗ ∏_{j∈ℕ^{∗}} \binom{Y}{j}^{ϕ(j)}

where ϕ ranges over all the functions ϕ : ℕ^{∗} := ℕ \ {0} → ℕ subject to the condition that ∑_{j∈ℕ^{∗}} j · ϕ(j) = k, and \binom{X}{ϕ} := \frac{X(X−1)·...·(X−∑_{j∈ℕ^{∗}} ϕ(j)+1)}{∏_{j∈ℕ^{∗}} ϕ(j)!}. To show that \binom{X}{ϕ} ∈ 𝒢_{n}, one notices

that \binom{X}{ϕ} = ∏_{j∈ℕ^{∗}} \binom{X−∑_{i=1}^{j−1} ϕ(i)}{ϕ(j)} and then uses (i).

4.2.16. For the rest of this section we fix a prime number p and we let v_{p} : ℚ → ℤ ∪ {∞} be the p-adic valuation. Lemma 4.2.17. The ring 𝒢_{n,(p)} := 𝒢_{n} ⊗_{ℤ} ℤ_{(p)} is the ℤ_{(p)}-algebra generated by the polynomials X, \binom{X}{p},\binom{X}{p^{2}},...,\binom{X}{p^{k}}, where k is the unique integer such that p^{k} ≤ n < p^{k+1}.

Proof. We proceed by induction on n. It suffices to prove that \binom{X}{n} is contained in the ℤ_{(p)}- algebra R := ℤ_{(p)}[\binom{X}{p}, \binom{X}{p^{2}}, ..., \binom{X}{p^{k}}]. We will use the following (easily verified) identity which holds in ℚ[X] for every i, j ∈ ℕ :

(4.2.18)  \binom{X}{i + j} = \binom{X}{i} · \binom{X − i}{j} · \binom{i + j}{j}^{−1}.

Suppose first that n is a multiple of p^{k}, and write n = (b + 1)p^{k} for some b < p − 1. If b = 0, there is nothing to prove, so we can even assume that b > 0. We apply (4.2.18) with i = b · p^{k} and j = p^{k}. By remark 4.2.15(i), \binom{X−b·p^{k}}{p^{k}} is in R, and so is \binom{X}{b·p^{k}}, by induction. The claim

will therefore follow in this case, if we show that \binom{(b+1)p^{k}}{p^{k}} is invertible in ℤ_{(p)}. However, this is clear, since v_{p}(i) = v_{p}(i + b · p^{k}) for every i = 1, ..., p^{k}. Finally, it remains consider the case where n = b · p^{k} + a for some b > 0 and 0 < a < p^{k}. This is dealt with in the same way: apply (4.2.18)with i = b · p^{k} and j = a and use the previous case. □

Lemma 4.2.19. Let k ∈ ℕ. If R is a flat ℤ_{(p)}-algebra and f ∈ R, then the following two conditions are equivalent:

(i) \binom{f}{p^{i}} ∈ R for every i = 1, ..., k;

(ii) locally on SpecR there exists j ∈ ℤ such that f ≡ j (mod p^{k}).

Proof. We may assume that R is local. For k = 0 there is nothing to prove. For k = 1 we have \binom{f}{p} = u · p^{−1} · ∏_{i=0}^{p−1}(f − i) for a unit u of R. Then the assertion holds since all but possibly one of the f − i are invertible. For k > 1, by induction we can write f = i + p · g for some g ∈ R and 0 ≤ i < p. Since v_{p}(p^{k}!) = 1 + p + p^{2} + ... + p^{k−1}, we have

\binom{f}{p^{k}} = u · p^{−1−p−p^{2}−...−p^{k−1}} · ∏_{{j≡i (mod p) \atop 0≤j<p^{k}}}(f − j) = u′ · \binom{g}{p^{k−1}}

for some units u, u′ ∈ R. The claim follows. □

4.2.20. For every integer k ≥ 0, we construct a scheme X_{k} by gluing the affine schemes Spec ℤ_{(p)}[\frac{X−i}{p^{k}}] (0 ≤ i < p^{k}) along their general fibres. For every k ∈ ℕ and every i ∈ ℕ with 0 ≤ i < p^{k+1} there is an obvious imbedding ℤ_{(p)}[\frac{X−i}{p^{k}}] ⊂ ℤ_{(p)}[\frac{X−i}{p^{k+1}} ]. By gluing the duals of these imbeddings, we obtain, for every k ∈ ℕ, a morphism of schemes ρ_{k} : X_{k+1} → X_{k}. Let also ξ_{k} : Spec 𝒢_{p^{k+1}} → Spec 𝒢_{p^{k}} be the morphism which is dual to the imbedding 𝒢_{p^{k}} ⊂ 𝒢_{p^{k+1}}. Proposition 4.2.21. With the notation of (4.2.20)we have:

(i) For given n > 0, let k be the unique integer such that p^{k} ≤ n < p^{k+1}. Then there is a natural isomorphism of schemes: π_{k} : X_{k} ⟶^{∼} Spec 𝒢_{n} ⊗_{ℤ} ℤ_{(p)}.

(ii) For every k ∈ ℕ the diagram of schemes:

\xymatrix{X_{k+1} \ar[r]^{π_{k+1}} \ar[d]_{ρ_{k}} & Spec 𝒢_{p^{k+1}} ⊗_{ℤ} ℤ_{(p)} \ar[d]^{ξ_{k}⊗_{ℤ}ℤ_{(p)}} \\ X_{k} \ar[r]^{π_{k}} & Spec 𝒢_{p^{k}} ⊗_{ℤ} ℤ_{(p)}}

commutes.

Proof. By lemma 4.2.17 we may assume that n = p^{k}. By lemma 4.2.19, we see that both X_{k} and Spec 𝒢_{p^{k}} ⊗_{ℤ} ℤ_{(p)} represent the same functor from the category of flat ℤ_{(p)}-schemes to the category of sets. Since both schemes are flat over Spec ℤ_{(p)}, (i) follows. It is similarly clear that ξ_{k} ⊗_{ℤ} ℤ_{(p)} and ρ_{k} represent the same natural transformation of functors, so (ii) follows. □

Corollary 4.2.22. (i) For given n ∈ ℕ, let k be the unique integer such that p^{k−1} ≤ n < p^{k}. Then there is a natural ring isomorphism

ℤ/p^{k}ℤ ⟶^{∼} End (𝔾_{m,𝔽_{p}}(n)) i ↦ (1 + T)^{i} − 1

(ii) Let R be a ring such that 𝔽_{p} ⊂ R. Then there is a natural ring isomorphism

𝒞^{0}(Spec R, ℤ_{p}) ⟶^{∼} End_{R} (𝔾̂_{m,R}) β ↦ (1 + T)^{β} − 1.

Proof. (i): by lemma 4.2.17 we can assume n = p^{k} − 1. In this case, it is clear that the polynomials (1 + T)^{i} − 1 are all distinct for i = 0, ..., p^{k} − 1 and they form a subring of End(𝔾_{m,𝔽_{p}}(n)). However, an endomorphism of 𝔾_{m,𝔽_{p}}(n) corresponds to a unique point in Spec 𝒢_{n}(𝔽_{p}). From proposition 4.2.21(i) we derive that Spec 𝒢_{n}⊗_{ℤ}𝔽_{p} is the union of the special fibres of the affine schemes Spec ℤ[\frac{X−i}{p^{k−1}} ], for i = 0, ..., p^{k−1} − 1. Each of those contribute an affine line 𝔸_{𝔽_{p}}^{1}, so Spec 𝒢_{n} ⊗_{ℤ} 𝔽_{p} consists of exactly p^{k−1} connected components. In total, we have therefore exactly p^{k} points in Spec 𝒢_{n}(𝔽_{p}), so (i) follows.

(ii): to give an endomorphism of 𝔾̂_{m} is the same as giving a compatible system of endomorphisms of 𝔾_{m}(n), one for each n ∈ ℕ. In case 𝔽_{p} ⊂ R, lemma 4.2.17 shows that this is also equivalent to the datum of a compatible system of morphisms ϕ_{k} : SpecR → Spec 𝒢_{p^{k}} ⊗_{ℤ} 𝔽_{p}, for every k ≥ 0. From proposition 4.2.21(ii) we can further deduce that, under the morphism ξ_{k}, each of the p^{k+1} connected components of Spec 𝒢_{p^{k+1}} ⊗_{ℤ} 𝔽_{p} gets mapped onto one of the p^{k+1} rational points of Spec 𝒢_{p^{k}} ⊗_{ℤ} 𝔽_{p}. Since ϕ_{k−1} = ξ_{k} ∘ ϕ_{k}, we see that the image of ϕ_{k−1} is contained in Spec 𝒢_{p^{k−1}}(𝔽_{p}), for every k > 0. Taking (i) into account, we see that an endomorphism of 𝔾̂_{m,R} is the same as the datum of a compatible system of continuous maps SpecR → ℤ/p^{k}ℤ. Since the p-adic topology of ℤ_{p} is the inverse limit of the discrete topologies on the ℤ/p^{k}ℤ, the claim follows. □

4.3. Modules of almost finite rank. Let A be a V^{a}-algebra, P an almost finitely generated projective A-module and ϕ ∈ End_{A}(P).

4.3.1. We say that ϕ is Λ-nilpotent if there exists an integer i > 0 such that Λ_{A}^{i}ϕ = 0. Notice that the Λ-nilpotent endomorphisms of P form a bilateral ideal of the unitary ring End_{A}(P). Notice also that Λ_{A}^{i}P is an almost projective A-module for every i ≥ 0; indeed, this is easily shown by means of lemma 2.4.15. For a Λ-nilpotent endomorphism ϕ we introduce the notation

det(1_{P} + ϕ) := ∑_{i≥0} tr_{Λ_{A}^{i}P/A}(Λ_{A}^{i}ϕ).

Notice that the above sum consists of only finitely many non-zero terms, so that det(1_{P} + ϕ) is a well defined element of A_{∗}. Lemma 4.3.2. Let P be an almost finitely generated projective A-module.

(i) If ϕ is a Λ-nilpotent endomorphism of P and α : A → A′ is any morphism of V^{a}- algebras, set P′ := P ⊗_{A} A′. Then: det(1_{P′} + ϕ ⊗_{A} 1_{A′}) = α(det(1_{P} + ϕ)). (ii) Let ϕ, ψ ∈ End_{A}(P) such that ϕ∘ψ and ψ∘ϕ are Λ-nilpotent. Then: det(1_{P} +ϕ∘ψ) =

det(1_{P} + ψ ∘ ϕ).

Proof. (i) is a straightforward consequence of the definitions. As for (ii), it is clear that ψ ∘ ϕ is Λ-nilpotent and the stated identity follows directly from lemma 4.1.2(i). □

4.3.3. Now, let ϕ, ψ ∈ End_{A}(P) be two endomorphisms. Set B := A[X, Y]/(X^{n}, Y^{n}) and P_{B} := P ⊗_{A} B; ϕ and ψ induce endomorphisms of P_{B} that we denote again by the same letters. Clearly X ·ϕ and Y ·ψ are Λ-nilpotent; hence we get elements det(1_{P_{B}} +X ·ϕ), det(1_{P_{B}} +Y ·ψ) and det(1_{P_{B}} + X · ϕ + Y · ψ + XY · ψ ∘ ϕ) in B_{∗}. Notice that any element of B_{∗} can be written uniquely as an A_{∗}-linear combination of the monomials X^{i}Y^{j} with 0 ≤ i, j < n. Moreover, it is clear that det(1_{P_{B}} + X · ϕ) = ∑_{0≤i<n} tr_{Λ_{A}^{i}P/A}(Λ_{A}^{i}ϕ) · X^{i}, and similarly for ψ. Proposition 4.3.4. With the above notation, the following identity holds :

(4.3.5) det(1_{P_{B}} + X · ϕ) · det(1_{P_{B}} + Y · ψ) = det(1_{P_{B}} + X · ϕ + Y · ψ + XY · ψ ∘ ϕ).

Proof. First of all we remark that, when P is a free A-module of finite rank, the above identity is well-known, and easily verified by working with matrices with entries in A_{∗}. Suppose next that P is arbitrary, but ϕ = ε · ϕ′, ψ = ε · ψ′ for some ϕ′, ψ′ ∈ End_{A}(P) and ε ∈ 𝔪. Pick a free A-module F of finite rank, and morphisms u : P → F, v : F → P such that v ∘ u = ε · 1_{P}. Set ϕ_{ε} := u ∘ ϕ′ ∘ v : F → F and define similarly ψ_{ε}. Clearly det(1_{P_{B}} + X · ϕ) = det(1_{P_{B}} + X · ε · ϕ′) = det(1_{P_{B}} + X · v ∘ u ∘ ϕ′) = det(1_{P_{B}} + X · ϕ_{ε}) (by lemma 4.3.2(ii)) and similarly for the other terms appearing in (4.3.5). Thus we have reduced this case to the case of a free A-module. Finally, we deal with the general case. The foregoing shows that the sought identity is known at least when ϕ and ψ are replaced by ε · ϕ, resp. ε · ψ, for any ε ∈ 𝔪. Equivalently, consider the A-algebra endomorphism α : B → B defined by X ↦ ε · X, Y ↦ ε · Y and let C be the B-algebra structure on B determined by α; by lemma 4.3.2(i) we have

det(1_{P_{B}} + X · ε · ϕ) = det(1_{P_{C}} + (X · ϕ) ⊗_{B} 1_{C}) = α(det(1_{P_{B}} + X · ϕ))

and similarly for the other terms appearing in (4.3.5). Thus, the images under α of the two members of (4.3.5)coincide. But applying α to a monomial of the form a · X^{i}Y^{j} has the effect of multiplying it by ε^{i+j}; by (B), the (i + j)-powers of elements of 𝔪 generate 𝔪, hence the claim follows easily. □

Corollary 4.3.6. If ϕ, ψ ∈ End_{A}(P) are two Λ-nilpotent endomorphisms, then

det(1_{P} + ϕ) · det(1_{P} + ψ) = det(1_{P} + ϕ + ψ + ϕ ∘ ψ).

Proof. For an arbitrary α ∈ End_{A}(P), one can define P′ := P ⊗_{A}A[[X]] and det(1_{P′}+X·α) := ∑_{i≥0} tr_{Λ_{A}^{i}P/A}(Λ_{A}^{i}α)·X^{i} ∈ A_{∗}[[X]]. Then proposition 4.3.4 implies that the analogue of (4.3.5) holds in A_{∗}[[X, Y]]. But if α is Λ-nilpotent, the power series det(1_{P′} + X · α) is actually a polynomial in A_{∗}[X]; the claim then follows by evaluating the polynomials det(1_{P′} + X · ϕ), det(1_{P′} + Y · ψ) and det(1_{P′} + X · ϕ + Y · ψ + XY · ϕ ∘ ψ) for X = Y = 1. □

4.3.7. Next, for P as above, set

χ_{P} (X) := ∑_{i≥0} tr_{Λ_{A}^{i}P/A}(Λ_{A}^{i}1_{P}) · X^{i} ∈ A_{∗}[[X]]
ψ_{P} (X) := ∑_{i≥0} tr_{Sym_{A}^{i}P/A}(1_{Sym_{A}^{i}P}) · X^{i} ∈ A_{∗}[[X]].

Corollary 4.3.8. Let P be an almost finitely generated projective A-module. Then: (i) the power series χ_{P} (X) defines an endomorphism of the formal group 𝔾̂_{m,A_{∗}}. (ii) χ_{P} (X) · ψ_{P} (−X) = 1. (iii) χ_{P} (X) ∈ 1 + ℰ_{P/A∗}[[X]].

Proof. (i) is immediate. For (ii), recall that, for every n > 0 there is an acyclic Koszul complex (cp. [17, Ch.X, §9, n.3, Prop.3])

0 → Λ_{A}^{n}Q → (Λ_{A}^{n−1}Q) ⊗_{A} (Sym_{A}^{1}Q) → ... → (Λ_{A}^{1}Q) ⊗_{A} (Sym_{A}^{n−1}Q) → Sym_{A}^{n}Q → 0.

From proposition 4.1.4 we derive, by a standard argument, that the trace is an additive function on arbitrary bounded acyclic complexes. Then, taking into account lemma 4.1.3 we obtain: ∑_{i=0}^{n}(−1)^{i} · tr_{Λ_{A}^{n−i}P/A}(1_{Λ_{A}^{n−i}P}) · tr_{Sym_{A}^{i}P/A}(1_{Sym_{A}^{i}P}) = 0 for every n > 0. This is equivalent to the sought identity. To show (iii) we remark more precisely that ℰ_{Λ_{A}^{r}P/A} ⊂ ℰ_{P/A} for every r > 0. Indeed, set B := A/ℰ_{P/A}. Then, by proposition 2.4.28(i),(iii): ℰ_{Λ_{A}^{r}P/A} · B = ℰ_{Λ_{B}^{r}(P ⊗_{A}B)/B} = 0, whence the claim. □

Definition 4.3.9. Let P be an almost finitely generated projective A-module.

(i) The formal rank of P is the ring homomorphism f.rk_{A}(P) : 𝒢_{∞} := ℤ[α, \binom{α}{2}, ...] → A_{∗} associated to χ_{P} (X).

(ii) We say that P is of almost finite rank if, for every ε ∈ 𝔪, there exists an integer i ≥ 0

such that ε · Λ_{A}^{i}P = 0. (iii) We say that P is of finite rank if there exists an integer i ≥ 0 such that Λ_{A}^{i}P = 0. (iv) Let r ∈ ℕ; we say that P has constant rank equal to r if Λ_{A}^{r+1}P = 0 and Λ_{A}^{r}P is an

invertible A-module.

Remark 4.3.10. (i): It follows easily from lemma 2.3.7(vi) that every uniformly almost finitely generated projective A-module is of finite rank.

(ii): Notice that if P is of finite rank, then χ_{P} (X) is a polynomial, whence it defines an endomorphism of the algebraic group 𝔾_{m,A_{∗}}. In this case, it follows that χ_{P} (X) is of the form (1 + X)^{α}, where α : Spec A_{∗} → ℤ is a continuous function (where ℤ is seen as a discrete topological space). More precisely, there is an obvious injective ring homomorphism

(4.3.11)  𝒞^{0}(Spec A_{∗}, ℤ) → End_{A}(𝔾_{m,A_{∗}})  β ↦ (1 + X)^{β}

which allows to identify the continuous function α with the formal rank of P. Moreover, if Λ_{A}^{i}P = 0, it is clear that α(Spec A_{∗}) ⊂ {0, ..., i − 1}.

The main result of this section is theorem 4.3.28, which describes general modules of almost finite rank as infinite products of modules of finite rank. The first step is lemma 4.3.12, concerned with the case of an A-module of rank one.

Lemma 4.3.12. Let P be an almost finitely generated projective A-module such that Λ_{A}^{2}P = 0. There exists V^{a}-algebras A_{0}, A_{1} and an isomorphism of V^{a}-algebras A ≃ A_{0} × A_{1} such that P ⊗_{A} A_{0} = 0 and P ⊗_{A} A_{1} is an invertible A_{1}-module.

Proof. Since the natural map P × P → Λ_{A}^{2}P is universal for alternating A-bilinear maps on P × P, we have

(4.3.13)  f(p) · q = f(q) · p  for every f ∈ (P^{∗})_{∗} and p, q ∈ P_{∗}.

Using (4.3.13) we derive ω_{P/A}(p ⊗ f)(q) = tr_{P/A}(ω_{P/A}(p ⊗ f)) · q for every f ∈ (P^{∗})_{∗} and p, q ∈ P_{∗}. In other words, ω_{P/A}(p⊗f) = tr_{P/A}(ω_{P/A}(p⊗f))·1_{P}, for every f ∈ (P^{∗})_{∗}, p ∈ P_{∗}. By linearity we finally deduce

(4.3.14)  ϕ = tr_{P/A}(ϕ) · 1_{P}  for all ϕ ∈ End_{A}(P).

Now, by remark 4.3.10(ii), the hypothesis Λ_{A}^{2}P = 0 also implies that χ_{P} (X) = (1 + X)^{α}, for a continuous function α : Spec A_{∗} → {0, 1}. We can decompose accordingly A = A_{0} × A_{1}, so that α(Spec A_{i∗}) = i, which gives the sought decomposition. We can now treat separately the two cases A = A_{0} and A = A_{1}. In case f.rk_{A}(P) = 0, then tr_{P/A}(1_{P}) = 0, and then (4.3.14) implies that P = 0. In case f.rk_{A}(P) = 1, then tr_{P/A}(1_{P}) = 1 and (4.3.14) implies that the natural map A → End_{A}(P) : a ↦ a · 1_{P} is an inverse for tr_{P/A}, thus A ≃ P ⊗_{A} P^{∗}. □

The next step consists in analyzing the structure of A-modules of finite rank. To this purpose we need some preliminaries of multi-linear algebra.

4.3.15. For every n ≥ 0 let n := {1, ..., n}; for a subset I ⊂ n let |I| be the cardinality of I; for a given partition n = I ∪ J, let ≺ denote the total ordering on n that restricts to the usual ordering on I and on J, and such that i ≺ j for every i ∈ I, j ∈ J. Finally let ε_{IJ} be the sign of the unique order-preserving bijection (n, <) → (n, ≺).

Let M be any A-module. Given elements m_{1}, m_{2}, ..., m_{n} in M_{∗}, and I ⊂ n a subset of elements i_{1} < i_{2} < ... < i_{|I|}, let m_{I} := m_{i_{1}} ∧ ... ∧ m_{i_{|I|}} ∈ Λ_{A}^{|I|}M_{∗} (with the convention that m_{∅} = 1 ∈ A_{∗} = Λ_{A}^{0}M_{∗}).

4.3.16. Let M, N be any two A-modules. For every i, j ≥ 0 there is a natural morphism

(4.3.17)  Λ_{A}^{i}M ⊗_{A} Λ_{A}^{j}N → Λ_{A}^{i+j}(M ⊕ N)

determined by the rule:

m_{1} ∧ ... ∧ m_{i} ⊗ n_{1} ∧ ... ∧ n_{j} ↦ (m_{1}, 0) ∧ (m_{2}, 0) ∧ ... ∧ (0, n_{1}) ∧ ... ∧ (0, n_{j})

for all m_{1}, ..., m_{i} ∈ M_{∗} and n_{1}, ..., n_{j} ∈ N. The morphisms (4.3.17) assemble to an isomorphism of A-modules

(4.3.18)  Λ_{A}^{•}M ⊗_{A} Λ_{A}^{•}N → Λ_{A}^{•}(M ⊕ N).

Clearly, there is a unique graded A-algebra structure on Λ_{A}^{•}M ⊗_{A} Λ_{A}^{•}N such that (4.3.18) is an isomorphism of (graded-commutative) A-algebras. Explicitly, given x_{i} ∈ Λ_{A}^{a_{i}}M, y_{i} ∈ Λ_{A}^{b_{i}}N (i = 1, 2) one verifies easily that the product on Λ_{A}^{•}M ⊗_{A} Λ_{A}^{•}N is fixed by the rule

(4.3.19)  (x_{1} ⊗ y_{1}) · (x_{2} ⊗ y_{2}) = (−1)^{a_{2}b_{1}} · (x_{1} ∧ x_{2}) ⊗ (y_{1} ∧ y_{2}).

Then Λ_{A}^{•}M ⊗_{A} Λ_{A}^{•}N is even a bigraded A-algebra, if we let Λ_{A}^{i}M ⊗_{A} Λ_{A}^{j}N be the graded component of bidegree (i, j).

4.3.20. Next, let δ : M → M ⊕ M be the diagonal morphism m ↦ (m, m) (for all m ∈ M_{∗}). It induces a morphism Λ_{A}^{•}δ : Λ_{A}^{•}M → Λ_{A}^{•}(M ⊕ M) of A-algebras. We let Δ : Λ_{A}^{•}M → Λ_{A}^{•}M ⊗_{A} Λ_{A}^{•}N be the composition of the morphism Λ_{A}^{•}δ and the inverse of the isomorphism (4.3.18). For every a, b ≥ 0 we also let Δ_{a,b} : Λ_{A}^{•}M → Λ_{A}^{a}M ⊗_{A} Λ_{A}^{b}N be the composition of Δ and the projection onto the graded component of bidegree (a, b). The morphisms Δ_{a,b} are usually called ”co-multiplication morphisms”. An easy calculation shows that:

(4.3.21)  Δ_{a,b}(x_{1} ∧ x_{2} ∧ ... ∧ x_{a+b}) = ∑_{I,J} ε_{IJ} · x_{I} ⊗ x_{J}

where the sum ranges over all the partitions a + b = I ∪ J such that |I| = a. Let now x_{1}, ..., x_{a}, y_{1}, ..., y_{b} ∈ M_{∗}. Since Δ is a morphism of A-algebras, we have Δ(x_{a} ∧ y_{b}) = Δ(x_{a}) · Δ(y_{b}). Hence, using (4.3.19)and (4.3.21)one deduces easily:

(4.3.22)  Δ_{a,b}(x_{a} ∧ y_{b}) = ∑_{I,J,K,L} ε_{IJ} · ε_{KL} · (−1)^{|J|} · (x_{I} ∧ y_{K}) ⊗ (x_{J} ∧ y_{L}).

where the sum runs over all partitions I ∪ J = a, K ∪ L = b such that |J| = |K|. Lemma 4.3.23. Suppose that Λ_{A}^{a+1}M = 0 for some integer a ≥ 0. Let 0 < b ≤ a and x_{1}, ..., x_{a}, y_{1}, ..., y_{b} ∈ M_{∗}. Then the following identity holds in Λ_{A}^{a}M ⊗_{A} Λ_{A}^{b}M :

x_{a} ⊗ y_{b} = ∑_{I,J} ε_{JI} · (x_{J} ∧ y_{b}) ⊗ x_{I}

where the sum ranges over all the partitions a = I ∪ J such that |I| = b.

Proof. For a given subset B ⊂ b we let

γ(y_{B}) := ∑_{I,J} ε_{IJ} · (x_{I} ∧ y_{B}) ⊗ x_{J} − x_{a} ⊗ y_{B}

where the sum is taken over all the partitions I ∪ J = a such that |J| = |B|. Notice that γ(y_{∅}) = 0. We have to show that γ(y_{b}) = 0. To this purpose we show the following: Claim 4.3.24. If |B| > 0, then

(4.3.25)  Δ_{a,|B|}(x_{a} ∧ y_{B}) = ∑_{K,L} ε_{KL} · (−1)^{|K|} · γ(y_{K}) ∧ y_{L}

where the sum ranges over all the partitions K ∪ L = B. Proof of the claim: Using (4.3.22), the difference between the two sides of (4.3.25)is seen to be equal to ∑_{K,L} ε_{KL} · (−1)^{|K|} · x_{a} ⊗ (y_{K} ∧ y_{L}) = ∑_{K,L} ·(−1)^{|K|} · x_{a} ⊗ y_{B}, where the sum runs over all partitions K ∪ L = B. A standard combinatorial argument shows that this expression can be rewritten as x_{a} ⊗ y_{B} · ∑_{k=0}^{|B|}(−1)^{k} · \binom{|B|}{k}, which vanishes if |B| > 0.

To conclude the proof of the lemma, we remark that Δ_{a,b} vanishes if b > 0 because by assumption Λ_{A}^{a+1}M = 0; then the claim follows by induction on |B|, using claim 4.3.24. □

Lemma 4.3.26. Let P be an A-module such that Λ_{A}^{n+1}P = 0 and assume that either P is flat or 2 is invertible in A_{∗}. Then Λ_{A}^{2}(Λ_{A}^{n}P) = 0.

Proof. For any A-module M and r ≥ 0 there exists an antisymmetrizer operator (cp. [15, Ch. III, §7.4, Remarque])

a_{r} : M^{⊗r} → M^{⊗r} m_{1} ⊗ ... ⊗ m_{r} ↦ ∑_{σ∈S_{r}} sgn(σ) · m_{σ(1)} ⊗ ... ⊗ m_{σ(r)}.

Clearly a_{r} factors thorugh Λ_{A}^{r}M, and in case M is free of finite rank, it is easy to check (just by arguing with basis elements) that the induced map \overline{a}_{r} : Λ_{A}^{r}M → Im(a_{r}) is an isomorphism. This is still true also in case r! is invertible in A_{∗}, since in that case one checks that a_{r}/r! is idempotent (see loc. cit.). More generally, if M is flat then, by [51, Ch.I, Th.1.2], M_{!} is the filtered colimit of a direct system of free A_{∗}-modules of finite rank, so also in this case \overline{a}_{r} is an isomorphism. Notice that, again by [51, Ch.I, Th.1.2], if P is flat, then Λ_{A}^{k}P is also flat, for every k ≥ 0. Hence, to prove the lemma, it suffices to verify that Im(a_{2} : (Λ_{A}^{n}P)^{⊗2} → (Λ_{A}^{n}P)^{⊗2}) = 0 when Λ_{A}^{n+1}P = 0. However, this follows easily from lemma 4.3.23. □

We are now ready to return to A-modules of finite rank. Proposition 4.3.27. Let P be an almost projective A-module of finite rank; say that Λ_{A}^{r}P = 0. There exists a natural decomposition A ≃ A_{0} × A_{1} × ... × A_{r−1} such that P_{i} := P ⊗_{A} A_{i} is an A_{i}-module of constant rank equal to i for every i = 0, ..., r − 1.

Proof. We proceed by induction on r; the case r = 2 is covered by lemma 4.3.12. By lemma 4.3.26 we have Λ_{A}^{2}(Λ_{A}^{r−1}P) = 0, so by lemma 4.3.12, there is a decomposition A ≃ A′_{r−1} × A′_{r−2}, such that for P_{i} := P ⊗_{A} A′_{i} (i = r − 2, r − 1) the following holds. Λ_{A′_{r−2}}^{r−1}(P_{r−2}) = 0 and Λ_{A′_{r−1}}^{r−1}(P_{r−1}) is an invertible A_{r−1}-module. It follows in particular that χ_{P_{r−1}}(X) is a polynomial of degree r −1, and its leading coefficient is invertible in A_{r−1}. Hence χ_{P_{r−1}}(X) = (1+X)^{r−1}. By induction, A′_{r−2} admits a decomposition A′_{r−2} ≃ A_{r−2} × ... × A_{0} with the stated properties; it suffices then to take A_{r−1} := A′_{r−1}. □

Theorem 4.3.28. Let P be an almost projective A-module of almost finite rank. Then there exists a natural decomposition A ≃ ∏_{i=0}^{∞} A_{i} such that:

(i) lim_{i→∞}Ann_{V^{a}}(A_{i}) = V^{a} (for the uniform structure of definition 2.3.1); (ii) for i ∈ ℕ, let P_{i} := P ⊗_{A} A_{i}; then P ≃ ∏_{i=0}^{∞} P_{i} and every P_{i} is an A_{i}-module of finite

constant rank equal to i.

Proof. Let {𝔪_{λ}}_{λ∈I} be the filtered family of finitely generated subideals of 𝔪. For every λ ∈ I, let A_{λ} := A/Ann_{A}(𝔪_{λ}). By hypothesis, P_{λ} := P ⊗_{A}A_{λ} is an A_{λ}-module of finite rank; say that the rank is r(λ). By proposition 4.3.27 we have natural decompositions A_{λ} ≃ A_{λ,0}×...×A_{λ,r(λ)} such that P ⊗_{A} A_{λ,i} is an A_{λ,i}-module of constant rank equal to i for every i ≤ r(λ). The naturality of the decomposition means that for every λ, μ ∈ I such that 𝔪_{λ} ⊂ 𝔪_{μ}, we have A_{μ,i} ⊗_{A} A_{λ} ≃ A_{λ,i}. for every i ≤ r(μ). In particular, 𝔪_{λ}A_{μ,i} = 0 for every i > r(λ). By considering the short exact sequence of cofiltered systems of A-modules:

0 → (Ann_{A}(𝔪_{λ}))_{λ∈I} → (A)_{λ∈I} → (A_{λ})_{λ∈I} → 0

we deduce easily that A ≃ lim_{λ∈I}A_{λ} and therefore we obtain a decomposition A ≃ ∏_{i=0}^{∞} A_{i}, with A_{i} := lim_{λ∈I}A_{λ,i} for every i ∈ ℕ. Notice that, for every i ∈ ℕ and every λ ∈ I, the natural morphism A_{i} → A_{λ,i} is surjective with kernel killed by 𝔪_{λ}. It follows easily that Λ_{A_{i}}^{i+1}(P ⊗_{A} A_{i}) = 0 and Λ_{A_{i}}^{i}(P ⊗_{A} A_{i}) is an invertible A_{i}-module. Furthermore, for every λ ∈ I, 𝔪_{λ} · A_{i} = 0 for all i > r(λ), which implies (i). Finally, for every λ ∈ I, 𝔪_{λ} kills the kernel of the projection P → ∏_{i=0}^{r(λ)} P_{i}, so P is isomorphic to the infinite product of the P_{i}. □

4.4. Localisation in the flat site. Throughout this section P denotes an almost finitely generated projective A-module. The following definition introduces the main tool used in this section. Definition 4.4.1. The splitting algebra of P is the A-algebra:

Split(A, P) := Sym_{A}^{•}(P ⊕ P^{∗})/(1 − ζ_{P}).

We endow Sym_{A}^{•}(P ⊕ P^{∗}) with the structure of graded algebra such that P is placed in degree one and P^{∗} in degree −1. Then ζ_{P} is a homogeneous element of degree zero, and consequently Split(A, P) is also a graded A-algebra.

4.4.2. We define a functor S : A-Alg → Set by assigning to every A-algebra B the set S(B) of all pairs (x, ϕ) where x ∈ (P ⊗_{A} B)_{∗}, ϕ : P ⊗_{A} B → B such that ϕ(x) = 1. Lemma 4.4.3. (i) Split(A, P) is a flat A-algebra.

(ii) Split(A′, P ⊗_{A} A′) ≃ Split(A, P) ⊗_{A} A′ for every A-algebra A′.

(iii) Split(A, P) represents the functor S.

Proof. For every k ≥ 0 we have ζ_{P}^{k} ∈ (Sym_{A}^{k}P) ⊗_{A} (Sym_{A}^{k}P^{∗}) ⊂ gr^{0}(Sym_{A}^{2k}(P ⊕ P^{∗})). It is easy to verify the formula:

(4.4.4)  gr^{k}Split(A, P) ≃ colim_{j∈ℤ}(Sym_{A}^{k+j}P) ⊗_{A} (Sym_{A}^{j}P^{∗})

where the transition maps in the direct system are given by multiplication by ζ_{P}. In particular, it is clear that gr^{k}Split(A, P) is a flat A-module, so (i) holds. (ii) is immediate. To show (iii), let us introduce the functor T : A-Alg → Set that assigns to every A-algebra B the set of all pairs (x, ϕ) where x ∈ (P ⊗_{A} B)_{∗} and ϕ : P ⊗_{A} B → B. So S is a subfunctor of T. Claim 4.4.5. The functor T is represented by the A-algebra Sym_{A}^{•}(P ⊕ P^{∗}). Proof of the claim: Indeed, there are natural bijections:

(P ⊗_{A} B)_{∗} ⟶^{∼} Hom_{B}(P^{∗} ⊗_{A} B, B) ⟶^{∼} Hom_{A}(P^{∗}, B) ⟶^{∼} Hom_{A-Alg}(Sym_{A}^{•}P^{∗}, B)

which show that the functor B ↦ (P ⊗_{A}B)_{∗} is represented by the A-algebra Sym_{A}P^{∗}. Working out the definitions, one finds that the composition of these bijections assigns to an element x ∈ (P ⊗_{A} B)_{∗} the unique A-algebra morphism f_{x} : Sym_{A}^{•}P^{∗} → B such that f_{x}(ψ) = ψ(x) for every ψ ∈ Hom_{A}(P, A). Similarly, the functor B ↦ Hom_{B}(P ⊗_{A} B, B) is represented by Sym_{A}^{•}P, and again, one checks that the bijection assigns to ϕ : P ⊗_{A} B → B the unique A-algebra morphism g_{ϕ} : Sym_{A}^{•}P → B such that g_{ϕ}(p) = ϕ(p) for every p ∈ P_{∗}. It follows that T is represented by (Sym_{A}^{•}P) ⊗_{A} (Sym_{A}^{•}P^{∗}) ≃ Sym_{A}^{•}(P ⊕ P^{∗}).

For every A-algebra B we have a natural map: α_{B} : T (B) → B_{∗} given by (x, ϕ) ↦ ϕ(x). This defines a natural transformation of functors α : T → (−)_{∗}. Moreover, let us consider the trivial map β_{B} : T (B) → B_{∗} that sends everything onto the element 1 ∈ B_{∗}. β is another natural transformation from T to the almost elements functor. Clearly :

S(B) = Equal( \xymatrix{T (B) \ar[r]^{α_{B}} \ar[r]_{β_{B}} & B_{∗}} ).

We remark that the functor B ↦ B_{∗} on A-algebras, is represented by A[X] := Sym_{A}^{•}A. Therefore there are morphisms α^{∗}, β^{∗} : A[X] → Sym_{A}^{•}(P ⊕ P^{∗}) that represent these natural transformations. It follows that S is represented by the A-algebra

Coequal( \xymatrix{A[X] \ar[r]^{α^{∗}} \ar[r]_{β^{∗}} & Sym_{A}^{•}(P ⊕ P^{∗})} ).

To determine α^{∗} and β^{∗} it suffices to calculate them on the element X ∈ A[X]_{∗}. It is easy to see that α^{∗}(X) = 1. To conclude the proof it suffices therefore to show: Claim 4.4.6. β^{∗}(X) = ζ_{P}. Proof of the claim: In view of the definitions, and using the notation of the proof of claim 4.4.5, the claim amounts to the identity: (g_{ϕ} ⊗ f_{x})(ζ_{P}) = ϕ(x) for every (x, ϕ) ∈ T (B). By naturality, it suffices to show this for B = A. Now, for every ε ∈ 𝔪, we can write ε · ζ_{P} = ∑_{i} q_{i} ⊗ ψ_{i} for some q_{i} ∈ P, ψ_{i} ∈ P^{∗} and we have ∑_{i} q_{i} · ψ_{i}(b) = ε · b for all b ∈ P_{∗}. Hence (g_{ϕ} ⊗ f_{x})(∑_{i} q_{i} ⊗ ψ_{i}) = ∑_{i} ϕ(q_{i}) · ψ_{i}(x) = ϕ(∑_{i} q_{i} · ψ_{i}(x)) = ϕ(εx) and the claim follows.

□

Remark 4.4.7. The construction of the splitting algebra occurs already, in a tannakian context, in Deligne’s paper [23] : see the proof of lemma 7.15 in loc.cit.

4.4.8. We recall that for every k ≥ 0 there are natural morphisms

Sym_{A}^{k}P ⟶^{α_{P}} Γ_{A}^{k}P ⟶^{β_{P}} Sym_{A}^{k}P

such that β_{P} ∘ α_{P} = k! · 1_{Sym_{A}^{k}P} and α_{P} ∘ β_{P} = k! · 1_{Γ_{A}^{k}P}. (to obtain the morphisms, one can consider the flat A_{∗}-module P_{!}, thus one can assume that P is a module over a usual ring; then α_{P} is obtained by extending multiplicatively the identity morphism Sym_{A}^{1}P = P → P = Γ_{A}^{1}P, and β_{P} is deduced from the homogeneous degree k polynomial law P ⊗_{A} B → (P ⊗_{A} B)^{⊗k} defined by x ↦ x^{⊗k}). Moreover (Γ_{A}^{k}P)^{∗} ≃ Sym_{A}^{k}P^{∗}. Lemma 4.4.9. With the notation of (4.4.8)we have: (α_{P} ⊗_{A} 1_{Sym_{A}^{k}P^{∗}})(ζ_{P}^{k}) = k! · ζ_{Γ_{A}^{k}P}.

Proof. Suppose first that P is a free A-module, let e_{1}, ..., e_{n} be a base of P_{∗} and e_{1}^{∗}, ..., e_{n}^{∗} the dual base of P^{∗}. Then Γ_{A}^{k}P is the free A-module generated by the basis e_{1}^{[n_{1}]} · ... · e_{k}^{[n_{k}]} where 0 ≤ n_{i} ≤ k for i = 1, ..., k and ∑_{j} n_{j} = k. The dual of this basis is the basis of Sym_{A}^{k}P^{∗} consisting of the elements e_{1}^{∗n_{1}} · ... · e_{k}^{∗n_{k}}. Furthermore, ζ_{P} = ∑_{i} e_{i} ⊗ e_{i}^{∗} and therefore ζ_{P}^{k} = ∑\underline{n} \binom{k}{\underline{n}}(e_{1}^{n_{1}} · ... · e_{k}^{n_{k}}) ⊗ (e_{1}^{∗n_{1}} · ... · e_{k}^{∗n_{k}}), where \underline{n} := (n_{1}, ..., n_{k}) ranges over the multi-indices submitted to the above conditions and \binom{k}{\underline{n}} := \frac{k!}{n_{1}!·...·n_{k}!}. Then the claim follows straightforwardly

from the identity: α_{P} (e_{1}^{n_{1}} · ... · e_{k}^{n_{k}}) = n_{1}! · ... · n_{k}! · e_{1}^{[n_{1}]} · ... · e_{k}^{[n_{k}]}. For the general case we shall use the following Claim 4.4.10. Let M be an almost finitely generated projective A-module and pick, for a given ε ∈ 𝔪, morphisms u : M → F and v : F → M with v ∘ u = ε · 1_{M}. Then v ⊗ u^{∗}(ζ_{F}) = ε · ζ_{M}.

Proof of the claim: We have a commutative diagram

\xymatrix{F ⊗_{A} F^{∗} \ar[r]^{v⊗u^{∗}} \ar[d] & M ⊗_{A} M^{∗} \ar[d] \\ End_{A}(F) \ar[r] & End_{A}(M)}

where the vertical morphisms are the natural ones, and where the bottom morphism is given by ϕ ↦ v ∘ ϕ ∘ u. Then the claim follows by an easy diagram chase.

Pick morphisms u : P → F and v : F → P with v∘u = ε·1_{P}. We consider the commutative diagram

\xymatrix{Sym_{A}^{k}F ⊗_{A} Sym_{A}^{k}F^{∗} \ar[r]^{Sym_{A}^{k}v⊗Sym_{A}^{k}u^{∗}} \ar[d]_{α_{F} ⊗1_{Sym^{k}_{A}F^{∗}}} & Sym_{A}^{k}P ⊗_{A} Sym_{A}^{k}P^{∗} \ar[d]^{α_{P} ⊗1_{Sym^{k}_{A}P^{∗}}} \\ Γ_{A}^{k}F ⊗_{A} Sym_{A}^{k}F^{∗} \ar[r]^{Γ_{A}^{k}v⊗Sym_{A}^{k}u^{∗}} & Γ_{A}^{k}P ⊗_{A} Sym_{A}^{k}P^{∗}.}

By claim 4.4.10, we have v ⊗ u^{∗}(ζ_{F}) = ε · ζ_{P}; whence Sym_{A}^{k}v ⊗ Sym_{A}^{k}u^{∗}(ζ_{F}^{k}) = ε^{k} · ζ_{P}^{k}. Moreover, we remark that (Γ_{A}^{k}v) ∘ (Sym_{A}^{k}u^{∗})^{∗} = (Γ_{A}^{k}v) ∘ (Γ_{A}^{k}u) = ε^{k} · 1_{Γ_{A}^{k}P}, therefore claim 4.4.10 (applied for M = Γ_{A}^{k}P) yields Γ_{A}^{k}v⊗Sym_{A}^{k}u^{∗}(ζ_{Γ_{A}^{k}F}) = ε^{k}·ζ_{Γ_{A}^{k}P}. Since we already know the lemma for F, a simple diagram chase shows that (α ⊗_{A} 1_{Sym_{A}^{k}P^{∗}})(ε^{k} · ζ_{P}^{k}) = ε^{k} · k! · ζ_{Γ_{A}^{k}P}. Since the k-powers of elements of 𝔪 generate 𝔪, the claim follows. □

Lemma 4.4.11. Let P be as above and suppose that ζ_{P} is nilpotent in Sym_{A}^{•}(P ⊕ P^{∗}). (i) If ℚ ⊂ A_{∗}, then χ_{P} (X) = (1 + X)^{−α} for some continuous function α : Spec A_{∗} → ℕ.

(ii) If 𝔽_{p} ⊂ A_{∗}, then ℰ_{P/A} is a Frobenius-nilpotent ideal.

Proof. (i): Since ℚ ⊂ A_{∗}, then k! is invertible in A_{∗} for every k ≥ 0; by lemma 4.4.9 it follows that Sym_{A}^{k}P = 0, thus ψ_{P} (X) is a polynomial. By corollary 4.3.8, ψ_{P} (−X) defines an endomorphism of 𝔾_{m,A_{∗}}, therefore ψ_{P} (X) = (1 + X)^{α} for some continuous function α : Spec A_{∗} → ℕ; then the claim follows by corollary 4.3.8(ii).

(ii): Let (f, p) ∈ (P^{∗} ⊕ P)_{∗}; in the notation of the proof of lemma 4.4.3, we can write (f, p) ∈ T (A). It follows that (f, p) corresponds to a morphism of A-algebras Sym_{A}^{•}(f, p)^{∗} : Sym_{A}^{•}(P ⊕ P^{∗}) → A. In particular Sym_{A}^{•}(f, p)^{∗}(ζ_{P}^{k}) = Sym_{A}^{•}(f, p)^{∗}(ζ_{P})^{k} for every k ≥ 0. By inspecting the proof of claim 4.4.6, we deduce Sym_{A}^{•}(f, p)^{∗}(ζ_{P}) = f(p) for every f ∈ P^{∗} and p ∈ P. By hypothesis, ζ_{P}^{p^{n}} = 0 for every sufficiently large n. It follows that ℰ_{P/A} is Frobenius nilpotent. □

Lemma 4.4.12. Let R_{0} be a noetherian commutative ring, R an R_{0}-algebra and M a flat Rmodule. Then M = 0 if and only if M ⊗_{R_{0}} κ = 0 for every residue field κ of R_{0} (i.e., for every field κ of the form Frac(R_{0}/𝔭), where 𝔭 is some prime ideal of R_{0}).

Proof. Clearly we have only to show the direction ⇐. It suffices to show that M_{𝔭} = 0 for every prime ideal of R_{0}. Hence we can assume that R_{0} is local, in particular of finite Krull dimension. We proceed by induction on the dimension of R_{0}. If dim R_{0} = 0, then R_{0} is a local artinian ring, hence a power of its maximal ideal 𝔪 is equal to 0. By assumption, M/𝔪M = 0, i.e. M = 𝔪M. Then M = 𝔪^{k}M for every k ≥ 0, so M = 0. Next, suppose that dim R_{0} = d and the lemma already known for all rings of dimension strictly less than d. Assume first that R_{0} is an integral domain and pick f ∈ S := R_{0} \ {0}. Then R_{0}/fR_{0} has dimension strictly less than d, so by induction we have M/fM = 0, i.e. M = fM. Due to the flatness of M, we have: Ann_{M}(f) = Ann_{R}(f) · M = Ann_{R}(f) · fM = 0. This implies that the kernel of the natural map M → S^{−1}M is trivial. On the other hand, by hypothesis S^{−1}M = 0, whence M = 0 in this case. For a general R_{0} of dimension d, notice that the above argument implies that, for every minimal prime ideal 𝔭 of R_{0}, we have 𝔭M = M. But the product of all (finitely many) minimal prime ideals is contained in the nilpotent radical ℜ of R_{0}, whence ℜM = M, and finally M = 0 as claimed. □

4.4.13. Let now 𝔭 ∈ Spec A_{∗}. By composing f.rk_{A}(P) with the map A_{∗} → A_{∗}/𝔭, we obtain a ring homomorphism

f.rk_{A}(P, 𝔭) : 𝒢_{∞} → A_{∗}/𝔭.

In case (A_{∗}/𝔭)^{a} ≠ 0, we can interpret f.rk_{A}(P, 𝔭) as the formal rank of P ⊗_{A} (A/𝔭^{a}). More precisely, let π : A → A/𝔭^{a} be the natural projection; then π_{∗} factors through a map π′ : A_{∗}/𝔭 → (A/𝔭^{a})_{∗} and we have: f.rk_{A/𝔭^{a}}(P ⊗_{A} (A/𝔭^{a})) = π′ ∘ f.rk_{A}(P, 𝔭).

Even if (A_{∗}/𝔭)^{a} = 0, the morphism f.rk_{A}(P, 𝔭) can still be interpreted as the map associated to an endomorphism of 𝔾̂_{m,A_{∗}/𝔭}, so it still makes sense to ask whether f.rk_{A}(P, 𝔭) is an integer, as indicated in remark 4.3.10(ii).

Lemma 4.4.14. Let P be an almost finitely generated projective A-module and 𝔭 ∈ Spec A_{∗}. If B is any A_{∗𝔭}^{a}-algebra and 𝔮 ∈ Spec B_{∗} such that r(𝔮) := f.rk_{B}(P ⊗_{A} B, 𝔮) is an integer, then r(𝔭) := f.rk_{A}(P, 𝔭) is also an integer and r(𝔭) = r(𝔮).

Proof. Indeed, let us consider the natural maps 𝒢_{∞} → A_{∗} → B_{∗}; under the assumptions, the contraction of 𝔮 in A_{∗} is contained in 𝔭. Since the image of \binom{α}{i} in B_{∗}/𝔮 is \binom{r(𝔮)}{i}, it follows that the same holds in A_{∗}/𝔭. □

Definition 4.4.15. We say that an A-module P admits infinite splittings if there is an infinite chains of decompositions of the form: P ≃ A ⊕ P_{1}, P_{1} ≃ A ⊕ P_{2}, P_{2} ≃ A ⊕ P_{3}, ...

Theorem 4.4.16. Let P be an almost finitely generated projective A-module. The following conditions are equivalent:

(i) P is of almost finite rank.

(ii) For all A-algebras B ≠ 0, we have: ⋂_{r>0} ℰ_{Λ_{B}^{r}(P ⊗_{A}B)/B} = 0.

(iii) For all A-algebras B ≠ 0, P_{B} := P ⊗_{A} B does not admit infinite splittings, and moreover if P_{B} ≃ B^{n} ⊕ Q for some B-module Q and χ_{Q}(X) = (1 + X)^{−α} for some continuous function α : Spec B_{∗} → ℕ, then Q = 0.

(iv) For all A-algebras B ≠ 0, P_{B} does not admit infinite splittings, and moreover if P_{B} ≃ B^{n} ⊕ Q for some B-module Q, then :

(a) If 𝔽_{p} ⊂ B_{∗} and Q = IQ for a Frobenius-nilpotent ideal I ⊂ B, then Q = 0;

(b) If ℚ ⊂ B_{∗} and Sym_{B}^{r}Q = 0 for some r ≥ 1, then Q = 0.

Proof. (i) ⇒ (ii) : indeed, from proposition 2.4.28(iii) one sees that, for every A-module of almost finite rank, every A-algebra B and every ε ∈ 𝔪, there exists r ≥ 0 such that ε · ℰ_{Λ_{B}^{r}(P ⊗_{A}B)/B} = 0.

(ii) ⇒ (iii) : let B ≠ 0 be an A-algebra; by hypothesis, there exists r ≥ 0 such that J_{r} := ℰ_{Λ_{B}^{r}(P ⊗_{A}B)/B} ≠ B. Suppose that P ⊗_{A}B admits infinite splittings. The B/J_{r}-module P_{B}/J_{r}P_{B} has rank < r, and at the same time it admits infinite splittings, a contradiction.

Suppose next, that there is a decomposition P ⊗_{A} B ≃ B^{n} ⊕Q; then Q is obviously of almost finite rank. Suppose that χ_{Q}(X) = (1 + X)^{α} has the shape described in (ii). We reduce easily to the case where α is a constant function. However, χ_{Q/J_{r}Q}(X) is a polynomial of degree < r, thus α = 0, and then Q = 0 by theorem 4.3.28(ii).

(iii) ⇒ (iv) : suppose that 𝔽_{p} ⊂ B_{∗} and Q = IQ for some Frobenius-nilpotent ideal I. Then χ_{Q}(X) ∈ 1 + I_{∗}[[X]], which means that the image of χ_{Q}(X) in End(𝔾̂_{m,B_{∗}/I_{∗}}) is the trivial endomorphism. But we have a commutative diagram

\xymatrix{𝒞^{0}(Spec B_{∗}, ℤ_{p}) \ar[r] \ar[d] & End(𝔾̂_{m,B_{∗}}) \ar[d] \\ 𝒞^{0}(Spec B_{∗}/I_{∗}, ℤ_{p}) \ar[r] & End(𝔾̂_{m,B_{∗}/I_{∗}})}

where the horizontal maps are those defined in remark 4.3.10(ii), and are bijective by corollary 4.2.22. The left vertical map is induced by restriction to the closed subset Spec B_{∗}/I_{∗}, and since I is Frobenius-nilpotent, it is a bijection as well. It follows that the right vertical map is bijective, whence f.rk_{B}(Q) = 0, and finally Q = 0 by (iii).

Next, consider the case when ℚ ⊂ B_{∗} and Sym_{B}^{r}Q = 0 for some r ≥ 1. It follows that ζ_{Q} is nilpotent in Sym_{B}^{•}(Q ⊕ Q^{∗}). By lemma 4.4.11(i), χ_{Q}(X) = (1 + X)^{−α} for some continuous α : Spec B_{∗} → ℕ. Then (iii) implies that Q = 0.

To show that (iv) ⇒ (i), we will use the following: Claim 4.4.17. Assume (iv). Then: Split(B, Q) = 0 ⇒ Q = 0. Proof of the claim: Suppose Q ≠ 0 and Split(B, Q) = 0; then (4.4.4) implies that for every ε ∈ 𝔪 there exists j ≥ 0 such that ε · ζ_{Q}^{j} = 0. We have εQ ≠ 0 for some ε ∈ 𝔪. From the flatness of Q, we derive Ann_{Sym_{B}^{•}(Q⊕Q^{∗})}(ε) = Ann_{B}(ε) · Sym_{B}^{•}(Q ⊕ Q^{∗}), hence we can replace B by B/Ann_{B}(ε), Q by Q/Ann_{B}(ε) · Q, thereby achieving that ζ_{Q} is nilpotent in Sym_{B}^{•}(Q ⊕ Q^{∗}) and still Q ≠ 0. Using lemma 4.4.12 (and the functoriality of Split(B, Q) for base extensions B → B′) we can further assume that B_{∗} contains either ℚ or one of the finite fields 𝔽_{p}. If ℚ ⊂ B_{∗}, then k! is invertible in B_{∗} for every k ≥ 0; by lemma 4.4.9 it follows that Sym_{B}^{k}Q = 0, whence Q = 0 by (iv), a contradiction. If 𝔽_{p} ⊂ B_{∗}, then by lemma 4.4.11(ii), ℰ_{Q/B} is Frobenius-nilpotent. However, from proposition 2.4.28(i) it follows easily that Q = ℰ_{Q/B} · Q, whence Q = 0, again by (iv), and again a contradiction. In either case, this shows that Split(B, Q) ≠ 0, as claimed.

Claim 4.4.18. Assume (iv). Then A/ℰ_{P/A} is a flat A-algebra. Proof of the claim: It suffices to show that A_{𝔭}/(ℰ_{P/A})_{𝔭} is a flat A_{𝔭}-algebra for every prime ideal 𝔭 ⊂ A_{∗}. If ℰ_{P/A∗} is not contained in 𝔭, then (ℰ_{P/A})_{𝔭} = A_{𝔭}, so there is nothing to prove in this case. We assume therefore that

(4.4.19)  ℰ_{P/A∗} ⊂ 𝔭.

We will show that P_{𝔭} = 0 in such case, whence A_{𝔭}/(ℰ_{P/A})_{𝔭} = A_{𝔭}, so the claim will follow. From (4.4.19)and corollary 4.3.8(iii) we know already that

(4.4.20)  f.rk_{A}(P, 𝔭) = 0.

Suppose that P_{𝔭} ≠ 0; then there exists ε ∈ 𝔪 such that εP_{𝔭} ≠ 0. Define inductively A_{0} := A_{𝔭}, Q_{0} := P_{𝔭}, A_{i+1} := Split(A_{i}, Q_{i}) and Q_{i+1} as an A_{i+1}-module such that Q_{i} ⊗_{A_{i}} A_{i+1} ≃ A_{i+1} ⊕ Q_{i+1}, for every i ≥ 0 (the existence of Q_{i+1} is assured by lemma 4.4.3(iii)). Then colim_{n∈ℕ}A_{n} ≃ 0, since, after base change to this V^{a}-algebra, P admits infinite splittings. This implies that there exists n ∈ ℕ such that εA_{n+1} = 0 and εA_{n} ≠ 0. However, since A_{n+1} is flat over A_{n}, we have: A_{n+1} = Ann_{A_{n+1}}(ε) = Ann_{A_{n}}(ε) · A_{n+1}. Set A′ := A_{n}/Ann_{A_{n}}(ε); then Split(A′, Q_{n} ⊗_{A_{n}} A′) = 0, so Q_{n} ⊗_{A_{n}} A′ = 0 by claim 4.4.17. By flatness of Q_{n}, this means that εQ_{n} = 0; in particular, n > 0. By definition, Q_{n−1} ⊗_{A_{n−1}} A_{n} ≃ A_{n} ⊕ Q_{n}; it follows that Q′ := Q_{n−1} ⊗_{A_{n−1}} A′ ≃ A′, in particular f.rk_{A′}(Q′) = 1 and consequently f.rk_{A′}(P ⊗_{A} A′) = n > 0; in view of lemma 4.4.14, this contradicts (4.4.20), therefore P_{𝔭} = 0, as required.

Claim 4.4.21. Assuming (iv), the natural morphism ϕ : A → (A/ℰ_{P/A}) × Split(A, P) is faithfully flat. Proof of the claim: The flatness is clear from claim 4.4.18. Hence, to prove the claim, it suffices to show that ((A/ℰ_{P/A}) × Split(A, P)) ⊗_{A} (A/J) ≠ 0 for every proper ideal J ⊂ A,. But the construction of ϕ commutes with arbitrary base changes A → A′, therefore we are reduced to verify that (A/ℰ_{P/A}) × Split(A, P) ≠ 0 when A ≠ 0. By claim 4.4.17, this can fail only when P = 0; but in this case ℰ_{P/A} = 0, so the claim follows.

We can now conclude the proof of the theorem: define inductively as in the proof of claim 4.4.18: A_{0} := A, Q_{0} := P, A_{i+1} := Split(A_{i}, Q_{i}) and Q_{i+1} as an A_{i+1}-module such that Q_{i}⊗_{A_{i}} A_{i+1} = A_{i+1} ⊕ Q_{i+1}, for every i ≥ 0. The same argument as in loc. cit. shows that, for every ε ∈ 𝔪, there exists n ∈ ℕ such that εA_{n} = 0. We may assume that εA_{n−1} ≠ 0. Moreover, by claim 4.4.21 (and an easy induction), B := A_{0}/ℰ_{Q_{0}/A_{0}}×A_{1}/ℰ_{Q_{1}/A_{1}}×...×A_{n−1}/ℰ_{Q_{n−1}/A_{n−1}}×A_{n} is a faithfully flat A-algebra. However, one checks easily by induction that P ⊗_{A} (A_{i}/ℰ_{Q_{i}/A_{i}}) is a free A_{i}/ℰ_{Q_{i}/A_{i}}-module of rank i, for every i < n. Hence, Λ_{B}^{n}(P ⊗_{A} B) ≃ Λ_{A_{n}}^{n}(P ⊗_{A} A_{n}), which is therefore killed by ε. By faithful flatness, so is Λ_{A}^{n}P. The proof is concluded. □

Proposition 4.4.22. If P is a faithfully flat almost projective A-module of almost finite rank, then Split(A, P) is faithfully flat over A.

Proof. If A = 0 there is nothing to prove, so we assume that A ≠ 0. In this case, it suffices to show that Split(A, P) ⊗_{A} A/I ≠ 0 for every proper ideal I of A. However, Split(A, P) ⊗_{A} A/I ≃ Split(A/I, P/IP), and since P is faithfully flat, P/IP ≠ 0; hence we are reduced to showing that Split(A, P) ≠ 0 when P is faithfully flat. Suppose that Split(A, P) = 0; then (4.4.4) implies that for every ε ∈ 𝔪 there exists j ≥ 0 such that ε · ζ_{P}^{j} = 0. Since P ≠ 0, we have εP ≠ 0 for some ε ∈ 𝔪. From the flatness of P, we derive Ann_{Sym_{A}^{•}(P ⊕P^{∗})}(ε) = Ann_{A}(ε)·Sym_{A}^{•}(P ⊕P^{∗}), hence we can replace A by A/Ann_{A}(ε), P by P/Ann_{A}(ε)·P, which allows us to assume that ζ_{P} is nilpotent in Split(A, P). Using lemmata 4.4.12 and 4.4.3(ii), we can further assume that A_{∗} contains either ℚ or one of the finite fields 𝔽_{p}. If ℚ ⊂ A_{∗}, then k! is invertible in A_{∗} for every k ≥ 0; by lemma 4.4.9 it follows that Sym_{A}^{k}P = 0, whence P = 0 by theorem 4.4.16(iv), which contradicts our assumptions, so the proposition is proved in this case. Finally, suppose that 𝔽_{p} ⊂ A_{∗}, then by lemma 4.4.11(ii), ℰ_{P/A} is Frobenius-nilpotent. However, since P is faithfully flat, proposition 2.4.28(iv) says that ℰ_{P/A} = A, so A = 0, which again contradicts our assumptions. □

4.4.23. For any V^{a}-algebra A we have a (large) fpqc site on the category (A-Alg)^{o} (in some fixed universe!); as usual, this site is defined by the pretopology whose covering families are the finite families {Spec C_{i} → Spec B | i = 1, ..., n} such that the induced morphism B → C_{1} × ... × C_{n} is faithfully flat (notation of (3.3.3)). Theorem 4.4.24. Every almost projective A-module of finite rank is locally free of finite rank in the fpqc topology of (A-Alg)^{o}.

Proof. We iterate the construction of Split(A, P) to split off successive free submodules of rank one. We use the previous characterization of modules of finite rank (proposition 4.3.27) to show that this procedure stops after finitely many iterations. By proposition 4.4.22, the output of this procedure is a faithfully flat A-algebra. □

Theorem 4.4.24 allows to prove easily results on almost projective modules of finite rank, by reduction to the case of free modules. Here are a few examples of this method. Lemma 4.4.25. Let P be an almost projective A-module of constant rank equal to r ∈ ℕ. Then, for every integer 0 ≤ k ≤ r, the natural morphism

(4.4.26)  Λ_{A}^{k}P ⊗_{A} Λ_{A}^{r−k}P → Λ_{A}^{r}P  x ⊗ y ↦ x ∧ y

is a perfect pairing.

Proof. By theorem 4.4.24, there exists a faithfully flat A-algebra B such that P_{B} := P ⊗_{A} B is a free B-module of rank r. It suffices to prove the assertion for the B-module P_{B}, in which case the claim is well known. □

4.4.27. Keep the assumptions of lemma 4.4.25. Taking k = 1 in (4.4.26), we derive a natural isomorphism

β_{P} : (Λ_{A}^{r−1}P)^{∗} →^{∼} P ⊗_{A} (Λ_{A}^{r}P)^{∗}.

Now, let us consider an A-linear morphism ϕ : P → Q of A-modules of constant rank equal to r. We set

ψ := β_{P} ∘ (Λ_{A}^{r−1}ϕ)^{∗} ∘ β_{Q}^{−1} : Q ⊗_{A} (Λ_{A}^{r}Q)^{∗} → P ⊗_{A} (Λ_{A}^{r}P)^{∗}.

Proposition 4.4.28. With the notation of (4.4.27), we have:

ψ ∘ (ϕ ⊗_{A} 1_{(Λ_{A}^{r}Q)^{∗}}) = 1_{P} ⊗_{A} (Λ_{A}^{r}ϕ)^{∗} and (ϕ ⊗_{A} 1_{(Λ_{A}^{r}P)^{∗}}) ∘ ψ = 1_{Q} ⊗_{A} (Λ_{A}^{r}ϕ)^{∗}.

Especially, ϕ is an isomorphism if and only if the same holds for Λ_{A}^{r}ϕ.

Proof. After faithfully flat base change, we can assume that P and Q are free modules of rank r. Then we recognize Cramer’s rule in the above identities. □

4.4.29. Keep the assumption of lemma 4.4.25 and let ϕ be an A-linear endomorphism of P. Set B := A[X, X^{−1}] and P_{B} := P ⊗_{A} B. Obviously ϕ induces a Λ-nilpotent endomorphism of P_{B}, which we denote by the same letter. Hence we can define

χ_{ϕ}(X) := X^{r} · det(1_{P_{B}} − X^{−1} · ϕ) ∈ A_{∗}[X]

(notation of (4.3.1)). Proposition 4.4.30. With the notation of (4.4.29), we have χ_{ϕ}(ϕ) = 0 in End_{A}(P).

Proof. Again, we can reduce to the case of a free A-module of constant finite rank, in which case we conclude by Cayley-Hamilton. □

Corollary 4.4.31. Keep the assumptions of (4.4.29), and suppose that ϕ is integral over a subring S ⊂ A_{∗}. Then the coefficients of χ_{ϕ} are integral over S.

Proof. The assumption means that ϕ satisfies an equation of the kind ϕ^{n} + ∑_{i=0}^{n−1} a_{i}ϕ^{i} = 0, where a_{i} ∈ S for every i = 0, ..., n − 1. We can assume that P is free, in which case we reduce to the case of an endomorphism of a free R-module of finite rank, where R is a usual commutative ring containing S; we can further suppose that R is of finite type over ℤ, and it is easily seen that we can even replace R by its associated reduced ring R/nil(R). Then R injects into a finite product of fields ∏_{i} K_{i}, and we can replace S by its integral closure in ∏_{i} K_{i}, which allows us to reduce to the case where R is a field. In this case the coefficients of χ_{ϕ} are elementary symmetric polynomials in the eigenvalues of ϕ, so it suffices to show that these eigenvalues e_{1}, ..., e_{r} are integral over S. But this is clear, since we have more precisely e_{j}^{n} + ∑_{i=0}^{n−1} a_{i}e^{i} = 0 for every j ≤ r. □

To conclude this section, we want to apply the previous results to analyze in some detail the structure of invertible modules : it turns out that the notion of invertibility is rather more subtle than for usual modules over rings. Definition 4.4.32. Let M be an invertible A-module. Clearly M ⊗_{A} M is invertible as well, consequently the map A → End_{A}(M ⊗_{A} M) → A : a ↦ a · 1_{M⊗_{A}M} is an isomorphism (by the proof of lemma 4.1.5(iii)). Especially, for the transposition endomorphism θ_{M|M} of M ⊗_{A} M : x ⊗ y ↦ y ⊗ x, there exists a unique element u_{M} ∈ A_{∗} such that θ_{M|M} = u_{M} · 1_{M⊗_{A}M}. Clearly u_{M}^{2} = 1. We say that M is strictly invertible if u_{M} = 1. Lemma 4.4.33. For an invertible A-module the following are equivalent:

(i) M is strictly invertible;

(ii) Λ_{A}^{2}M = 0;

(iii) M is of almost finite rank; (iv) there exists a faithfully flat A-algebra B such that M ⊗_{A} B ≃ B.

Proof. (i) ⇒ (ii): indeed, the condition u_{M} = 1 says that the antisymmetrizer operator a_{2} : M^{⊗2} → M^{⊗2} vanishes (cp. the proof of lemma 4.3.26); since M is flat, (ii) follows.

(ii) ⇒ (iii) and (iv) ⇒ (i) are obvious. To show that (iii) ⇒ (iv) let us set B := Split(A, M); by proposition 4.4.22 B is faithfully flat over A, and B ⊗_{A} M ≃ B ⊕ X for some B-module X. Clearly B ⊗_{A} M is an invertible B-module, therefore, by lemma 4.1.5(ii), the evaluation morphism gives an isomorphism (B ⊕X)⊗_{A} (B ⊕X^{∗}) ≃ B ⊕X ⊕X^{∗} ⊕(X ⊗_{A} X^{∗}) ≃ B. By inspection, the restriction of the latter morphism to the direct summand B equals the identity of B; hence X = 0 and (iv) follows. □

Lemma 4.4.34. If M is invertible, then tr_{M/A}(1_{M}) = u_{M}.

Proof. Pick arbitrary f ∈ M_{∗}^{∗}, m, n ∈ M_{∗}. Then, directly from the definition of u_{M} we deduce that f(m) · n = u_{M} · f(n) · m. In other words, ω_{M/A}(n ⊗ f) = u_{M} · ev_{M/A}(n ⊗ f) · 1_{M}. By linearity we deduce that ϕ = u_{M} · tr_{M/A}(ϕ) · 1_{M} for every ϕ ∈ End_{A}(M). By letting ϕ := 1_{M}, and taking traces on both sides, we obtain: tr_{M/A}(1_{M}) = u_{M} · tr_{M/A}(1_{M})^{2}. But since M is invertible, tr_{M/A}(1_{M}) is invertible in A_{∗}, whence u_{M} · tr_{M/A}(1_{M}) = 1, which is equivalent to the sought identity. □

Proposition 4.4.35. Let M be an invertible A-module. Then:

(i) M ⊗_{A} M is strictly invertible.

(ii) There exists a natural decomposition A ≃ A_{1} × A_{−1} where M ⊗_{A} A_{1} is strictly invertible, A_{−1∗} is a ℚ-algebra and Sym_{A_{−1}}^{2}(M ⊗_{A} A_{−1}) = 0.

Proof. (i): it is clear that M^{⊗n} is invertible for every n. Let σ ∈ S_{n} be any permutation; it is easy to verify that the morphism σ_{M} : M^{⊗n} → M^{⊗n} : x_{1} ⊗ x_{2} ⊗ ... ⊗ x_{n} ↦ x_{σ(1)} ⊗ x_{σ(2)} ⊗ ... ⊗ x_{σ(n)} equals u_{M}^{sgn(σ)} · 1_{M^{⊗n}}. Especially, the transposition operator on (M ⊗_{A} M)^{⊗2} acts via the permutation: x ⊗ y ⊗ z ⊗ w ↦ z ⊗ w ⊗ x ⊗ y whose sign is even. Therefore u_{M⊗_{A}M} = 1, which is (i).

(ii): It follows from (i) that the antisymmetrizer operator a_{2} on (M ⊗_{A} M)^{⊗2} vanishes; a fortiori it vanishes on the quotient (Λ_{A}^{2}M)^{⊗2}, therefore Λ_{A}^{2}(Λ_{A}^{2}M) ≃ Im(a_{2} : Λ_{A}^{2}M → Λ_{A}^{2}M) = 0. Then lemma 4.3.12 says that there exists a natural decomposition A ≃ A_{1} × A_{−1} such that (Λ_{A}^{2}M)⊗_{A}A_{1} = 0 and (Λ_{A}^{2}M)⊗_{A}A_{−1} is invertible. To show that A_{−1∗} is a ℚ-algebra, it is enough to show that A_{−1}/pA_{−1} = 0 for every prime p. Up to replacing A by A/pA, we reduce to verifying that, if 𝔽_{p} ⊂ A_{∗} and M is invertible, then M is of almost finite rank. To this aim, it suffices to verify that the equivalent condition (iv) of theorem 4.4.16 is satisfied. If B ≠ 0 and M_{B} := M ⊗_{A} B ≃ B ⊕ X, then the argument in the proof of lemma 4.4.33 shows that X = 0 and therefore M_{B} does not admit infinite splittings. Finally, it remains only to verify condition (a) of loc. cit. So suppose that M_{B} ≃ B^{n} ⊕ Q. If n > 0, we have just seen that Q = 0; if n = 0, and Q/IQ = 0 for some ideal I, then by the faithfulness of M (lemma 4.1.5(iii)) we must have I = B; if I is Frobenius nilpotent it follows that B = 0. Finally, set M_{−1} := M ⊗_{A} A_{−1}; notice that, since A_{−1∗} is a ℚ-algebra, the endomorphism group of 𝔾̂_{m,A_{−1∗}} is isomorphic to A_{−1∗}, and therefore χ_{M_{−1}}(X) = (1 + X)^{α}, where α is an element of A_{−1∗} which can be determined by looking at the coefficient of χ_{M_{−1}}(X) in degree 1. One finds α = tr_{M_{−1}/A_{−1}}(1_{M_{−1}}). In view of lemma 4.4.34, we can rewrite α = u_{M_{−1}}; therefore

(4.4.36)  tr_{Λ_{A_{−1}}^{2}M_{−1}/A_{−1}}(1_{Λ_{A_{−1}}^{2}M_{−1}}) = \binom{u_{M_{−1}}}{2}.

On the other hand, since Λ_{A_{−1}}^{2}M_{−1} is an invertible A_{−1}-module of finite rank, we know that the left-hand side of (4.4.36) equals 1; consequently u_{M_{−1}} = −1. This means that, in M_{−1}^{⊗2}, the identity x ⊗ y = −y ⊗ x holds for every x, y ∈ M_{−1∗}; therefore, the kernel of the projection M_{−1}^{⊗2} → Sym_{A_{−1}}^{2}M_{−1} contains all the elements of the form 2 · x ⊗ y; in other words, multiplication by 2 is the zero morphism in Sym_{A_{−1}}^{2}M_{−1}; since A_{−1∗} is a ℚ-algebra, this at last shows that Sym_{A_{−1}}^{2}M_{−1} vanishes, and concludes the proof of the proposition. □

4.5. Construction of quotients by flat equivalence relations. We will need to recall some generalities on groupoids, which we borrow from [25, Exp. V].

4.5.1. If 𝒞 is any category admitting fibred products and a final object, a 𝒞 -groupoid is the datum of two objects X_{0}, X_{1} of 𝒞, together with ”source” and ”target” morphisms s, t : X_{1} → X_{0}, an ”identity” morphism ι : X_{0} → X_{1} and a further ”composition” morphism c : X_{2} → X_{1}, where X_{2} is the fibre product in the cartesian diagram:

\xymatrix{X_{2} \ar[r]^{t′} \ar[d]_{s′} & X_{1} \ar[d]^{s} \\ X_{1} \ar[r]^{t} & X_{0}.}

The datum (X_{0}, X_{1}, s, t, c, ι) is subject to the following condition. For every object S of 𝒞, the set X_{0}(S) := Hom_{𝒞} (S, X_{0}) is a groupoid, with set of morphisms given by X_{1}(S), and for every ϕ ∈ X_{1}(S), the source and target of ϕ are respectively s(ϕ) := ϕ ∘ s and t(s) := ϕ ∘ s; furthermore the composition law in X_{0}(S) is given by c(S) : X_{1}(S) ×_{X_{0}(S)} X_{1}(S) → X_{1}(S). The above conditions amount to saying that s ∘ ι = t ∘ ι = 1_{X_{0}} and the commutative diagrams

(4.5.2)  \xymatrix{X_{2} \ar[r]^{c} \ar[r]_{t′} \ar[d]_{s′} & X_{1} \ar[d]^{s} & X_{2} \ar[r]^{c} \ar[d]_{t′} & X_{1} \ar[d]^{t} \\ X_{1} \ar[r]^{s} \ar[r]_{t} & X_{0} & X_{1} \ar[r]^{t} & X_{0}}

are cartesian in 𝒞 both for the square made up from the upper arrows and for the square made up from the lower arrows (cp. [25, Exp. V §1]).

One says that the groupoid G := (X_{0}, X_{1}, s, t, c, ι) has trivial automorphisms, if the morphism (s, t) : X_{1} → X_{0} × X_{0} is a (categorical) monomorphism. (This translates in categorical terms the requirement that for every object S of 𝒞, and every x ∈ X_{0}(S), the automorphism group of x in X_{0}(S) is trivial).

It is sometimes convenient to denote by X ×_{(α,β)} Z the fibre product of two morphisms α : X → Y and β : Z → Y.

4.5.3. Given a groupoid G, and a morphism X_{0} → X′_{0}, we obtain a new groupoid G ×_{X_{0}} X′_{0} by taking the datum (X′_{0}, X_{1} ×_{X_{0}} X′_{0}, s ×_{X_{0}} 1_{X′_{0}}, t ×_{X_{0}} 1_{X′_{0}}, c ×_{X_{0}} 1_{X′_{0}}, ι ×_{X_{0}} 1_{X′_{0}}).

Moreover, suppose that 𝒞 admits finite coproducts and that all such coproducts are disjoint universal (cp. [4, Exp.II, Def.4.5]). Denote by Y ⨿ Z the coproduct of two objects Y and Z of 𝒞. Let G′ := (X′_{0}, X′_{1}, s′, t′, c′, ι′) be another groupoid; one can define a groupoid G ⨿ G′ by taking the datum (X_{0} ⨿ X′_{0}, X_{1} ⨿ X′_{1}, s ⨿ s′, t ⨿ t′, c ⨿ c′, ι ⨿ ι′).

4.5.4. In the following we will be concerned with groupoids in the category A-Alg^{o} of Aschemes, where A is any V^{a}-algebra. We will use the general terminology for almost schemes introduced in (3.3.3), and complemented by the following: Definition 4.5.5. Let ϕ : X → Y be a morphism of A-schemes, G := (X_{0}, X_{1}, s, t, c, ι) a groupoid in the category of A-schemes.

(i) We say that ϕ is a closed imbedding (resp. is almost finite, resp. is étale, resp. is flat, resp. is almost projective) if the corresponding morphism ϕ^{♯} : 𝒪_{Y} → 𝒪_{X} is an epimorphism of 𝒪_{Y} -modules (resp. enjoys the same property). We say that ϕ is an open and closed imbedding if it induces an isomorphism X →^{∼} Y_{1} onto one of the factors of a decomposition Y = Y_{1} ⨿ Y_{2}.

(ii) We say that G is a closed equivalence relation if the morphism (s, t) : X_{1} → X_{0}×X_{0} is a a closed imbedding. We say that G is flat (resp. étale, resp. almost finite, resp. almost projective) if the morphism s : X_{1} → X_{0} enjoys the same property. We say that G is of finite rank if 𝒪_{X_{1}} is an almost projective 𝒪_{X_{0}}-module of finite rank. Furthermore, we set X_{0}/G := Spec 𝒪_{X_{0}}^{G}, where 𝒪_{X_{0}}^{G} ⊂ 𝒪_{X_{0}} is the equalizer of the morphisms s^{♯} and t^{♯}.

4.5.6. Let G := (X_{0}, X_{1}, s, t, c, ι) be a groupoid of finite rank in A-Alg^{o}. by assumption 𝒪_{X_{1}} is an almost projective 𝒪_{X_{0}}-module of finite rank, hence, by proposition 4.3.27, there is a decomposition 𝒪_{X_{0}} ≃ ∏_{i=0}^{r} B_{i} such that C_{i} := 𝒪_{X_{1}} ⊗_{𝒪_{X_{0}}} B_{i} is of constant rank equal to i, for i = 0, ..., r. Set X_{0,i} := Spec B_{i}. Lemma 4.5.7. In the situation of (4.5.6), there is a natural isomorphism of groupoids:

G ≃ (G ×_{X_{0}} X_{0,1}) ⨿ ... ⨿ (G ×_{X_{0}} X_{0,r}).

Proof. For every i ≤ r, let α_{i} : X_{0,i} → X_{0} be the open and closed imbedding defined by (4.5.6). Set X_{1,i} := X_{0,i} ×_{(α_{i},s)} X_{1} (so X_{1,i} = Spec C_{i}). Moreover, let X′_{1,i} := X_{0,i} ×_{(α_{i},t)} X_{1}, β_{i} : X′_{1,i} → X_{1} the open and closed imbedding (obtained by pulling back α_{i}), X_{2,i} := X_{1,i}×_{(β_{i},s′)}X_{2} and X_{2,i} := X′_{1,i} ×_{(β_{i},s′)} X_{2}. There follow natural decompositions X_{1} ≃ X′_{1,1} ⨿ ... ⨿ X′_{1,r} and X_{2} ≃ X′_{2,1} ⨿ ... ⨿ X′_{2,r}, such that s′ decomposes as a coproduct of morphisms X′_{2,i} → X′_{1,i}. By the construction of X_{0,i}, it is clear that X_{2,i}^{′o} has rank equal to i as an X_{1,i}^{′o}-module, for every i ≤ r. In other words, the above decompositions fulfill the conditions of proposition 4.3.27. Similarly, we obtain decompositions X_{1} ≃ X_{1,1} ⨿ ... ⨿ X_{1,r} and X_{2} ≃ X_{2,1} ⨿ ... ⨿ X_{2,r} which fulfill the same conditions. However, these conditions characterize uniquely the factors occuring in it, thus X_{1,i} = X′_{1,i} for i ≤ r. The claim follows easily. □

Lemma 4.5.8. Let G := (X_{0}, X_{1}, s, t, c, ι) be a groupoid of finite rank in A-Alg^{o}. If G has trivial automorphisms, then it is a closed equivalence relation.

Proof. Using lemma 4.5.7 we reduce easily to the case where 𝒪_{X_{1}} is of constant rank, say equal to r ∈ ℕ. Let Y := X_{0} ×X_{0}; since G has trivial automorphisms, the morphism (s, t) : X_{1} → Y is a monomorphism; equivalently, the natural projections pr_{1}, pr_{2} : X_{1} ×_{Y} X_{1} → X_{1} are isomorphisms. Let D := Im((s, t)^{♯} : 𝒪_{Y} → 𝒪_{X_{1}}); it follows that the natural morphisms pr_{1}^{♯}, pr_{2}^{♯} : 𝒪_{X_{1}} → 𝒪_{X_{1}} ⊗_{D} 𝒪_{X_{1}} are isomorphisms and consequently,

(4.5.9)  (𝒪_{X_{1}}/D) ⊗_{D} 𝒪_{X_{1}} = 0.

We need to show that 𝒪_{X_{1}} = D, or equivalently, that 𝒪_{X_{1}}/D = 0. However, by theorem 4.4.24, we can find a faithfully flat 𝒪_{X_{0}}-algebra B such that C := B ⊗_{𝒪_{X_{0}}} 𝒪_{X_{1}} ≃ B^{r}. Let D′ := B ⊗_{B} D; it follows that C is a faithful finitely generated D′-module. It suffices to show that C/D′ = 0, and we know already from (4.5.9)that (C/D′)⊗_{D′} C = 0. By proposition 3.4.4 it follows that C/D′ is a flat D′-module; consequently C/D′ ⊂ (C/D′) ⊗_{D′} C, and the claim follows. □

4.5.10. Let B be an A-algebra, P an almost finitely generated projective B-module. For every integer i ≥ 0, we define a B-linear morphism

(4.5.11)  Γ_{B}^{i}(End_{B}(P)^{a}) → B

as follows (see (8.1.13) for the definition of the functor Γ_{B}^{i} : B-Mod → B-Mod). Let R be any B_{∗}-algebra; we remark that the natural map β_{P} : End_{B}(P)_{!}^{a} ⊗_{B_{∗}} R → End_{R^{a}}(R^{a} ⊗_{A} P)_{!}^{a} is an isomorphism. Hence, for every i ≥ 0, we can define a map of sets

λ_{R}^{i} : End_{B}(P)_{!}^{a} ⊗_{B_{∗}} R → End_{B}(Λ_{B}^{i}P)_{!}^{a} ⊗_{B_{∗}} R

by letting ϕ ↦ β_{Λ_{B}^{i}P}^{−1} (Λ_{R^{a}}^{i}β_{P} (ϕ)^{a}). In the terminology of [61], the system of maps λ_{R}^{i} forms a homogeneous polynomial law of degree i from End_{B}(P)_{!}^{a} to End_{B}(Λ_{B}^{i}P)_{!}^{a}, so it induces a B_{∗}-linear map \overline{λ^{i}} : Γ_{B_{∗}}^{i}(End_{B}(P)_{!}^{a}) → End_{B}(Λ_{B}^{i}P)_{!}^{a}. After passing to almost modules, we obtain a B-linear morphism

(4.5.12)  Γ_{B}^{i}(End_{B}(P)^{a}) → End_{B}(Λ_{B}^{i}P)^{a}.

Then (4.5.11)is defined as the composition of (4.5.12)and the trace morphism tr_{Λ_{B}^{i}P/B}.

4.5.13. Let C be an almost finite projective B-algebra. Define μ : C → End_{B}(C)^{a} as in (4.1.7). By composition of Γ_{B}^{i}μ and (4.5.11)we obtain a B-linear morphism

Γ_{B}^{i}C → B

characterized by the condition: c^{[i]} ↦ σ_{i}(c) := tr_{C/B}(Λ_{C}^{i}μ(c)).

4.5.14. The construction of (4.5.13) applies especially to an almost finite projective groupoid G := (X_{0}, X_{1}, s, t, c, ι). In such case one verifies, using the cartesian diagrams (4.5.2), that σ_{i}(t^{♯}(f)) ∈ 𝒪_{X_{0}}^{G} for every f ∈ 𝒪_{X_{0∗}} and every i ≤ r: the argument is the same as in the proof of [25, Exp.V, Th.4.1]. In this way one obtains 𝒪_{X_{0}}^{G}-linear morphisms

(4.5.15)  T_{G,i} : Γ_{𝒪_{X_{0}}^{G}}^{i} 𝒪_{X_{0}} ⟶^{Γ^{i}t^{♯}} Γ_{𝒪_{X_{0}}}^{i} 𝒪_{X_{1}} → 𝒪_{X_{0}}^{G} f^{[i]} ↦ σ_{i}(t^{♯}(f)).

Theorem 4.5.16. Let G := (X_{0}, X_{1}, s, t, c, ι) be an étale almost finite and closed equivalence relation in A-Alg^{o}. Then G is effective and the natural morphism X_{0} → X_{0}/G is étale and almost finite projective.

Proof. See [25, Exp.IV, §3.3] for the definition of effective equivalence relation. By (4.5.2), we have an identification X_{2} ≃ X_{1} ×_{(s,s)} X_{1}.; therefore, the natural diagonal morphism X_{1} → X_{1} ×_{(s,s)} X_{1} gives a section δ : X_{1} → X_{2} of the morphism s′ : X_{2} → X_{1}. Furthermore, since X_{2} = X_{1} ×_{(s,t)} X_{1}, the pair of morphisms (1_{X_{1}}, ι ∘ s) : X_{1} → X_{1} induces another morphism ψ_{0} : X_{1} → X_{2}; similarly, let ψ_{1} : X_{1} → X_{2} be the morphism induced by the pair (ι ∘ t, 1_{X_{1}}) (these are the degeneracy maps of the simplicial complex associated to G: cp. [25, Exp. V, §1]). By arguing with T -points (and exploiting the interpretation (4.5.1) of X_{0}(T), X_{1}(T), etc.) one checks easily, first that ψ_{1} = δ, and second, that the two commutative diagrams

(4.5.17)  \xymatrix{X_{0} \ar[r]^{ι} \ar[d]_{ι} & X_{1} \ar[d]^{ψ_{1}} & X_{1} \ar[r]^{t} \ar[d]_{ψ_{1}} & X_{0} \ar[d]^{ι} \\ X_{1} \ar[r]^{ψ_{0}} & X_{2} & X_{2} \ar[r]^{t′} & X_{1}}

are cartesian. Since by assumption s is étale, corollary 3.1.9 implies that δ is an open and closed imbedding; consequently the same holds for ι. Let e_{0} ∈ 𝒪_{X_{1∗}} (resp. e_{1} ∈ 𝒪_{X_{2∗}}) be the idempotent corresponding to the open and closed imbedding ι (resp. δ); since G is a closed equivalence relation, for every ε ∈ 𝔪 we can write ε · e_{0} = ∑_{i}^{n} s^{♯}(b_{i}) · t^{♯}(b′_{i}) for some b_{i}, b′_{i} ∈ 𝒪_{X_{1∗}}. In view of (4.5.17)we deduce that ε·e_{1} = ∑_{i}^{n}(t^{′♯} ∘s^{♯}(b_{i}))·(t^{′♯} ∘t^{♯}(b′_{i})). However, s ∘ t′ = t ∘ s′ and t ∘ t′ = t ∘ c, consequently

ε · e_{1} = ∑_{i}^{n}(s^{′♯} ∘ t^{♯}(b_{i})) · (c^{♯} ∘ t^{♯}(b′_{i})).

Finally, thanks to remark 4.1.17, and again (4.5.2), we can write:

(4.5.18)  ε · f = ∑_{i}^{n} s^{♯} ∘ Tr_{X_{1}/X_{0}}(f · t^{♯}(b_{i})) · t^{♯}(b′_{i}) for every f ∈ 𝒪_{X_{1∗}}.

If we now let f := t^{♯}(g) in (4.5.18) we deduce: ε · t^{♯}(g) = ∑_{i} s^{♯}(T_{G,1}(g · b_{i})) · t^{♯}(b′_{i}) = ∑_{i} t^{♯}(T_{G,1}(g · b_{i})) · t^{♯}(b′_{i}) for every g ∈ 𝒪_{X_{0∗}}. Since t^{♯} is injective, this means that:

(4.5.19)  ε · g = ∑_{i}^{n} T_{G,1}(g · b_{i}) · b′_{i}  for every g ∈ 𝒪_{X_{0∗}}.

It follows easily that 𝒪_{X_{0}} is an almost finitely generated projective 𝒪_{X_{0}}^{G}-module. Furthermore, let us introduce the bilinear pairing t_{G} := T_{G} ∘ μ_{𝒪_{X_{0}}/𝒪_{X_{0}}^{G}} : 𝒪_{X_{0}} ⊗_{𝒪_{X_{0}}^{G}} 𝒪_{X_{0}} → 𝒪_{X_{0}}^{G}.

Claim 4.5.20. t_{G} is a perfect pairing. Proof of the claim: We have to show that the associated 𝒪_{X_{0}}-linear morphism

τ_{G} : 𝒪_{X_{0}} → 𝒪_{X_{0}}^{∗} := alHom_{𝒪_{X_{0}}^{G}} (𝒪_{X_{0}}, 𝒪_{X_{0}}^{G})

is an isomorphism. From (4.5.19)it follows easily that τ_{G} is a monomorphism. Let ϕ : 𝒪_{X_{0}} → 𝒪_{X_{0}}^{G} be a 𝒪_{X_{0}}^{G}-linear morphism; it remains only to show that, for every ε ∈ 𝔪, there exists b ∈ 𝒪_{X_{0∗}} such that τ_{G}(b) = ε · ϕ. Let α : 𝒪_{X_{1}} → 𝒪_{X_{0}}^{n}, β : 𝒪_{X_{0}}^{n} → 𝒪_{X_{1}} be defined by the rules: c ↦ (Tr_{X_{1}/X_{0}}(c · b_{1}), ..., Tr_{X_{1}/X_{0}}(c · b_{n})) and (x_{1}, ..., x_{n}) ↦ ∑_{i}^{n} x_{i} · b′_{i} for all c ∈ 𝒪_{X_{1∗}}, x_{1}, ..., x_{n} ∈ 𝒪_{X_{0∗}}. We remark that Im(α ∘ t^{♯}) ⊂ (𝒪_{X_{0}}^{G})^{n} and Im(β ∘ (t^{♯})^{n}) ⊂ 𝒪_{X_{0}}, so that we deduce, by restriction, morphisms α_{0} : 𝒪_{X_{0}} → (𝒪_{X_{0}}^{G})^{n} and β_{0} : (𝒪_{X_{0}}^{G})^{n} → 𝒪_{X_{0}}. Let ψ := ((ϕ ∘ β_{0}) ⊗_{𝒪_{X_{0}}^{G}} 1_{𝒪_{X_{0}}} ) ∘ α : 𝒪_{X_{1}} → 𝒪_{X_{0}}. By theorem 4.1.14 we can find, for every ε ∈ 𝔪, an element c ∈ 𝒪_{X_{1∗}} such that ε · ψ = τ_{X_{1}/X_{0}}(c). Using (4.5.18) we derive easily ε · c = ε · ∑_{i}^{n} ψ(t^{♯}(b_{i})) · t^{♯}(b′_{i}) = ε · ∑_{i}^{n} ϕ ∘ β_{0} ∘ α_{0}(b_{i}) · t^{♯}(b′_{i}) = ε · ∑_{i}^{n} ϕ(ε · b_{i}) · t^{♯}(b_{i}). In particular, ε · c = t^{♯}(b) for some b ∈ B, so the claim follows.

By assumption, the morphism π : C := 𝒪_{X_{0}} ⊗_{𝒪_{X_{0}}^{G}} 𝒪_{X_{0}} → 𝒪_{X_{1}} induced by the pair (s^{♯}, t^{♯}) is an epimorphism. Moreover, by construction, we have the identity:

Tr_{X_{1}/X_{0}} ∘ π = 1_{𝒪_{X_{0}}} ⊗_{𝒪_{X_{0}}^{G}} T_{G,1}.

By claim 4.5.20 we see that 1_{𝒪_{X_{0}}} ⊗_{𝒪_{X_{0}}^{G}} T_{G,1} induces a perfect pairing C ⊗_{𝒪_{X_{0}}} C → 𝒪_{X_{0}}; on the other hand, Tr_{X_{1}/X_{0}} is already a perfect pairing, by theorem 4.1.14. It then follows that π must be a monomorphism, hence C ≃ 𝒪_{X_{1}}, which shows that G is effective; then it is easy to verify that T_{G,1} is actually the trace of the 𝒪_{X_{0}}^{G}-algebra 𝒪_{X_{0}}, which is consequently étale over 𝒪_{X_{0}}^{G}. □

Proposition 4.5.21. Let G = (X_{0}, X_{1}, s, t, c, ι) be a groupoid of finite rank. Then 𝒪_{X_{0∗}} is integral over 𝒪_{X_{0∗}}^{G}.

Proof. By lemma 4.5.7 we can reduce to the case where the rank of 𝒪_{X_{1}} is constant, say equal to r. The assertion is then a direct consequence of the following: Claim 4.5.22. Let f ∈ 𝒪_{X_{0∗}}. With the notation of (4.5.14)we have:

(t^{♯}(f))^{r} + T_{G,1}(f) · (t^{♯}(f))^{r−1} + T_{G,2}(f) · (t^{♯}(f))^{r−2} + ... + T_{G,r}(f) = 0.

Proof of the claim: We apply proposition 4.4.30 (i.e. Cayley-Hamilton’s theorem) to the endomorphism t^{♯}(f) · 1_{𝒪_{X_{1}}} : 𝒪_{X_{1}} → 𝒪_{X_{1}}. □

Proposition 4.5.23. Let G be an étale closed equivalence relation of finite rank. Then G is universally effective and the morphism X_{0} → X_{0}/G is étale, faithfully flat and almost finite projective.

Proof. Everything is known by theorem 4.5.16, except for the faithfulness, which follows from the following: Claim 4.5.24. Under the assumptions of proposition 4.5.21, let I ⊂ 𝒪_{X_{0}}^{G} be an ideal such that I · 𝒪_{X_{0}} = 𝒪_{X_{0}}. Then I = 𝒪_{X_{0}}^{G}. Proof of the claim: First of all, let 𝒪_{X_{0}} ≃ ∏_{i=0}^{r} B_{i} be the decomposition as in (4.5.6); one derives easily a corresponding decomposition 𝒪_{X_{0}}^{G} ≃ ∏_{i=0}^{r} B_{i}^{G}, so we can assume that the rank of 𝒪_{X_{1}} is constant, equal to r. Let J ⊂ 𝒪_{X_{0}}^{G} be any ideal, and set C := 𝒪_{X_{0}}^{G}/J. We have a natural isomorphism

Γ_{C}^{i}(𝒪_{X_{0}} ⊗_{𝒪_{X_{0}}^{G}} C) ≃ Γ_{𝒪_{X_{0}}^{G}}^{i} (𝒪_{X_{0}}) ⊗_{𝒪_{X_{0}}^{G}} C.

Composing with (4.5.15)⊗_{𝒪_{X_{0}}^{G}} C, we derive a C-linear morphism: ψ_{i} : Γ_{C}^{i}(𝒪_{X_{0}} ⊗_{𝒪_{X_{0}}^{G}} C) → C. By inspecting the construction, one shows easily that ψ_{i}(1^{[i]}) = \binom{r}{i} (indeed, by flat base change one reduces easily to the case where 𝒪_{X_{1}} is a free B-module of rank r, in which case the result is obvious). Let us now take J = I. Then Γ_{C}^{i}(𝒪_{X_{0}} ⊗_{𝒪_{X_{0}}^{G}} C) = 0 for every i > 0, whence 1 = ψ_{r}(1^{[r]}) = 0 in C, and the claim follows. □

Proposition 4.5.25. Keep the assumptions of proposition 4.5.23. If X_{0} is an almost finite (resp. almost finitely presented, resp. flat, resp. almost projective, resp. weakly unramified, resp. unramified, resp. weakly étale, resp. étale) Spec A-scheme, then the same holds for the Spec Ascheme X_{0}/G.

Proof. By proposition 4.5.23, 𝒪_{X_{0}} is a faithful almost finitely generated projective 𝒪_{X_{0}}^{G}-module, hence ℰ_{𝒪_{X_{0}}/𝒪_{X_{0}}^{G}} = 𝒪_{X_{0}}^{G} by proposition 2.4.28(iv). It follows easily that, for every ε ∈ 𝔪 there exists n ∈ ℕ such that ε · 1_{𝒪_{X_{0}}^{G}} factors as a composition of 𝒪_{X_{0}}^{G}-linear morphisms:

(4.5.26)  𝒪_{X_{0}}^{G} → 𝒪_{X_{0}}^{n} → 𝒪_{X_{0}}^{G}.

The assertions for “almost finite” and for “almost projective” are immediate consequences. To prove the assertion for “almost finitely presented” we use the criterion of proposition 2.3.15(ii). Indeed, let (N_{λ}, ϕ_{λμ} | λ) be a filtered system of A-modules; we apply the natural transformation (2.3.16) to the sequence of morphisms (4.5.26) : since 𝒪_{X_{0}} is almost finitely presented, so is 𝒪_{X_{0}}^{n}, hence the claim follows by a little diagram chase. The assertions for “flat” and “weakly unramified” are easy and shall be left to the reader. To conclude, it suffices to consider the assertion for “unramified”. Now, by proposition 4.5.23 it follows that 𝒪_{X_{0}} ⊗_{A} 𝒪_{X_{0}} is an almost finitely generated projective 𝒪_{X_{0}}^{G} ⊗_{A} 𝒪_{X_{0}}^{G}-module; since by assumption 𝒪_{X_{0}} is an almost projective 𝒪_{X_{0}} ⊗_{A} 𝒪_{X_{0}}-module, we deduce from lemma 2.4.7 that 𝒪_{X_{0}} is an almost projective 𝒪_{X_{0}}^{G} ⊗_{A} 𝒪_{X_{0}}^{G}-module. Using (4.5.26)we deduce that 𝒪_{X_{0}}^{G} is almost projective over 𝒪_{X_{0}}^{G} ⊗_{A} 𝒪_{X_{0}}^{G} as well. □

5. Henselization and completion of almost algebras

This chapter deals with more advanced aspects of almost commutative algebras : we begin with the definitions of Jacobson radical rad(A) of an almost algebra A, of henselian pair and henselization of a pair (A, I), where I ⊂ A is an ideal contained in rad(A). These notions are especially well behaved when I is a tight ideal (definition 5.1.5), in which case we can also prove a version of Nakayama’s lemma (lemma 5.1.6).

In section 5.3 we explain what is a linear topology on an A-module and an A-algebra; as usual one is most interested in the case of I-adic topologies. In case A is I-adically complete and I is tight, we show that the functor B ↦ B/IB from almost finitely presented étale A-algebras to almost finitely presented étale A/I-algebras is an equivalence (theorem 5.3.27). For the proof we need some criteria to ensure that an A-algebra is unramified under various conditions : such results are collected in section 5.2, especially in theorem 5.2.12 and its corollary 5.2.15.

In section 5.5, theorem 5.3.27 is further generalized to the case where the pair (A, I) is tight henselian (see theorem 5.5.6, that also contains an analogous statement concerning almost finitely generated projective A-modules). The proof is a formal patching argument, which can be outlined as follows. First one reduces to the case where I is principal, say generated by f, and since I is tight, one can assume that f ∈ 𝔪; hence, we want to show that a given étale almost finitely presented A/fA-algebra B_{0} lifts uniquely to an A-algebra B of the same type; in view of section 5.3 one can lift B_{0} to an étale algebra B^{∧} over the f-adic completion A^{∧} of A; on the other hand, the almost spectrum Spec A is a usual scheme away from the closed subscheme defined by I, so we can use standard algebraic geometry to lift B_{0}[f^{−1}] to an étale algebra B′ over A[f^{−1}]. Finally we need to show that B^{∧} and B′ can be patched in a unique way over Spec A; this amounts to showing that certain commutative diagrams of functors are 2-cartesian (proposition 5.5.5).

The techniques needed to construct B′ are borrowed from Elkik’s paper [31]; for our purpose we need to extend and refine slightly Elkik’s results, to deal with non-noetherian rings. This material is presented in section 5.4; its usefulness transcends the modest applications to almost ring theory presented here.

The second main thread of the chapter is the study of the smooth locus of an almost scheme; first we consider the affine case: as usual, an affine scheme X over S := Spec A can be identified with the fpqc sheaf that it represents; then the smooth locus X_{sm} of X is a certain natural subsheaf, defined in terms of the cotangent complex 𝕃_{X/S}. To proceed beyond simple generalities one needs to impose some finiteness conditions on X, whence the definition of almost finitely presented scheme over S. For such affine S-schemes we can show that the smooth locus enjoys a property which we could call almost formal smoothness. Namely, suppose that I ⊂ A is a tight ideal such that the pair (A, I) is henselian; suppose furthermore that σ_{0} : S_{0} := Spec A/I → X is a section lying in the smooth locus of X; in this situation it does not necessarily follow that σ_{0} extends to a full section σ : S → X, however σ always exists if σ_{0} extends to a section over some closed subscheme of the form Spec A/𝔪_{0}I (for a finitely generated subideal 𝔪_{0} ⊂ 𝔪).

Next we consider quasi-projective almost schemes; if X is such a scheme, the invertible sheaf 𝒪_{X}(1) defines a quasi-affine 𝔾_{m}-torsor Y → X, and we define the smooth locus X_{sm} just as the projection of the smooth locus of Y. This is presumably not the best way to define X_{sm}, but anyway it suffices for the applications of section 5.8. In the latter we consider again a tight henselian pair (A, I), and we study some deformation problems for G-torsors, where G is a closed subgroup scheme of GL_{n} defined over Spec A and fulfilling certain general assumptions (see (5.8.4)). For instance, theorem 5.8.21 says that every G-torsor over the closed subscheme Spec A/I extends to a G-torsor over the whole of Spec A; the extension is however not unique, but any two such extensions are close in a precise sense (theorem 5.8.19) : here the almost formal smoothness of the quasi-projective almost scheme (GL_{n}/G)^{a} comes into play and accounts for the special quirks of the situation.

5.1. Henselian pairs. Definition 5.1.1. Let A be a V^{a}-algebra. The Jacobson radical of A is the ideal rad(A) := rad(A_{∗})^{a} (where, for a ring R, we have denoted by rad(R) the usual Jacobson ideal of R). Lemma 5.1.2. Let R be a V -algebra, I ⊂ R an ideal. Then I^{a} ⊂ rad(R^{a}) if and only if 𝔪I ⊂ rad(R).

Proof. Let us remark the following: Claim 5.1.3. If S is any ring, J ⊂ S an ideal, then J ⊂ rad(S) if and only if, for every x ∈ J there exists y ∈ J such that (1 + x) · (1 + y) = 1. Proof of the claim: Suppose that J ⊂ rad(S) and let x ∈ J; then 1 + x is not contained in any maximal ideal of S, so it is invertible. Find some u ∈ S with u · (1 + x) = 1; setting y := u − 1, we derive y = −x − xy ∈ J. Conversely, suppose that the condition of the claim holds for all x ∈ J. Let x ∈ J; we have to show that x ∈ rad(S). If this were not the case, there would be a maximal ideal 𝔮 ⊂ S such that x ∉ 𝔮; then we could find a ∈ S such that x · a ≡ −1 (mod 𝔮), so 1 + x · a ∈ 𝔮, especially 1 + a · x is not invertible, which contradicts the assumption.

Let ϕ : 𝔪I → R → R_{∗}^{a} be the natural composed map. Claim 5.1.4. Im ϕ is an ideal of R_{∗}^{a} and 𝔪I ⊂ rad(R) if and only if Im ϕ ⊂ rad(R_{∗}^{a}). Proof of the claim: The first assertion is easy to check, and clearly we only have to verify the ”if” direction of the second assertion, so suppose that Im ϕ ⊂ rad(R_{∗}^{a}). Notice that Ker ϕ is a square-zero ideal of R. Then, using claim 5.1.3, we deduce that for every x ∈ 𝔪 · I there exists z ∈ 𝔪 · I such that (1 + x) · (1 + z) = 1 + a, where a ∈ Ker ϕ, so a^{2} = 0. Consequently (1 + x) · (1 + z) · (1 − a) = 1, so the element y := z − a − z · a fulfills the condition of claim 5.1.3.

On the other hand, it is clear that I^{a} ⊂ rad(R^{a}) := rad(R_{∗}^{a})^{a} if and only if Im ϕ ⊂ rad(R_{∗}^{a}), so the lemma follows from claim 5.1.4. □

Given a V^{a}-algebra A and an ideal I ⊂ rad(A), one can ask whether the obvious analogue of Nakayama’s lemma holds for almost finitely generated A-modules. As stated in lemma 5.1.6, this is indeed the case, at least if the ideal I has the property singled out by the following definition, which will play a constant role in the sequel. Definition 5.1.5. Let I be an ideal of a V^{a}-algebra A. We say that I is tight if there exists a finitely generated subideal 𝔪_{0} ⊂ 𝔪 and an integer n ∈ ℕ such that I^{n} ⊂ 𝔪_{0}A. Lemma 5.1.6. Let A be a V^{a}-algebra, I ⊂ rad(A) a tight ideal. If M is an almost finitely generated A-module with IM = M, we have M = 0.

Proof. Under the assumptions of the lemma we can find a finitely generated A-module Q such that 𝔪_{0}M ⊂ Q ⊂ M. It follows that M = I^{n}M ⊂ 𝔪_{0}M ⊂ Q, so M = Q and actually M is finitely generated. Let M_{0} ⊂ M_{∗} be a finitely generated A_{∗}-submodule with M_{0}^{a} = M; clearly 𝔪M_{0} ⊂ I_{∗}M_{0} and 𝔪(I_{∗})^{n} ⊂ 𝔪_{0}A_{∗}. Therefore:

𝔪_{0}M_{0} ⊂ 𝔪^{n+2}M_{0} ⊂ I_{∗}𝔪^{n+1}M_{0} ⊂ ... ⊂ I_{∗}^{n+1}𝔪M_{0} ⊂ I_{∗}𝔪 · 𝔪_{0}M_{0} ⊂ 𝔪_{0}M_{0}

and consequently 𝔪M_{0} = 𝔪_{0}M_{0} = 𝔪I_{∗} · 𝔪_{0}M_{0}. By lemma 5.1.2 we have 𝔪I_{∗} ⊂ rad(A_{∗}), hence 𝔪_{0}M_{0} = 0 by Nakayama’s lemma, thus finally 𝔪M_{0} = 0, i.e. M = 0, as claimed. □

Corollary 5.1.7. Let A, I be as in lemma 5.1.6; suppose that ϕ : N → M is an A-linear morphism of almost finitely generated projective A-modules such that ϕ⊗_{A} 1_{A/I} is an isomorphism. Then ϕ is an isomorphism.

Proof. From the assumptions we derive that Coker(ϕ) ⊗_{A} (A/I) = 0, hence Coker ϕ = 0, in view of lemmata 2.3.17(i) and 5.1.6. By lemma 2.3.17(ii) it follows that Ker ϕ is almost finitely generated; moreover, since M is flat, Ker(ϕ)⊗_{A} (A/I) = Ker(ϕ⊗_{A} 1_{A/I}) = 0, whence Ker ϕ = 0, again by lemma 5.1.6. □

Definition 5.1.8. Let A be a V^{a}-algebra, I ⊂ rad(A) an ideal. We say that (A, I) is a henselian pair if (A_{∗}, 𝔪 · I_{∗}) is a henselian pair. If in addition, I is tight, we say that (A, I) is a tight henselian pair. Remark 5.1.9. For the convenience of the reader, we recall without proofs a few facts about henselian pairs.

(i) For a ring R, let R_{red} := R/nil(R) and denote by \sqrt{I} the radical of the ideal I. Then the pair (R, I) is henselian if and only if the same holds for the pair (R_{red}, \sqrt{I · R_{red}}).

(ii) Suppose that I ⊂ rad(R); then (R, I) is a henselian pair if and only if the same holds for the pair (ℤ ⊕ I, I), where ℤ ⊕ I is endowed with the ring structure such that (a, x) · (b, y) := (ab, ay + bx + xy) for every a, b ∈ ℤ, x, y ∈ I. Indeed, this follows easily from the following criterion (iii), which is shown in [60, Ch.XI, §2, Prop.1].

(iii) Assume I ⊂ rad(R); then the pair (R, I) is henselian if and only if every monic polynomial p(X) ∈ R[X] such that p(X) ≡ (X^{2} − X)^{m} (mod I[X]) decomposes as a product p(X) = q(X) · r(X) where q(X), r(X) are monic polynomials in R[X] and q(X) ≡ X^{m} (mod I[X]), r(X) ≡ (X − 1)^{m} (mod I[X]).

(iv) Let J ⊂ I be a subideal. If the pair (R, I) is henselian, the same holds for (R, J).

(v) If (R, I) is a henselian pair, and R → S is an integral ring homomorphism, then the pair (S, IS) is henselian ([60, Ch.XI, §2, Prop. 2]). Lemma 5.1.10. Let R be a V -algebra, I ⊂ rad(R) an ideal. Then the pair (R^{a}, I^{a}) is henselian if and only the same holds for the pair (R, 𝔪I).

Proof. It comes down to checking that (R, 𝔪I) is henselian if and only if (R_{∗}^{a}, 𝔪I_{∗}^{a}) is. To this aim, let ϕ : R → R_{∗}^{a} be the natural map. Let S := Im ϕ and J := ϕ(𝔪I) ⊂ S. Since Ker ϕ is a square-zero ideal in R, it follows from remark 5.1.9(i) that (R, 𝔪I) is henselian if and only if (S, J) is. However, it is clear that the induced map J → 𝔪I_{∗}^{a} is bijective, hence remark 5.1.9(ii) and lemma 5.1.2 say that (S, J) is henselian if and only if (R_{∗}^{a}, 𝔪I_{∗}^{a}) is. □

5.1.11. Suppose that (R, I) is a henselian pair, with R a V -algebra. Then, in view of lemma 5.1.10 and remark 5.1.9(iv) we see that (R^{a}, I^{a}) is also henselian. This gives a way of producing plenty of henselian pairs. Lemma 5.1.12. Let B be an almost finite A-algebra, I ⊂ A an ideal.

(i) The induced ring homomorphism A_{∗} → ϕ(A_{∗}) + 𝔪B_{∗} is integral.

(ii) If I ⊂ rad(A), then IB ⊂ rad(B).

(iii) If (A, I) is a henselian pair, the same holds for the pair (B, IB).

Proof. (i): for a given finitely generated subideal 𝔪_{0} ⊂ 𝔪, pick a finitely generated submodule Q ⊂ B_{∗} with 𝔪_{0}B_{∗} ⊂ Q; notice that (𝔪_{0}Q)^{2} ⊂ 𝔪_{0}Q, hence ϕ(A_{∗}) + 𝔪_{0}Q is a subring of B_{∗}, finite over A_{∗}. As ϕ(A_{∗}) + 𝔪B_{∗} is a filtered union of such subrings, the assertion follows. (ii) follows from (i) and from lemma 5.1.2. (iii) is a direct consequence of (i), of remark 5.1.9(v) and of (5.1.11). □

5.1.13. Given a V^{a}-algebra A and an ideal I ⊂ A, we define the henselization of the pair (A, I) as the unique pair (up to unique isomorphism) (A^{h}, I^{h}) which satisfies the (almost version of the) usual universal property (cp. [60, Ch.XI, §2, Déf.4]). Suppose A = R^{a} and I = J^{a} for a V -algebra R and an ideal J ⊂ R, and let (R′, J′) be a henselization of the pair (R, 𝔪J); one can easily check that (R^{′a}, J^{′a}) is a henselization of (A, I). It follows in particular that, if I ⊂ rad(A) and (A^{h}, I^{h}) is a henselization of (A, I), then the morphism A → A^{h} is faithfully flat ([60, Ch.XI, §2]). Lemma 5.1.14. Let 𝔪_{0} ⊂ 𝔪 be a finitely generated subideal. Then there exists an integer n := n(𝔪_{0}) > 0 such that ((𝔪_{0}A)_{∗})^{n} ⊂ 𝔪_{0}A_{∗} for any V^{a}-algebra A.

Proof. We proceed by induction on the number k of generators of 𝔪_{0}. To start out, let B be any V^{a}-algebra and f ∈ B_{∗} any almost element. The endomorphism B → B : x ↦ f · x induces an isomorphism α : \overline{B} := B/Ann_{B}(f) →^{∼} fB, and a commutative diagram:

\xymatrix{\overline{B}_{∗} ⊗_{V} \overline{B}_{∗} \ar[r]^{μ_{\overline{B}∗}} \ar[d]_{α_{∗}⊗α_{∗}} & \overline{B}_{∗} \ar[d]^{f·α_{∗}} \\ (fB)_{∗} ⊗_{V} (fB)_{∗} \ar[r]^{μ_{B∗}} & (fB)_{∗}.}

It follows that

(5.1.15)  f · (fB)_{∗} = ((fB)_{∗})^{2}

as ideals of B_{∗}, which takes care of the case k = 1. Suppose now that k > 1; let us write 𝔪_{0} = x_{1}·V +𝔪_{1}, where 𝔪_{1} is generated by k−1 elements. We apply (5.1.15)with B := A/𝔪_{1}A and f = x_{1} to deduce:

(𝔪_{0}A/𝔪_{1}A)_{∗} · (𝔪_{0}A/𝔪_{1}A)_{∗} = x_{1} · (𝔪_{0}A/𝔪_{1}A)_{∗} ⊂ Im((x_{1}A)_{∗} → (A/𝔪_{1}A)_{∗}).

Therefore ((𝔪_{0}A)_{∗})^{2} ⊂ (x_{1}A)_{∗} +(𝔪_{1}A)_{∗}. After raising the latter inclusion to some high power, the inductive assumption on 𝔪_{1} allows to conclude. □

Corollary 5.1.16. Let A be a V^{a}-algebra, I ⊂ rad(A) a tight ideal. Then:

(i) I_{∗} ⊂ rad(A_{∗}).

(ii) If (A, I) is a henselian pair, then the same holds for the pair (A_{∗}, I_{∗}).

Proof. For integers n, m large enough we can write: (I_{∗})^{nm} ⊂ (I_{∗}^{n})^{m} ⊂ (𝔪_{0}A)_{∗}^{m} ⊂ 𝔪_{0}A_{∗}, thanks to lemma 5.1.14. Then by lemma 5.1.2 we deduce (I_{∗})^{nm+1} ⊂ rad(A_{∗}), which implies (i). Similarly, we deduce from lemma 5.1.14 that \sqrt{𝔪I} = \sqrt{I}, so (ii) follows from remark 5.1.9(i). □

Proposition 5.1.17. Let (A, I) be a tight henselian pair. The natural morphism A → A/I induces a bijection from the set of idempotents of A_{∗} to the set of idempotents of (A/I)_{∗}.

Proof. Pick an integer m > 0 and a finitely generated subideal 𝔪_{0} ⊂ 𝔪 such that I^{m} ⊂ 𝔪_{0} · A. We suppose first that 𝔪̃ has homological dimension ≤ 1. By corollary 5.1.16(ii), the quotient map A_{∗} → A_{∗}/I_{∗} induces a bijection on idempotents. So we are reduced to showing that the natural injective map A_{∗}/I_{∗} → (A/I)_{∗} induces a surjection on idempotents. To this aim, it suffices to show that an idempotent almost element \overline{e} : V^{a} → A/I always lifts to an almost element e : V^{a} → A; indeed, the image of e inside A_{∗}/I_{∗} will then necessarily agree with \overline{e}. Now, the obstruction to the existence of e is a class ω_{1} ∈ Ext_{V^{a}}^{1}(V^{a}, I). On the other hand, proposition 2.5.13(i) ensures that, for every integer n > 0, \overline{e} admits a unique idempotent lifting \overline{e}_{n} : V^{a} → A/I^{n}. Let more generally ω_{n} ∈ Ext_{V^{a}}^{1}(V^{a}, I^{n}) be the obstruction to the existence of a lifting of \overline{e}_{n} to an almost element of A; the imbedding I^{n} ⊂ I induces a map

(5.1.18)  Ext_{V^{a}}^{1}(V^{a}, I^{n}) → Ext_{V^{a}}^{1}(V^{a}, I)

and clearly the image of ω_{n} under (5.1.18) agrees with ω_{1}. Thus, the proposition will follow in this case from the following: Claim 5.1.19. For n sufficiently large, the map (5.1.18)vanishes identically.

Proof of the claim: We will prove more precisely that the natural map

(5.1.20)  Ext_{V^{a}}^{1}(V^{a}, 𝔪_{0}I) → Ext_{V^{a}}^{1}(V^{a}, I)

vanishes; the claim will follow for n := m + 1. A choice of generators ε_{1}, ..., ε_{k} for 𝔪_{0} determines an epimorphism ϕ : I^{⊕k} → 𝔪_{0}I; notice that Ext_{V^{a}}^{2}(V^{a}, Ker ϕ) = 0 due to lemma 2.4.14(i),(ii) and the assumption on the homological dimension of 𝔪̃. Thus, the induced map

(5.1.21)  Ext_{V^{a}}^{1}(V^{a}, I^{⊕k}) → Ext_{V^{a}}^{1}(V^{a}, 𝔪_{0}I)

is surjective. To prove the claim, it suffices therefore to show that the composition of (5.1.21) and (5.1.20)vanishes, which is obvious, since the modules in question are almost zero.

Finally suppose that 𝔪 is arbitrary; by theorem 2.1.12(i.b),(ii.b), 𝔪 is the colimit of a filtered family of subideals (𝔪_{α} ⊂ 𝔪 | λ ∈ Λ), such that 𝔪_{0} ⊂ 𝔪_{λ} = 𝔪_{λ}^{2} and 𝔪̃_{λ} := 𝔪_{λ} ⊗_{V} 𝔪_{λ} is V -flat for every λ ∈ Λ, and moreover each 𝔪̃_{λ} has homological dimension ≤ 1. We may suppose that (A, I) = (R, J)^{a} for a henselian pair (R, I), where R is a V -algebra and J^{m} ⊂ 𝔪_{0}R. Each pair (V, 𝔪_{λ}) is a basic setup, and we denote by (A_{λ}, I_{λ}) the tight henselian pair corresponding to (R, J) in the almost category associated to (V, 𝔪_{λ}) (so A_{λ} is the image of R under the localization functor V -Alg → (V, 𝔪_{λ})^{a}-Alg). Let \overline{e} be an idempotent almost element of A/I (which is an object of (V, 𝔪)^{a}-Alg); \overline{e} is represented by a unique V -linear map f : 𝔪̃ → R/J and by the foregoing, for every λ ∈ Λ the restriction f_{λ} : 𝔪̃_{λ} → R/J lifts to a unique idempotent map g_{λ} : 𝔪̃_{λ} → R. By uniqueness, the maps g_{λ} glue to a map colim_{λ∈Λ} g_{λ} : 𝔪̃ → R which is the sought lifting of \overline{e}. □

5.2. Criteria for unramified morphisms. The following lemma generalizes a case of [42, Partie II, lemma 1.4.2.1]. Lemma 5.2.1. Let A → C be a morphism of V^{a}-algebras, f ∈ A_{∗} any almost element and M a C-module. Suppose that M[f^{−1}] is a flat C[f^{−1}]-module, M/fM is a flat C/fC-module, Tor_{1}^{A}(C, A/fA) = 0 and Tor_{i}^{A}(M, A/fA) = 0 for i = 1, 2. Then M is a flat C-module.

Proof. Let N be any C-module; We need to show that Tor_{1}^{C}(M, N) = 0. To this aim we consider the short exact sequence 0 → K → N →^{j} N[f^{−1}] → L → 0 where j is the natural morphism. We have Tor_{1}^{C}(M, N[f^{−}1]) = Tor_{1}^{C[f^{−1}]}(M[f^{−1}], N[f^{−1}]) = 0, therefore we are reduced to showing : Claim 5.2.2. Tor_{1}^{C}(M, K) = Tor_{2}^{C}(M, L) = 0. Proof of the claim: Notice that K = ⋃_{k>0} Ann_{K}(f^{k}), and similarly for L, so it suffices to show that Tor_{i}^{C}(M, Q) = 0 for i = 1, 2 and every C-module Q such that f^{k}Q = 0 for some integer k > 0. By considering the short exact sequence 0 → Ann_{Q}(f^{k−1}) → Q → Q′ → 0, an easy induction on k further reduces to showing that Tor_{i}^{C}(M, Q) = 0 for i = 1, 2, in case fQ = 0. However, the morphisms C → C/fC and A → C determine base change spectral sequences (cp. [67, Th.5.6.6])

E_{pq}^{2} :=Tor_{p}^{C/fC}(Tor_{q}^{C}(M, C/fC), Q) ⇒ Tor_{p+q}^{C}(M, Q)
F_{pq}^{2} :=Tor_{p}^{C}(Tor_{q}^{A}(C, A/fA), M) ⇒ Tor_{p+q}^{A}(M, A/fA).

The spectral sequence F yields an exact sequence:

[figure: Tor_{2}^{A}(M, A/fA); Tor_{2}^{C}(C/fC, M); Tor_{1}^{A}(C, A/fA) ⊗_{C} M; Tor_{1}^{A}(M, A/fA); Tor_{1}^{C}(C/fC, M); 0]

which implies Tor_{i}^{C}(M, C/fC) = 0 for i = 1, 2. Thus E_{pq}^{2} = 0 for q = 1, 2 and we get

Tor_{i}^{C}(M, Q) ≃ Tor_{i}^{C/fC}(M/fM, Q) for i = 1, 2.

Since M/fM is a flat C/fC-module, the claim follows. □

Lemma 5.2.3. Let M be an A-module, f ∈ A_{∗} an almost element, and denote by A^{∧} := lim_{n∈ℕ}(A/f^{n}A) the f-adic completion of A. Then we have:

(i) Ann_{A}(M[f^{−1}]) · Ann_{A}(M ⊗_{A} A^{∧}) ⊂ Ann_{A}(M).

(ii) If M is almost finitely generated and M/fM = M[f^{−1}] = 0 then M = 0.

Proof. (i): let a ∈ Ann_{A}(M[f^{−1}])_{∗}, b ∈ Ann_{A}(M ⊗_{A} A^{∧})_{∗} and denote by μ_{a}, μ_{b} : M → M the scalar multiplication morphisms. From a·M[f^{−1}] = 0 we deduce that aM ⊂ ⋃_{n>0} Ann_{M}(f^{n}); it follow that the natural morphism aM → (aM) ⊗_{A} A^{∧} is an isomorphism. Now the claim follows by inspecting the commutative diagram:

\xymatrix{M \ar[r] \ar[d]_{μ_{a}} & M ⊗_{A} A^{∧} \ar[r]^{0=μ_{b}⊗1_{A^{∧}}} & M ⊗_{A} A^{∧} \ar[d]^{μ_{a}⊗1_{A^{∧}}} \\ aM \ar[r]^{μ_{b}} & (aM) ⊗_{A} A^{∧} \ar[r]^{∼} & (aM) ⊗_{A} A^{∧}.}

(ii): for a given finitely generated subideal 𝔪_{0} ⊂ 𝔪, pick a finitely generated A-module Q ⊂ M such that 𝔪_{0}M ⊂ Q. By assumption, Q[f^{−1}] = 0; hence there exists an integer n ≥ 0 such that 𝔪_{0} · f^{n}Q = 0, whence 𝔪_{0}^{2} · f^{n}M = 0. However M = fM by assumption, thus 𝔪_{0}^{2}M = 0, and finally M = 0, since 𝔪_{0} is arbitrary. □

Theorem 5.2.4. Let A be a V^{a}-algebra, f ∈ A_{∗} any almost element and I ⊂ A an almost finitely generated ideal with I^{2} = I. Suppose that both I[f^{−1}] ⊂ A[f^{−1}] and (I + fA)/fA ⊂ A/fA are generated by idempotents in A[f^{−1}]_{∗} and respectively (A/fA)_{∗}. Then I is generated by an idempotent of A_{∗}.

Proof. We start out with the following: Claim 5.2.5. Let I, J ⊂ A be two ideals such that J is nilpotent and I^{2} = I. If \overline{I} := Im(I → A/J) is generated by an idempotent of (A/J)_{∗}, then I is generated by an idempotent of A_{∗}. Proof of the claim: We apply proposition 2.5.13 to the non-unitary extension 0 → J ∩ I → I → \overline{I} → 0 to derive that the idempotent \overline{e} that generates \overline{I} lifts uniquely to an idempotent e ∈ I_{∗}. Then e induces decompositions A ≃ A_{0} × A_{1} and I = (IA_{0}) × (IA_{1}) such that e = 0 (resp. e = 1) on A_{0} (resp. on A_{1}). We can therefore reduce to the cases A = A_{0} or A = A_{1}. If A = A_{0}, then I ⊂ J, hence I = I^{n} = 0 for n sufficiently large. If A = A_{1} then 1 ∈ I_{∗}, so I = A; in either case, the claim holds.

Claim 5.2.6. In the situation of (3.4.7), suppose that I ⊂ A_{0} is an ideal such that IA_{1} and IA_{2} are generated by idempotents. Then I is generated by an idempotent of A_{0∗}. Proof of the claim: By applying termwise the functor M ↦ M_{∗} to (3.4.8), we obtain a commutative diagram (3.4.8)_{∗} which is still cartesian. By assumption, we can find idempotents e_{1} ∈ A_{1∗}, e_{2} ∈ A_{2∗} with e_{i}A_{i} = IA_{i} (i = 1, 2). It is then easy to see that the images of e_{1} and e_{2} agree in A_{3∗}; consequently there exists a unique element e_{0} ∈ A_{0∗} whose image in A_{i} agrees with e_{i} for i = 1, 2. Such element is necessarily an idempotent; moreover, since the natural A_{0}-linear morphism I → (IA_{1}) × (IA_{2}) is a monomorphism, we deduce easily that (1 − e_{0}) · I = 0, whence I ⊂ e_{0}A_{0}. Let M := (e_{0}A_{0})/I; clearly M ⊗_{A_{0}} A_{i} = 0 for i = 1, 2. Using lemma 3.4.18(i) we deduce that M = 0, whence e_{0} ∈ I, as stated.

Suppose next that f is a regular element of A_{∗}. Denote by A^{∧} the f-adic completion of A. One verifies easily that the natural commutative diagram

(5.2.7)  \xymatrix{A \ar[r] \ar[d] & A[f^{−1}] \ar[d] \\ A^{∧} \ar[r] & A^{∧}[f^{−1}]}

is cartesian. Claim 5.2.8. In the situation of (5.2.7), let I ⊂ A be an ideal such that both I[f^{−1}] ⊂ A[f^{−1}] and I · A^{∧} ⊂ A^{∧} are generated by idempotents. Then I is generated by an idempotent. Proof of the claim: As in the proof of claim 5.2.6, we find in A_{∗} an idempotent e such that eA^{∧} = IA^{∧} and eA[f^{−1}] = I[f^{−1}], and deduce that I ⊂ eA. Let M := (eA)/I; on one hand we have M ⊗_{A} A^{∧} = 0. On the other hand, M[f^{−1}] = 0, i.e. M = ⋃_{n∈ℕ} Ann_{M}(f^{n}), which implies that M ⊗_{A} A^{∧} = M. Hence M = 0, and the claim follows.

Claim 5.2.9. The theorem holds in case A is f-adically complete. Proof of the claim: Say that Im(I → A/fA) = \overline{e}_{1} · (A/fA), for an idempotent \overline{e}_{1} ∈ (A/fA)_{∗}. For every n > 0 set \overline{I}_{n} := Im(I → A/f^{n}A); applying proposition 2.5.13 to the non-unitary extensions

0 → (f^{n}A/f^{n+1}A) ∩ \overline{I}_{n+1} → \overline{I}_{n+1} → \overline{I}_{n} → 0

we construct recursively a compatible system of idempotents \overline{e}_{n} ∈ \overline{I}_{n∗}, and by claim 5.2.5, \overline{e}_{n} generates \overline{I}_{n} for every n > 0; whence an element e ∈ lim_{n∈ℕ}(A/f^{n}A)_{∗} ≃ (lim_{n∈ℕ}A/f^{n}A)_{∗} ≃ A_{∗}.

Clearly e is again an idempotent, and it induces decompositions A ≃ A_{0} × A_{1}, I ≃ (IA_{0}) × (IA_{1}), such that e = 0 (resp. e = 1) in A_{0} (resp. in A_{1}). We can thus assume that either A = A_{0} or A = A_{1}. If A = A_{0}, then \overline{e}_{n} = 0 for every n > 0, so I ⊂ ⋂_{n>0} f^{n}A = 0. If A = A_{1}, then \overline{e}_{n} = 1 for every n > 0, i.e. \overline{I}_{n} = A/f^{n}A for every n > 0, that is, I is dense in A. It follows easily that Ann_{A}(I) = 0. Using lemma 2.4.6 we deduce Ann_{A[f^{−1}]}(I[f^{−1}]) = 0; since I[f^{−1}] is generated by an idempotent, this means that I[f^{−1}] = A[f^{−1}]. Set M := A/I; the foregoing shows that M fulfills the hypotheses of lemma 5.2.3(ii), whence M = 0, which implies the claim.

Claim 5.2.10. The theorem holds in case f is a non-zero-divisor of A_{∗}. Proof of the claim: According to claim 5.2.8, it suffices to prove that the ideal IA^{∧} ⊂ A^{∧} is generated by an idempotent. However, since A^{∧}/fA^{∧} ≃ A/fA, this follows directly from our assumptions and from claim 5.2.9.

After these preparations, we are ready to prove the theorem. Let T := ⋃_{n>0} Ann_{A}(f^{n}); by claim 5.2.10, we know already that the ideal I · (A/T) is generated by an idempotent, and the same holds for the ideal IA^{∧}. Since T ⊗_{A} A^{∧} ≃ T, it follows that the natural morphism ϕ : T → T A^{∧} is an epimorphism; one verifies easily that J := Ker ϕ = T ∩ (⋂_{n>0} f^{n}A), and then it is clear that J^{2} = 0. Furthermore, it follows that the natural commutative diagram

\xymatrix{A/J \ar[r] \ar[d] & A/T \ar[d] \\ A^{∧} \ar[r] & A^{∧}/T A^{∧}}

is cartesian. Taking claim 5.2.6 into account, we deduce that the image of I in A/J is generated by an idempotent. Lastly, we invoke claim 5.2.5 to show that I is indeed generated by an idempotent. □

Remark 5.2.11. One may wonder whether every almost finitely generated ideal I ⊂ A such that I = I^{2}, is generated by an idempotent. We do not know the answer to this question. Theorem 5.2.12. Let ϕ : A → B be a morphism of V^{a}-algebras, I, J ⊂ A any two ideals and f ∈ A_{∗} any almost element.

(i) If ϕ⊗_{A} 1_{A/I} and ϕ⊗_{A} 1_{A/J} are weakly unramified, then the same holds for ϕ⊗_{A} 1_{A/IJ}. (ii) If ϕ ⊗_{A} 1_{A/I} and ϕ ⊗_{A} 1_{A/J} are unramified, then the same holds for ϕ ⊗_{A} 1_{A/IJ}. (iii) If ϕ is flat and both ϕ ⊗_{A} 1_{A/fA} and ϕ ⊗_{A} 1_{A[f^{−1}]} are weakly unramified, then ϕ is

weakly unramified. (iv) If ϕ is almost finite and both ϕ ⊗_{A} 1_{A/fA} and ϕ ⊗_{A} 1_{A[f^{−1}]} are unramified, then ϕ is

unramified.

Proof. To start out, we remark the following: Claim 5.2.13. Let A′ be any A-algebra and set B′ := B ⊗_{A} A′; then for every B′ ⊗_{A′} B′-module M we have a natural isomorphism: Tor_{1}^{B⊗_{A}B}(B, M) ≃ Tor_{1}^{B′⊗_{A′}B′}(B′, M). Proof of the claim: Notice that the short exact sequence of B ⊗_{A} B-modules

(5.2.14)  0 → I_{B/A} → B ⊗_{A} B → B → 0.

is split exact as a sequence of B-modules (and a fortiori as a sequence of A-modules); hence (5.2.14)⊗_{A}N is again exact for every A-module N. We deduce easily that Tor_{1}^{B⊗_{A}B}(B, N ⊗_{A} B ⊗_{A} B) = 0 for every A-module N. On the other hand, the morphism B ⊗_{A} B → B′ ⊗_{A′} B′ determines a base change spectral sequence (cp. [67, Th.5.6.6])

E_{pq}^{2} := Tor_{p}^{B′⊗_{A′}B′}(Tor_{q}^{B⊗_{A}B}(B, B′ ⊗_{A′} B′), M) ⇒ Tor_{p+q}^{B⊗_{A}B}(B, M)

for every B′ ⊗_{A} B′-module M. The foregoing yields E_{p1}^{2} = 0 for every p ∈ ℕ, so the claim follows.

In order to show (i) we may replace A by A/IJ and therefore assume that IJ = 0; then we have to prove that Tor_{1}^{B⊗_{A}B}(B, N) = 0 for every B ⊗_{A} B-module N. By a simple devissage we can further reduce to the case where either IN = 0 or JN = 0. Say IN = 0; we apply claim 5.2.13 with A′ := A/I, B′ := B/IB to deduce Tor_{1}^{B⊗_{A}B}(B, N) ≃ Tor_{1}^{B′⊗_{A′}B′}(B′, N); however the latter module vanishes by our assumption on ϕ ⊗_{A} 1_{B′}.

For (ii) we can again assume that IJ = 0. We need to show that B is an almost projective B ⊗_{A} B-module, and we know already that it is flat by (i). Set I′ := I(B ⊗_{A} B), J′ := J(B ⊗_{A} B); by assumption B/IB (resp. B/JB) is almost projective over B ⊗_{A} B/I′ (resp. B ⊗_{A} B/J′); in light of proposition 3.4.21 we deduce that B/(IB ∩ JB) is almost projective over B ⊗_{A} B/(I′ ∩ J′). Then, since (I′ ∩ J′)^{2} ⊂ I′J′ = 0, the assertion follows from lemma 3.2.25(i).

Next we remark that, under the assumptions of (iii), the hypotheses of lemma 5.2.1 are fulfilled by C := B ⊗_{A} B and M := B, so assertion (iii) holds. Finally, suppose that ϕ is almost finite; hence Ω_{B/A} is an almost finitely generated A-module, and by assumption we have Ω_{B/A} ⊗_{A} (A/fA) = Ω_{B/A}[f^{−1}] = 0, therefore Ω_{B/A} = 0, in view of lemma 5.2.3(ii). Set I := Ker(B ⊗_{A} B → B); it follows that I^{2} = I, and I is almost finitely generated as an A-module, so a fortiori as a B ⊗_{A} B-module. Moreover, from proposition 3.1.4 and our assumption on ϕ ⊗_{A} 1_{A/fA} and ϕ[f^{−1}], it follows that both I[f^{−1}] ⊂ B ⊗_{A} B[f^{−1}] and Im(I → B ⊗_{A} (B/fB)) are generated by idempotents. Then theorem 5.2.4 says that also I is generated by an idempotent of (B ⊗_{A} B)_{∗}, so that B is almost projective as a B ⊗_{A} B-module, by remark 3.1.8, which proves (iv). □

Corollary 5.2.15. Let I ⊂ rad(A) be a tight ideal. If B is an almost finite A-algebra and B/IB is unramified over A/I, then B is unramified over A.

Proof. Under the stated assumptions, Ω_{B/A} is an almost finitely generated A-module such that Ω_{B/A} ⊗_{A} A/I = 0; by lemma 5.1.6 it follows that

(5.2.16)  Ω_{B/A} = 0.

Set I_{B/A} := Ker(μ_{B/A} : B ⊗_{A} B → B); we derive that I_{B/A} = I_{B/A}^{2}. Pick n > 0 and a finitely generated subideal 𝔪_{0} ⊂ 𝔪 with I^{n} ⊂ 𝔪_{0}A; let ε_{1}, ..., ε_{k} be a set of generators for 𝔪_{0}. Claim 5.2.17. I_{B/A}[ε_{i}^{−1}] is generated by an idempotent of B ⊗_{A} B[ε_{i}^{−1}], for every i ≤ k. Proof of the claim: Notice that A[ε_{i}^{−1}] is a (usual) V -algebra (that is, the localization functor A[ε_{i}^{−1}]_{∗}-Mod → A[ε_{i}^{−1}]-Mod is an equivalence) and B[ε_{i}^{−1}] is a finite A[ε_{i}^{−1}]-algebra; from (5.2.16) and [40, Exp.I, Prop.3.1] we deduce that B[ε_{i}^{−1}] is unramified over A[ε_{i}^{−1}], which implies the claim.

Claim 5.2.18. I_{B/A}/𝔪_{0}I_{B/A} is generated by an idempotent as well. Proof of the claim: We apply theorem 5.2.12(ii) with J = I^{m} to deduce, by induction on m, that B/I^{m+1}B is an unramified A/I^{m+1}-algebra for every m ∈ ℕ; it follows that B/𝔪_{0}B is unramified over A/𝔪_{0}A, whence the claim.

Using theorem 5.2.12(iv) and claims 5.2.17, 5.2.18, the assertion is now easily verified by induction on the number k of generators of 𝔪_{0}, □

The following lemma shows that morphisms of unramified A-algebras can be read off from their ”graphs”. This result will be useful in section 5.5.

5.2.19. Let A → B be an unramified morphism of V^{a}-algebras, and C any A-algebra. Suppose that e ∈ (B ⊗_{A} C)_{∗} is an idempotent; to e we can attach a morphism of A-algebras:

Γ(e) : C → B ⊗_{A} C → e · (B ⊗_{A} C) c ↦ 1 ⊗ c ↦ e · (1 ⊗ c) (c ∈ C_{∗})

and, in case Γ(e) is an isomorphism, we obtain a morphism f_{e} : B → C after composing Γ(e)^{−1} and the natural morphism

Δ(e) : B → B ⊗_{A} C → e · (B ⊗_{A} C) b ↦ b ⊗ 1 ↦ e · (b ⊗ 1) (b ∈ B_{∗})

Conversely, To a given a morphism ϕ : B → C of A-algebras, we can associate an idempotent e_{ϕ} ∈ (B ⊗_{A} C)_{∗}, by setting e_{ϕ} := (1_{B} ⊗_{A} ϕ)(e_{B/A}), where e_{B/A} is the idempotent provided by proposition 3.1.4. Then, since the commutative diagram

\xymatrix{B ⊗_{A} B \ar[r]^{1_{B}⊗ϕ} \ar[d]_{μ_{B/A}} & B ⊗_{A} C \ar[d]^{μ_{C/A}∘(ϕ⊗1_{C})} \\ B \ar[r]^{ϕ} & C}

is cocartesian, we deduce from proposition 3.1.4 that the corresponding morphism Γ(e_{ϕ}) is an isomorphism. In fact, the following holds: Lemma 5.2.20. The rule ϕ ↦ e_{ϕ} of (5.2.19)induces a natural bijection from Hom_{A-Alg}(B , C) onto the set of idempotents e ∈ (B ⊗_{A} C)_{∗} such that Γ(e) is an isomorphism. Its inverse is the rule e ↦ ϕ_{e}.

Proof. Let ϕ : B → C be given and set e := e_{ϕ}. We need to show that f_{e} = ϕ, or equivalently Δ(e) = Γ(e) ∘ ϕ. This translates into the equality:

(5.2.21)  e · (b ⊗ 1) = e · (1 ⊗ ϕ(b)) for every b ∈ B_{∗}.

The latter can be rewritten as (1 ⊗ ϕ)(e_{B/A} · (1 ⊗ b − b ⊗ 1)) = 0 which follows from identity (iii) in proposition 3.1.4. Conversely, for e given such that Γ(e) is an isomorphism, set ϕ := ϕ_{e}; we have to check that e_{ϕ} = e. By construction of ϕ we have Δ(e) = Γ(e) ∘ ϕ, which means that (5.2.21) holds for the pair (e, ϕ). Let J ⊂ B ⊗_{A} C be the ideal generated by the elements of the form b ⊗ 1 − 1 ⊗ ϕ(b), for all b ∈ B_{∗}; we know that eJ = 0, which means that the natural morphism B⊗_{A}C → e(B⊗_{A}C) factors through the natural morphism B⊗_{A}C → e_{ϕ}(B⊗_{A}C). Since both Γ(e) and Γ(e_{ϕ}) are isomorphisms, it follows that e(B ⊗_{A} C) = e_{ϕ}(B ⊗_{A} C), so e = e_{ϕ}. □

5.3. Topological algebras and modules. Definition 5.3.1. Let A be a V^{a}-algebra, M an A-module.

(i) A linear topology on M is a non-empty family ℒ of submodules of M that satisfies the following conditions. If I, J ∈ ℒ, then I ∩ L ∈ ℒ; if I ∈ ℒ and I ⊂ J, then J ∈ ℒ. We say that a submodule I is open if I ∈ ℒ.

(ii) Let ℱ be a family of submodules of M. The topology generated by ℱ is the smallest

linear topology containing ℱ. (iii) Let ℒ be a linear topology on M and I ⊂ M a submodule. The closure of I is the

submodule \overline{I} := ⋂_{J∈ℒ} (I + J). (iv) We say that the topology ℒ is complete if the natural morphism M → lim_{I∈ℒ} M/I is an

isomorphism.

(v) Let ℒ_{M}, ℒ_{A} be topologies on M, resp. on A, and I an open ideal of A. We say that the topology ℒ_{M} is I-preadic (resp. I-c-preadic) if the family of submodules (I^{n}M | n ∈ ℕ) (resp. (\overline{\overline{I^{n}} · M} | n ∈ ℕ)) is a cofinal subfamily in ℒ_{M}. In either case, we say that I is an ideal of definition for ℒ_{M}. Furthermore, we say that the topology ℒ_{M} is I-adic if it is I-preadic and complete. We similarly define an I-c-adic topology.

5.3.2. One defines as usual the notions of continuous, resp. open morphism of A-modules, of induced topology on submodules of a linearly topologized module and of completion of a topological module. Notice that, if A is an I-c-preadic topological V^{a}-algebra and M an I-c-preadic A-module (for some open ideal I ⊂ A) then we have \overline{I^{n}M} = \overline{\overline{I^{n}} · M} for every n ∈ ℕ. Lemma 5.3.3. Let M, N be two A-modules endowed with descending filtrations ℱ_{M} := (M_{n} | n ∈ ℕ), ℱ_{N} := (N_{n} | n ∈ ℕ) so that M_{0} = M, N_{0} = N and M (resp. N) is complete for the topology generated by ℱ_{M} (resp. by ℱ_{N}). Suppose furthermore that ϕ : M → N is a morphism of A-modules which respect the filtrations, and such that the induced morphisms gr^{i}ϕ : gr^{i}M → gr^{i}N are epimorphisms for every i ∈ ℕ. Then ϕ is an epimorphism.

Proof. Under the assumptions, the induced morphism ϕ_{n} : M/M_{n} → N/N_{n} is an epimorphism for every n ∈ ℕ. Clearly it suffices to show then that lim_{n∈ℕ}^{1} Ker(ϕ_{n}) vanishes. However,

from Coker(gr^{i}ϕ) = 0 one deduces by snake lemma that the induced morphism Ker(ϕ_{i+1}) → Ker(ϕ_{i}) is an epimorphism, so the claim follows. □

5.3.4. Let (A_{n}; ϕ_{n} : A_{n+1} → A_{n} | n ∈ ℕ) be a projective system of V^{a}-algebras. Then A_{∞} := lim_{n∈ℕ}A_{n} is naturally endowed with a linear topology, namely the topology ℒ generated

by the family (Ker(A_{∞} → A_{n}) | n ∈ ℕ). We call ℒ the projective topology defining A_{∞}. Lemma 5.3.5. Let (A_{n}; ϕ_{n} : A_{n+1} → A_{n} | n ∈ ℕ) be a projective system of V^{a}-algebras, A_{∞} its projective limit, and suppose that:

(a) ϕ_{n} is an epimorphism for every n ∈ ℕ.

(b) Ker ϕ_{n} = Ker(ϕ_{0} ∘ ... ∘ ϕ_{n} : A_{n+1} → A_{0})^{n+1} for every n ∈ ℕ. Then we have:

(i) Let I := Ker(A_{∞} → A_{0}). The projective topology of A_{∞} is I-c-preadic. This topology is complete and moreover \overline{I^{n+1}} = Ker(A_{∞} → A_{n}) for every n ∈ ℕ.

(ii) Conversely, suppose that A is any complete linearly topologized V^{a}-algebra and I ⊂ A an open ideal such that the topology of A is I-c-preadic. Set A_{n} := A/\overline{I^{n+1}} for every n ∈ ℕ. Then the projective system (A_{n} | n ∈ ℕ) satisfies conditions (a) and (b).

Proof. To show (i) it suffices to show that \overline{I^{n+1}} = Ker(A_{∞} → A_{n}) for every n ∈ ℕ. We endow \overline{I^{n+1}} (resp. Ker(A_{∞} → A_{n})) with the descending filtration F_{i} := \overline{I^{n+1+j}} (resp. G_{i} := Ker(A_{∞} → A_{n+j})) (i ∈ ℕ). Notice first that the natural morphism A_{∞} → A_{k+1} induces an isomorphism I/G_{k+1} →^{∼} Ker(A_{k+1} → A_{0}). By (b) we deduce isomorphisms

(5.3.6)  (I^{k+1} + G_{k+1})/G_{k+1} →^{∼} Ker ϕ_{k}  for every k ∈ ℕ.

Especially, I^{k+1} ⊂ Ker(A_{∞} → A_{k}). Since the latter is a closed ideal, we deduce \overline{I^{k+1}} ⊂ Ker(A_{∞} → A_{k}) for every k ∈ ℕ. Therefore, we obtain an imbedding \overline{I^{n+1}} ⊂ Ker(A_{∞} → A_{n}) that respects the filtrations F_{•} and G_{•}. Since gr^{i}G_{•} ≃ Ker ϕ_{i+n}, we derive easily from (5.3.6) that the induced morphism gr^{i}F_{•} → gr^{i}G_{•} is an epimorphism for every i ∈ ℕ, so (i) follows by lemma 5.3.3. Under the assumptions of (ii) it is obvious that condition (a) holds. To show (b) comes down to verifying the identity I^{n+1} + \overline{I^{n+2}} = \overline{I^{n+1}}, which is obvious since \overline{I^{n+2}} is an open ideal. □

Remark 5.3.7. Notice that, in the situation of lemma 5.3.5, the pair (A_{∞}, I) is henselian. One sees this as follows, using the criterion of remark 5.1.9(iii). Let R_{n} := A_{∗}/\overline{I^{n+1}}_{∗}, J_{n} := \overline{I^{n}}_{∗}/\overline{I^{n+1}}_{∗} for every n ∈ ℕ; let also R := (lim_{n∈ℕ}R_{n}) and denote by J the image of lim_{n∈ℕ}J_{n} in

R. Since J_{n} is a nilpotent ideal of R_{n}, the pair (R_{n}, J_{n}) is henselian. Let now p(X) ∈ R[X] and denote by p_{n}(X) the image of p(X) in R_{n}(X), for every n ∈ ℕ; suppose that p_{0}(X) decomposes in R_{0}[X] as p_{0}(X) = q_{0}(X) · r_{0}(X); by the cited criterion we derive a compatible family of decompositions p_{n}(X) = q_{n}(X)·r_{n}(X) in R_{n}[X] for every n ∈ ℕ; hence p(X) admits a similar decomposition in R[X], so (R, J) is a henselian pair, which implies the contention, in view of lemma 5.1.10. Lemma 5.3.8. Let A_{∞} be the projective limit of a system of V^{a}-algebras satisfying conditions (a) and (b) of lemma 5.3.5. Let J ⊂ A_{∞} be a finitely generated ideal such that \overline{J} is open. Then:

(i) J = \overline{J}.

(ii) Every finite system of generators x_{1}, ..., x_{r} of J defines an open morphism A_{∞}^{r} → J

(for the topologies on A_{∞}^{r} and J induced from A_{∞}). (iii) J^{k} is open for every k ∈ ℕ.

Proof. By lemma 5.3.5 the topology on A_{∞} is c-preadic for some defining ideal I. The assumption means that \overline{I^{n}} ⊂ J for a sufficiently large integer n. Claim 5.3.9. The natural morphism: (J ∩ \overline{I^{m}})/(J ∩ \overline{I^{m+1}}) → \overline{I^{m}}/\overline{I^{m+1}} is an isomorphism for every m ≥ n. Proof of the claim: It is an easy verification that shall be left to the reader.

Let x_{1}, ..., x_{r} ∈ J_{∗} be a system of generators for J, and f : A_{∞}^{r} → J the corresponding epimorphism. Set M_{k} := f^{−1}(J ∩ \overline{I^{k}}). To start out, M_{k} contains (\overline{I^{k}})^{⊕r}, whence M_{k} is open in A_{∞}^{r} for every k ∈ ℕ. This already shows that f is continuous. We define a descending filtration on M_{n} by setting F_{k} := \overline{I^{k}M_{n}} for every k ∈ ℕ. In view of claim 5.3.9, the morphism f induces an epimorphism M_{n} → \overline{I^{n}}/\overline{I^{n+1}}, hence for every k ∈ ℕ, the morphisms:

(5.3.10)  \overline{I^{k}} · M_{n} → (\overline{I^{k}} · \overline{I^{n}})/(\overline{I^{k}} · \overline{I^{n+1}}) → \overline{I^{n+k}}/\overline{I^{n+k+1}}

are epimorphisms as well. The composition of the morphisms (5.3.10) extends to an epimorphism F_{k} → \overline{I^{n+k}}/\overline{I^{n+k+1}}, for every k ∈ ℕ. In other words, if we endow \overline{I^{n}} with its I-c-preadic filtration, then f induces a morphism ϕ : M_{n} → \overline{I^{n}} of filtered modules, such that gr^{•}ϕ is an epimorphism of graded modules. By assumption \overline{I^{n}} is complete for its filtration. Similarly, M_{n} is complete for its filtration F_{•}; indeed, this follows by remarking that (\overline{I^{n+k}})^{⊕r} ⊂ F_{k} ⊂ (\overline{I^{k}})^{⊕r} for every k ∈ ℕ. Then lemma 5.3.3 says that ϕ is an epimorphism, i.e. f(M_{n}) = \overline{I^{n}}; it follows that J is an open ideal, hence equal to its closure. More generally, the same argument proves that for every k ∈ ℕ, f(F_{k}) = \overline{I^{n+k}}. Since by the foregoing, (F_{k} | k ∈ ℕ) is a cofinal system of open submodules of A_{∞}^{r}, we deduce that f is an open morphism. Finally the identity

f((J^{n})^{⊕r}) = J^{n+1} for every n ∈ ℕ

together with (ii) and an easy induction, yields (iii). □

5.3.11. Let I be an ideal of definition for the projective topology on a ring A_{∞} as in lemma 5.3.5(i). We wish now to give a criterion to ensure that the projective topology on A_{∞} is actually I-adic. This can be achieved in case I is tight, as shown by proposition 5.3.12, which generalizes [26, Ch.0, Prop.7.2.7]. Proposition 5.3.12. Let A_{∞} be as in lemma 5.3.8 and I an ideal of definition for the projective topology of A_{∞}. Assume that I is tight and moreover that I/\overline{I^{2}} is an almost finitely generated A_{∞}-module. Then :

(i) There exists n ∈ ℕ and a finitely generated ideal J ⊂ A_{∞} such that I^{n} ⊂ J ⊂ I. (ii) The topology of A_{∞} is I-adic.

Proof. Using the natural epimorphism (I/\overline{I^{2}})^{⊗k} → \overline{I^{k}}/\overline{I^{k+1}} we deduce that \overline{I^{k}}/\overline{I^{k+1}} is almost finitely generated for every k ≥ 0; then the same holds for I/\overline{I^{k+1}}. Let 𝔪_{0} ⊂ 𝔪 be a finitely generated subideal and n ∈ ℕ such that I^{n} ⊂ 𝔪_{0}; pick a finitely generated ideal 𝔪_{1} ⊂ 𝔪 with 𝔪_{0} ⊂ 𝔪_{1}^{2}; we can find a finitely generated A_{∞}-module Q ⊂ I/\overline{I^{n+2}} such that 𝔪_{1} · (I/\overline{I^{n+2}}) ⊂ Q; up to replacing Q by 𝔪_{1} ·Q, we can then achieve that Q is generated by the images of finitely many almost elements x_{1}, ..., x_{t} of I and moreover 𝔪_{0} ·(I/\overline{I^{n+2}}) ⊂ Q. Then, since \overline{I^{n+2}} is open we deduce

(5.3.13)  \overline{I^{n+1}}/\overline{I^{n+2}} ⊂ Q.

Let J ⊂ A_{∞} be the ideal generated by x_{1}, ..., x_{t}. From (5.3.13) we deduce that the natural morphism (J ∩ \overline{I^{n+1}})/(J ∩ \overline{I^{n+2}}) → \overline{I^{n+1}}/\overline{I^{n+2}} is an isomorphism. Since I^{k}(J ∩ \overline{I^{n+1}}) ⊂ J ∩ \overline{I^{n+1+k}}, the same holds more generally for the morphisms (J ∩ \overline{I^{n+1+k}})/(J ∩ \overline{I^{n+2+k}}) → \overline{I^{n+1+k}}/\overline{I^{n+2+k}}, for every k ∈ ℕ. This easily implies that J′ := J ∩ \overline{I^{n+1}} is dense in \overline{I^{n+1}}, i.e. \overline{J′} = \overline{I^{n+1}}; especially, \overline{J} is open in A_{∞} and therefore J = \overline{J} by lemma 5.3.8(i), which proves assertion (i). Assertion (ii) follows easily from (i) and lemma 5.3.8(iii). □

5.3.14. Let now (A_{n} | n ∈ ℕ) be an inverse system of V^{a}-algebras satisfying conditions (a) and (b) of lemma 5.3.5 and let I := Ker(A_{∞} → A_{0}). The induced functors A_{n+1}-Mod → A_{n}-Mod : M ↦ A_{n} ⊗_{A_{n+1}} M define an inverse system of categories (A_{n}-Mod | n ∈ ℕ). In such situation one can define a natural functor

(5.3.15)  A_{∞}-Mod_{top} → 2-lim_{n∈ℕ} A_{n}-Mod

from the category of topological A_{∞}-modules whose topology is I-c-adic (and of continuous A_{∞}-linear morphisms) to the 2-limit of the foregoing inverse system of categories (see [43, Ch.I] for generalities on 2-categories). Namely, let I := Ker(A_{∞} → A_{0}); then to an A_{∞}- module M one associates the compatible system (M/\overline{I^{n+1}M} | n ∈ ℕ). Lemma 5.3.16. The functor (5.3.15)is an equivalence of categories.

Proof. We claim that a quasi-inverse to (5.3.15) can be given by associating to any compatible system (M_{n} | n ∈ ℕ) the A_{∞}-module M_{∞} := lim_{n∈ℕ}M_{n}, with the linear topology generated by the submodules K_{n} := Ker(M_{∞} → M_{n}), for all n ∈ ℕ. In order to show that this topology on M_{∞} is I-c-adic (where I := Ker(A_{∞} → A_{0})), it suffices to verify that K_{n} = \overline{I^{n+1}M_{∞}} for every n ∈ ℕ. Clearly I^{n+1}M_{∞} ⊂ K_{n}, hence we are reduced to showing that K_{n} ⊂ K_{m} + I^{n+1}M_{∞} for every m > n or equivalently, that K_{n}/K_{m} equals the image of I^{n+1}M_{∞} inside M_{m}, which is obvious. □

In the following we will seek conditions under which (5.3.15)can be refined to equivalences between interesting subcategories, for instance to almost finitely generated, or almost projective modules. Lemma 5.3.17. Let M_{∞} be a topological A_{∞}-module whose topology is I-c-adic (for some ideal of definition I ⊂ A_{∞}). Suppose that N ⊂ M_{∞} is a finitely generated submodule such that \overline{N} is open in M_{∞}. Then :

(i) N = \overline{N}.

(ii) Every finite set of generators x_{1}, ..., x_{r} of N determines an open morphism A_{∞}^{r} → N. (iii) For every open ideal J ⊂ A_{∞}, the submodule JN is open in M_{∞}.

Proof. Mutatis mutandis, this is the same as the proof of lemma 5.3.8; the details can thus be safely entrusted to the reader. □

Lemma 5.3.18. Keep the notation of (5.3.14) and assume that I is a tight ideal. Let M be a topological A_{∞}-module whose topology is I-c-adic and let (M_{n} | n ∈ ℕ) be the image of M under (5.3.15). Suppose also that M_{0} is an almost finitely generated A_{0}-module. Then:

(i) M admits an open finitely generated submodule.

(ii) M is almost finitely generated.

(iii) M ⊗_{A_{∞}} A_{n} ≃ M_{n} for every n ∈ ℕ.

Proof. Using lemma 3.2.25(ii) and the hypothesis on M_{0} we deduce that M_{n} is almost finitely generated for every n ∈ ℕ. Choose finitely generated 𝔪_{0} ⊂ 𝔪 and n ≥ 0 such that I^{n} ⊂ 𝔪_{0} ·A; it follows easily that there are almost elements x_{1}, ..., x_{t} of M whose images in M_{n+1} generate a submodule Q such that 𝔪_{0} · M_{n+1} ⊂ Q. This implies that \overline{I^{n+1}M}/\overline{I^{n+2}M} ⊂ Q. Let N ⊂ M be the submodule generated by x_{1}, ..., x_{t}. Then, arguing as in the proof of proposition 5.3.12 we see that \overline{I^{n+1}M} ⊂ \overline{N}. Hence N = \overline{N} in view of lemma 5.3.17(i), so (i) holds. Furthermore, by lemma 5.3.17(iii), \overline{I^{n}} · M is open for every n ∈ ℕ (since it contains \overline{I^{n}} · N), so (iii) follows easily. Finally, M is almost finitely generated because the natural morphism M_{n} → M/N is an epimorphism. □

5.3.19. In the situation of (5.3.14), let M be an A_{∞}-module with an I-c-adic topology, and let (M_{n} | n ∈ ℕ) be its image under (5.3.15). We suppose now that M_{n} is an almost projective A_{n}-module for every n ∈ ℕ. Let us choose an epimorphism f : A_{∞}^{(S)} → M. We endow the free A_{∞}-module A_{∞}^{(S)} with the topology generated by the family of submodules (\overline{I^{n}} · A_{∞}^{(S)} | n ∈ ℕ). Notice that f is continuous for this topology, hence it extends to a continuous morphism f^{∧} : F_{∞} → M on the completion F_{∞} of A_{∞}^{(S)}. Lemma 5.3.20. With the notation of (5.3.19), we have:

(i) f^{∧} is topologically almost split, i.e., for every ε ∈ 𝔪 there exists a continuous morphism g : M → F_{∞} such that f^{∧} ∘ g = ε · 1_{M}.

(ii) If M is almost finitely generated, then M is almost projective.

Proof. By lemma 5.3.16 the set of continuous morphisms M → F_{∞} is in natural bijection with lim_{n∈ℕ} Hom_{A_{n}}(M_{n}, F_{n}), where F_{n} := F_{∞}/\overline{I^{n+1}F_{∞}} for every n ∈ ℕ. On the other hand, under the assumptions of the lemma, f induces an epimorphism:

lim_{n∈ℕ} alHom_{A_{n}}(M_{n}, F_{n}) → lim_{n∈ℕ} alHom_{A_{n}}(M_{n}, M_{n}).

Assertion (i) follows easily. Suppose now that M is almost finitely generated; then for every finitely generated subideal 𝔪_{0} ⊂ 𝔪 we can find m ≥ 0 and a morphism h : A_{∞}^{m} → M such that 𝔪_{0} · Coker h = 0. Let ε_{1}, ..., ε_{t} be a system of generators for 𝔪_{0}; then for every i, j ≤ t we can pick a morphism ϕ : A_{∞}^{(S)} → A_{∞}^{m} such that h ∘ ϕ = ε_{i} · ε_{j} · f; after taking completions, this relation becomes h ∘ ϕ^{∧} = ε_{i} · ε_{j} · f^{∧}. Choose g : M → F_{∞} as in (i); we deduce ε_{i} · ε_{j} · ε · 1_{M} = h ∘ ϕ^{∧} ∘ g, which as usual shows that ε_{i} · ε_{j} · ε annihilates alExt_{A_{∞}}^{1}(M, N) for every A_{∞}-module N, so (ii) holds. □

5.3.21. For a V^{a}-algebra A, let us denote by A-Mod_{afpr} the full subcategory of A-Mod consisting of all almost finitely generated projective A-modules. Let now (A_{n} | n ∈ ℕ) and I ⊂ A_{∞} be as in (5.3.14). We define natural functors

(5.3.22)  A_{∞}-Mod_{afpr} → 2-lim_{n∈ℕ} A_{n}-Mod_{afpr}

respectively (see (3.2.27)),

(5.3.23)  A_{∞}-Ét_{afp} → A_{0}-Ét_{afp}

by assigning to a given object M of A_{∞}-Mod_{afpr}, the compatible system (M ⊗_{A_{∞}} A_{n} | n ∈ ℕ) (resp. to an object B of A_{∞}-Ét_{afp}, the A_{0}-algebra B ⊗_{A_{∞}} A_{0}). Theorem 5.3.24. In the situation of (5.3.21), suppose that I is tight. Then (5.3.22)is an equivalence of categories.

Proof. We claim that a quasi-inverse to (5.3.22) is obtained by assigning to any compatible system (M_{n} | n ∈ ℕ) the A_{∞}-module M := lim_{n∈ℕ} M_{n}. Indeed, by the proof of lemma 5.3.16 there results that M is endowed with a natural I-c-adic topology; then by lemma 5.3.18(ii) we deduce that M is almost finitely generated, and finally lemma 5.3.20(ii) says that M is almost projective. From lemma 5.3.18(iii) it follows that the functor thus defined is a right quasiinverse to (5.3.22), so the latter is essentially surjective. Full faithfulness is a consequence of the following: Claim 5.3.25. Let P be an almost finitely generated projective A_{∞}-module. Then the natural morphism

(5.3.26)  P → lim_{n∈ℕ}(P ⊗_{A_{∞}} A_{n})

is an isomorphism. Proof of the claim: This is clear if P = A_{∞}^{r} for some r ≥ 0. For a general P one chooses, for every ε ∈ 𝔪, a sequence P → A_{∞}^{n} → P as in lemma 2.4.15 and applies to it the natural transformation (5.3.26); the claim follows by the usual diagram chase. □

Theorem 5.3.27. In the situation of (5.3.21), suppose that I is tight. Then (5.3.23)is an equivalence of categories.

Proof. It follows easily from theorem 3.2.28(ii) that the natural functor 2-lim_{n∈ℕ}A_{n}-Ét_{afp} →

A_{0}-Ét_{afp} is an equivalence. Hence we are reduced to showing that the natural functor

(5.3.28)  A_{∞}-Ét_{afp} → 2-lim_{n∈ℕ}A_{n}-Ét_{afp}

is an equivalence. To this aim, we claim that the rule (B_{n} | n ∈ ℕ) ↦ lim_{n∈ℕ}B_{n} defines a quasiinverse to (5.3.28). Taking into account theorem 5.3.24, this will follow once we have shown:

Claim 5.3.29. Let B be an almost finitely generated projective A_{∞}-algebra such that B_{0} := B ⊗_{A_{∞}} A_{0} is unramified. Then B is unramified. Proof of the claim: To start out, we apply theorem 5.2.12(ii) with J = I^{n} to deduce, by induction on n, that B/I^{n+1}B is unramified over A_{∞}/I^{n+1} for every n ∈ ℕ. Set B_{n} := B ⊗_{A_{∞}} A_{n}; it follows that B_{n} is unramified over A_{n} for every n ∈ ℕ. By proposition 3.1.4 we deduce that there exists a compatible system of idempotent almost elements e_{n} of B_{n} ⊗_{A_{n}} B_{n}, such that e_{n} · I_{B_{n}/A_{n}} = 0 and μ_{B_{n}/A_{n}}(e_{n}) = 1 for every n ∈ ℕ. Hence we obtain an idempotent in (lim_{n∈ℕ}B_{n} ⊗_{A_{n}} B_{n})_{∗} ≃ lim_{n∈ℕ}(B_{n} ⊗_{A_{n}} B_{n})_{∗}; however, the latter is isomorphic to (B ⊗_{A_{∞}} B)_{∗}, in view of claim 5.3.25. Then conditions (ii) and (iii) of proposition 3.1.4 follow easily by

remarking that μ_{B/A_{∞}} = lim_{n∈ℕ}μ_{B_{n}/A_{n}} and I_{B/A_{∞}} = lim_{n∈ℕ}I_{B_{n}/A_{n}}. □

5.4. Henselian approximation of structures over adically complete rings. This section reviews and complements some results of Elkik’s article [31]. Especially, we wish to show how the main theorems of loc.cit. generalize to the case of not necessarily noetherian rings. In principle, this is known ([31, Ch.III, §4, Rem.2, p.587] explains briefly how to adapt the proofs to make them work in some non-noetherian situations), but we feel that it is worthwhile to give more details. Definition 5.4.1. Let R be a ring, F := R[X_{1}, ..., X_{N}] a free R-algebra of finite type, J ⊂ F a finitely generated ideal, and let S := F/J. We define an ideal of F by setting:

H_{R}(F, J) := Ann_{F} 𝔼xt_{S}^{1}(𝕃_{S/R}, J/J^{2}).

Lemma 5.4.2. In the situation of definition 5.4.1, we have:

(i) For any R-algebra R′ let F′ := R′ ⊗_{R} F. Then H_{R}(F, J) · F′ ⊂ H_{R′}(F′, JF′). (ii) The open subset Spec S \ V (H_{R}(F, J) · S) is the smooth locus of S over R. (iii) H_{R}(F, J) annihilates 𝔼xt_{S}^{1}(𝕃_{S/R}, N) for every S-module N.

Proof. According to [46, Ch.III, Cor.1.2.9.1] there is a natural isomorphism in D(S-Mod):

(5.4.3)  τ_{[−1}𝕃_{S/R} ≃ (0 → J/J^{2} →^{∂} S ⊗_{F} Ω_{F/R} → 0)

where ∂ is induced by the universal derivation d : F → Ω_{F/R}. Since Ω_{F/R} is a free F -module, we derive a natural isomorphism:

(5.4.4)  𝔼xt_{S}^{1}(𝕃_{S/R}, J/J^{2}) ≃ End_{S}(J/J^{2})/∂^{∗}Hom_{F} (Ω_{F/R}, J/J^{2}).

Let now R′ be an R-algebra and h ∈ F; in view of (5.4.4), the condition h ∈ H_{R}(F, J) means precisely that the scalar multiplication h : J/J^{2} → J/J^{2} factors through ∂. It follows that h : (J/J^{2}) ⊗_{R} R′ → (J/J^{2}) ⊗_{R} R′ factors through ∂ ⊗_{S} 1_{S′}. However, the latter map admits a factorization:

∂ ⊗_{S} 1_{S′} : (J/J^{2}) ⊗_{R} R′ →^{α} J′/J^{′2} →^{∂′} S′ ⊗_{F′} Ω_{F′/R′}

where α is a surjective map. It follows easily that the scalar multiplication h : J′/J^{′2} → J′/J^{′2} factors through ∂′, which shows (i).

To show (ii), let us first pick any h ∈ H_{R}(F, J); we have to prove that S_{h} is smooth over R. However, the foregoing shows that multiplication by h on J/J^{2} factors through S ⊗_{F} Ω_{F/R}; if now h is invertible, this means that ∂ is a split imbedding, therefore S_{h} is formally smooth over R by the Jacobian criterion [28, Ch.0, Th.22.6.1]. Since S is of finite presentation over R, the assertion follows. Conversely, let h ∈ F be an element such that S_{h} is smooth over R; we wish to show that h^{n} ∈ H_{R}(F, J) for n large enough. One can either prove this directly using the definition of H_{R}(F, J), or else by using the following lemma 5.4.6 and the well known fact that h^{n} ∈ H_{S} for n large enough.

Finally we recall the natural isomorphism:

Hom_{D(S-Mod)}(𝕃_{S/R}, N[1]) ≃ Hom_{D(S-Mod)}(τ_{[−1}𝕃_{S/R}, N[1])

which, in view of (5.4.3), shows that every morphism τ_{[−1}𝕃_{S/R} → N[1] factors through a map τ_{[−1}𝕃_{S/R} → J/J^{2}[1], whence (iii). □

5.4.5. In the situation of definition 5.4.1, choose a finite system of generators f_{1}, ..., f_{q} for J; it is shown in [31, Ch.0, §2] how to construct an ideal, called H_{S} in loc.cit. with the same properties as in lemma 5.4.2(i),(ii). However, the definition of H_{S} depends explicitly on the choice of the generators f_{1}, ..., f_{q}. This goes as follows. For every integer p and every multiindex (α) = (α_{1}, ..., α_{p}) ∈ ℕ^{p} with 1 ≤ α_{1} < α_{2}... < α_{p} ≤ q, set |α| := p and let J_{α} ⊂ J be the subideal generated by f_{α_{1}}, ..., f_{α_{p}} and Δ_{α} ⊂ F the ideal generated by the determinants of the minors of order p of the Jacobian matrix (∂f_{α_{i}}/∂X_{j} | 1 ≤ i ≤ p, 1 ≤ j ≤ N). We set

H_{S} := ∑_{p≥0} ∑_{|α|=p} Δ_{α} · (J_{α} : J).

Though we won’t be using the ideal H_{S}, we want to explain how it relates to the more intrinsic H_{R}(F, J). This is the purpose of the following: Lemma 5.4.6. With the notation of (5.4.5), we have: H_{S} ⊂ H_{R}(F, J).

Proof. Let p ∈ ℕ, α a multi-index with |α| = p, δ ∈ Δ_{p} and x ∈ F such that xJ ⊂ J_{α}. We can suppose that δ = det(M), where M = (∂f_{α_{j}}/∂X_{β_{i}} | 1 ≤ i, j ≤ p) for a certain multiindex β with |β| = p. We consider the maps ϕ_{α} : R^{p} → J/J^{2} : e_{i} ↦ f_{α_{i}} (mod J^{2}) and π_{β} : S ⊗_{F} Ω_{F/R} → R^{p} : dX_{β_{j}} ↦ e_{j} for j = 1, ..., p and dX_{k} ↦ 0 if k ∉ {β_{1}, ..., β_{p}}. Then the matrix of the composed S-linear map π_{β} ∘ ∂ ∘ ϕ_{α} : S^{p} → S^{p} is none else than M. Let M′ be the adjoint matrix of M, so that M · M′ = δ · 1_{S^{p}}, and we can compute:

ϕ_{α} ∘ M′ ∘ π_{β} ∘ ∂ ∘ ϕ_{α} = ϕ_{α} ∘ (δ · 1_{S^{p}}) = (δ · 1_{J/J^{2}}) ∘ ϕ_{α}.

In other words, the maps ϕ_{α}∘M′∘π_{β}∘∂ and δ·1_{J/J^{2}} agree on Im ϕ_{α} = (J_{α}+J^{2})/J^{2}. Therefore: (x · ϕ_{α}) ∘ M′ ∘ π_{β} ∘ ∂ = x · δ · 1_{J/J^{2}}, i.e the scalar multiplication by x · δ on J/J^{2} factors through ∂; as in the proof of lemma 5.4.2 this implies that x · δ ∈ H_{R}(F, J), as claimed. □

5.4.7. Let R be a (not necessarily noetherian) ring, t ∈ R a non-zero-divisor, I ⊂ R an ideal, S an R-algebra of finite presentation, which we write in the form S = F/J for some finitely generated ideal J ⊂ F := R[X_{1}, ..., X_{N}]. For given a := (a_{1}, ..., a_{N}) ∈ R^{N}, let 𝔭_{a} ⊂ F be the ideal generated by (X_{1} − a_{1}, ..., X_{N} − a_{N}). Lemma 5.4.8. In the situation of (5.4.7), let n, h ≥ 0 be two integers and a ∈ R^{N} such that

(5.4.9)  t^{h} ∈ H_{R}(F, J) + 𝔭_{a}  J ⊂ 𝔭_{a} + t^{n}IF  n > 2h.

Then there exists b ∈ R^{N} such that

b − a ∈ t^{n−h}IR^{N} and J ⊂ 𝔭_{b} + (t^{n−h}I)^{2}F.

Proof. We are given a morphism σ : Spec R/t^{n}I → Spec S whose restriction to the closed subscheme Spec R/t^{n−h}I we denote by σ_{0}. The claim amounts to saying that there exists a lifting σ̃ : Spec R/(t^{n−h}I)^{2} → Spec S of σ_{0}. By proposition 3.2.16, the obstruction to the existence of an extension of σ to a morphism Spec R/(t^{n}I)^{2} → Spec S is a class ω ∈ 𝔼xt_{S}^{1}(𝕃_{S/R}, t^{n}I/(t^{n}I)^{2}). We have a commutative diagram:

\xymatrix{t^{n}I/(t^{n}I)^{2} \ar[r]^{α} \ar[d]_{β} & t^{n−h}I/(t^{n−h}I)^{2} \ar[d]^{γ} \\ t^{n}I/(t^{n}I)^{2} \ar[r]^{δ} & t^{n}I/t^{2n−h}I^{2}}

where α is induced by the inclusion t^{n}I ⊂ t^{n−h}I, β is the scalar multiplication by t^{h}, δ is the restriction to t^{n}I of the natural projection R/(t^{n}I)^{2} → R/t^{2n−h}I^{2}, and γ is an isomorphism induced by scalar multiplication by t^{h} : t^{n−h}I →^{∼} t^{n}I. Since the S-module structure on t^{n}I/(t^{n}I)^{2} is induced by extension of scalars via σ, it is clear that 𝔭_{a} · ω = 0; on the other hand, by lemma 5.4.2(iii) we know that H_{R}(F, J) · ω = 0, hence t^{h} · ω = 𝔼xt_{S}^{1}(𝕃_{S/R}, β)(ω) = 0, and consequently 𝔼xt_{S}^{1}(𝕃_{S/R}, α)(ω) = 0. Since the latter class is the obstruction to the existence of σ̃, the assertion follows. □

Lemma 5.4.8 is the basis of an inductive procedure that allows to construct actual sections of X := Spec S, starting from approximate solutions of the system of equations defined by the ideal J. The section thus obtained live in X(R^{∧}), where R^{∧} is the tI-adic completion of R. In case I is finitely generated, R^{∧} is (separated) complete for the (tI)^{∧}-adic topology, where (tI)^{∧} is the topological closure of tI in R^{∧}; however, in later sections we will find situations where the relevant ideal I is not finitely generated; in such case, R^{∧} is complete only for the (tI)^{∧}-c-adic topology (see definition 5.3.1(v)). In this section we carry out a preliminary study of some topologies on R-modules and on sets of sections of R-schemes; one of the main themes is to compare the topologies of, say Z(R) where Z is an R-scheme, and of Z(R^{∧}), where R^{∧} is an adic completion of R. For this reason, it is somewhat annoying that, in passing from R to its completion, one is forced to replace a preadic topology by a c-preadic one. That is why we prefer to use a slightly coarser topology on R, described in the following: Definition 5.4.10. Let R be any ring, t ∈ R a non-zero-divisor, I ⊂ R an ideal, M_{0} (resp. N) a finitely generated R-module (resp. R[t^{−1}]-module).

(i) The (t, I)-preadic topology of M_{0} is the linear topology that admits the family of submodules (t^{n}IM_{0} | n ∈ ℕ_{0}) as a cofinal system of open neighborhoods.

(ii) The (t, I)-preadic topology of N is the linear topology ℒ_{N} defined as follows. Choose a finitely generated R-module N_{0} with an R-linear map ϕ : N_{0} → N such that ϕ[t^{−1}] : N_{0}[t^{−1}] → N is onto; then ℒ_{N} is the finest topology such that ϕ_{0} becomes an open map when we endow N_{0} with the (t, I)-preadic topology. Therefore, the family of Rsubmodules (ϕ(t^{k}IN_{0}) | k ∈ ℕ) is a cofinal system of open neighborhoods of 0 ∈ N for the topology ℒ_{N}.

The advantage of the (t, I)-adic topology is that the (t, I)-adic completion R^{∧} of R is complete for a topology of the same type, namely for the (t, I^{∧})-topology (where I^{∧} ⊂ R^{∧} is the topological closure of I). Hence, for every R-scheme Z, the topologies on Z(R) and Z(R^{∧}) admit a uniform description. After this caveat, let us stress nevertheless that all of the results of this section hold as well for the c-adic topologies.

5.4.11. In the situation of definition 5.4.10, it is easy to verify that the (t, I)-preadic topology on N does not depend on the choice of N_{0} and ϕ. Indeed, if ϕ′ : N_{1} → N is another choice, then one checks that ϕ(t^{k}N_{0}) ⊂ ϕ′(N_{1}) for k ∈ ℕ large enough, and symmetrically ϕ_{1}(t^{k}N_{1}) ⊂ ϕ_{0}(N_{0}), which implies the assertion.

Moreover, it is easy to check that every R[t^{−1}]-linear map M → N of finitely generated R[t^{−1}]-modules is continuous for the respective (t, I)-preadic topologies. Lemma 5.4.12. Let R be a noetherian ring, ℐ ⊂ R an ideal, and suppose that the pair (R, ℐ) is henselian. Let \overline{R} be the ℐ -adic completion of R and set \overline{S} := \overline{R} ⊗_{R} S. Let U ⊂ Spec S be an open subset smooth over Spec R, and \overline{U} ⊂ Spec \overline{S} the preimage of U. Then, for every integer n and every \overline{R}-section \overline{σ} : Spec \overline{R} → Spec \overline{S} whose restriction to Spec \overline{R} \ V (ℐ \overline{R}) factors through \overline{U}, there exists an R-section σ : Spec R → Spec S congruent to \overline{σ} modulo ℐ^{n}, and whose restriction to Spec R \ V (ℐ) factors through U.

Proof. It is [31, Ch.II, Th.2 bis]. □

Proposition 5.4.13. Keep the notation of (5.4.7)and let I ⊂ R be an ideal such that (R, tI) is a henselian pair. Let a ∈ R^{N} and n, h ≥ 0 such that (5.4.9) holds. Then there exists b ∈ R^{N} such that b − a ∈ t^{n−h}IR^{N} and J ⊂ 𝔭_{b}.

Proof. We consider first the following special case:

Claim 5.4.14. The proposition holds if R is complete for the tI-adic topology.

Proof of the claim: We apply repeatedly lemma 5.4.8 to obtain a tI-adically convergent sequence of elements (a_{m} ∈ R^{N} | m ∈ ℕ), with a_{0} := a and such that a_{m} ≡ a (mod t^{n−h}IR^{N}) for every m ∈ ℕ. The limit b of the sequence (a_{m} | m ∈ ℕ) will do.

Let next R be a general ring; let H ⊂ H_{R}(F, J) be a finitely generated subideal such that t^{h} ∈ H. We can write R = ⋃_{λ∈Λ} R_{λ} for a filtered family of noetherian subrings (R_{λ} | λ ∈ Λ) such that t ∈ R_{λ} and a ∈ R_{λ}^{N} for every λ ∈ Λ, and we set I_{λ} := I ∩ R_{λ} for every λ ∈ Λ. We can also assume that the pair (R_{λ}, tI_{λ}) is henselian for every λ ∈ Λ. Let f_{1}, ..., f_{q} ∈ F be a finite set of generators for J and g_{1}, ..., g_{r} ∈ F a finite set of generators for H; up to restricting to a cofinal family, we can then assume that f_{i}, g_{j} ∈ F_{λ} := R_{λ}[X_{1}, ..., X_{N}] for every λ ∈ Λ and every i ≤ q, j ≤ r. Let J_{λ} ⊂ F_{λ} be the ideal generated by f_{1}, ..., f_{q} and set S_{λ} := F_{λ}/J_{λ}; let also H_{λ} ⊂ F_{λ} be the ideal generated by g_{1}, ..., g_{r}. Again after replacing Λ by a cofinal subfamily, we can achieve that H_{λ} ⊂ H_{R_{λ}}(F_{λ}, J_{λ}) for every λ ∈ Λ. Finally, let 𝔭_{λ,a} ⊂ F_{λ} be the ideal generated by X_{1} − a_{1}, ..., X_{N} − a_{N}; we can assume that t^{h} ∈ H_{λ} + 𝔭_{λ,a} and J_{λ} ⊂ 𝔭_{λ,a} + t^{n}I_{λ}F_{λ} for every λ ∈ Λ. With this setup, let \overline{R}_{λ} be the tI_{λ}-adic completion of R_{λ}; we can apply claim 5.4.14 to deduce that there exists c ∈ \overline{R}_{λ}^{N} such that c − a ∈ t^{n−h}I_{λ}\overline{R}_{λ}^{N} and such that J_{λ} ⊂ \overline{𝔭}_{λ,c} (where \overline{𝔭}_{λ,c} denotes the ideal generated by X_{1} − c_{1}, ..., X − c_{n} in \overline{R}_{λ}[X_{1}, ..., X_{N}]). Let U_{λ} := Spec S_{λ} \ V (H_{λ} · S_{λ}) and \overline{S}_{λ} := \overline{R}_{λ} ⊗_{R_{λ}} S; from lemma 5.4.2(ii) we know that U_{λ} is smooth over Spec R_{λ}, and by construction c determines a \overline{R}_{λ}-section of Spec \overline{S}_{λ} whose restriction to Spec \overline{R}_{λ} \ V (tI_{λ}) factors through the preimage of U_{λ}. Therefore lemma 5.4.12 ensures that there exists an R_{λ}-section σ : Spec R_{λ} → Spec S_{λ} that agrees with c modulo (tI_{λ})^{n−h} and whose restriction to Spec R_{λ}\V (tI_{λ}) factors through U. Let π_{λ} : F_{λ} → S_{λ} be the natural projection; there is a unique b ∈ R_{λ}^{N} such that 𝔭_{λ,b} = π_{λ}^{−1}(Ker σ^{♯} : S_{λ} → R_{λ}); this point b has the sought properties. □

5.4.15. Let now X be an affine scheme of finite type over Spec R[t^{−1}]; for every n ∈ ℕ let 𝔸_{R[t^{−1}]}^{n} be the n-dimensional affine space over Spec R[t^{−1}]; for n large enough we can find a closed imbedding j : X → 𝔸_{R[t^{−1}]}^{n} of Spec R[t^{−1}]-schemes. The choice of coordinates on 𝔸_{R[t^{−1}]}^{n}, yields a bijection 𝔸_{R[t^{−1}]}^{n}(R[t^{−1}]) ≃ R[t^{−1}]^{n}, and then j induces an injective map j_{∗} : X(R[t^{−1}]) ⊂ R[t^{−1}]^{n}. The (t, I)-adic topology of X(R[t^{−1}]) is defined as the subspace topology induced by j_{∗} (where R[t^{−1}]^{n} is endowed with its (t, I)-preadic topology). Lemma 5.4.16. Let X be as in (5.4.15); we have:

(i) The (t, I)-adic topology of X(R[t^{−1}]) is independent of the choice of closed imbedding j and of coordinates on 𝔸_{R[t^{−1}]}^{n}.

(ii) If Y is another affine R[t^{−1}]-scheme, then the natural map (X ×_{R[t^{−1}]} Y)(R[t^{−1}]) → X(R[t^{−1}]) × Y (R[t^{−1}]) is a homeomorphism for the (t, I)-adic topologies of the corresponding schemes.

(iii) If U ⊂ X is any open subset, and σ ∈ U(R[t^{−1}]), there exists f ∈ 𝒪_{X}(X) such that

D(f) := X \ V (f) ⊂ U and such that σ factors through D(f). (iv) If U ⊂ X is an affine open subset, then the (t, I)-adic topology on U(R[t^{−1}]) agrees

with the topology induced from the (t, I)-adic topology on X(R[t^{−1}]), and U(R[t^{−1}]) is an open subset of X(R[t^{−1}]).

Proof. (i): suppose j_{1} : X → 𝔸_{R[t^{−1}]}^{n} and j_{2} : X → 𝔸_{R[t^{−1}]}^{m} are two closed imbeddings, τ_{1} and τ_{2} the respective topologies on X(R[t^{−1}]); by symmetry it suffices to show that the identity map (X(R[t^{−1}]), τ_{1}) → (X(R[t^{−1}]), τ_{2}) is continuous. However, j_{2} can be extended to some morphism ϕ : 𝔸_{R[t^{−1}]}^{n} → 𝔸_{R[t^{−1}]}^{m}, and we come down to showing that the induced map ϕ_{∗} : R[t^{−1}]^{n} → R[t^{−1}]^{m} is continuous for the (t, I)-adic topology. We can further reduce to the case of m = 1, in which case ϕ is given by a single polynomial f ∈ R[t^{−1}, T_{1}, ..., T_{n}], and ϕ_{∗} is the map (a_{1}, ..., a_{n}) ↦ f(a_{1}, ..., a_{n}). Let x_{0} ∈ R[t^{−1}]^{n} and set y_{0} := f(x_{0}); we have to show that, for every k ∈ ℕ there exists h ∈ ℕ such that f(x_{0} + t^{h}I) ⊂ y + t^{k}I. The Taylor formula gives an identity of the form: f(T) = y_{0} + ∑_{r∈ℕ^{n}\{0}} a_{r} · (T − x_{0})^{r}, where a_{r} ∈ R[t^{−1}] for every r ∈ ℕ^{n} and a_{r} = 0 for all but finitely many r. Let s ∈ ℕ be an integer large enough so that t^{s}a_{r} ∈ R for every r ∈ ℕ^{n}; obviously h := k + s will do. (ii) is easily reduced to the corresponding statement for (t, I)-adic topologies on direct sums of R[t^{−1}]-modules; we leave the details to the reader.

(iii): we can write U = X \ V (J), where J ⊂ S := 𝒪_{X}(X), and a section σ ∈ U(R[t^{−1}]) induces a map ϕ : S → R[t^{−1}] such that ϕ(J)R = R. Let a_{i} ∈ R and f_{i} ∈ J, i = 1, ..., n with ∑_{i} a_{i} · ϕ(f_{i}) = 1 and set f := ∑_{i} f_{i}a_{i}; it follows that the image of σ is contained in the affine open subset Spec S[f^{−1}] ⊂ U. To show (iv), we can suppose, thanks to (iii), that U = X \V (f) for some f ∈ 𝒪_{X}(X). Let ϕ_{f} : X → 𝔸_{R[t^{−1}]}^{1} be the morphism defined by f, and Γ_{f} : X → X ×_{R[t^{−1}]} 𝔸_{R[t^{−1}]}^{1} its graph; choose a closed imbedding X ⊂ 𝔸_{R[t^{−1}]}^{n}. The composition j : X → X ×_{R[t^{−1}]} 𝔸_{R[t^{−1}]}^{1} → 𝔸_{R[t^{−1}]}^{n+1} is another closed imbedding, which induces the same topology on X(R[t^{−1}]) in view of (i). We have j^{−1}(𝔸_{R[t^{−1}]}^{n} ×_{R[t^{−1}]} 𝔾_{m,R[t^{−1}]}) = U, consequently we are reduced to showing the assertion for the open imbedding 𝔸_{R[t^{−1}]}^{n} ×_{R[t^{−1}]} 𝔾_{m,R[t^{−1}]} ⊂ 𝔸_{R[t^{−1}]}^{n+1}. Using (ii) we further reduce to considering the imbedding 𝔾_{m,R[t^{−1}]} ⊂ 𝔸_{R[t^{−1}]}^{1}. This case can be dealt with by explicit calculations. □

Lemma 5.4.17. Let R be any ring, X a quasi-projective R-scheme. Then every R-section Spec R → X of X factors through an open imbedding U ⊂ X, where U is an affine R-scheme.

Proof. By assumption X is a locally closed subset of ℙ_{R}^{n}, for some n ∈ N. Claim 5.4.18. The lemma holds if X is a closed subscheme of ℙ_{R}^{n}. Proof of the claim: Indeed, in this case we can assume X = ℙ_{R}^{n}. Therefore, let σ ∈ ℙ_{R}^{n}(R); by [27, Ch.II, Th.4.2.4] σ corresponds to a rank one locally free quotient L of R^{n+1}. We choose a section of the projection π_{1} : R^{n+1} → L, hence a decomposition R^{n+1} ≃ L ⊕ Ker π_{1}; the induced projection π_{2} : R^{n+1} → Ker π_{1} determines a closed imbedding ℙ(Ker π_{1}) → ℙ_{R}^{n} representing the transformation of functors that assigns to every rank one locally free quotient of Ker π_{1} the same module, seen as a quotient of R^{n+1} via the projection π_{2}. It is clear that the image of σ does not intersect the image of ℙ(Ker π_{1}), and the complement ℙ_{R}^{n} \ ℙ(Ker π_{1}) is affine (π_{1} is a section in Γ(ℙ_{R}^{n}, 𝒪_{ℙ_{R}^{n}}(1))).

Thanks to claim 5.4.18 we can assume that X is an open subscheme of an affine scheme Y of finite type over SpecR. In this case, the claim reduces to lemma 5.4.16(iii) (applied with t = 1). □

5.4.19. Let now X be a quasi-projective R[t^{−1}]-scheme; each affine open subset of X comes with a natural (t, I)-adic topology. By lemma 5.4.16(iv) these topologies agree on the intersections of any two such affine open subsets, and according to lemma 5.4.17 we have X(R[t^{−1}]) = ⋃_{U} U(R[t^{−1}]), where U ranges on the family of all affine open subsets U ⊂ X, so X(R[t^{−1}]) can be endowed with a well defined (t, I)-adic topology, independent of all choices. Lemma 5.4.20. The closed subscheme U_{n} of Spec ℤ[x_{11}, ..., x_{nn}] that classifies the n × n idempotent matrices is smooth over Spec ℤ.

Proof. Clearly U_{n} is of finite type over Spec ℤ, hence it suffices to show that U_{n} is formally smooth. Therefore, let R_{0} be a ring and I ⊂ R_{0} an ideal with I^{2} = 0; we need to show that the induced map U_{n}(R_{0}) → U_{n}(R_{0}/I) is surjective, i.e. that every n × n idempotent matrix \overline{M} with entries in R_{0}/I lifts to an idempotent matrix with entries in R_{0}. Pick an arbitrary matrix M ∈ M_{n}(R_{0}) that lifts \overline{M}; let E := R_{0}[M] ⊂ M_{n}(R_{0}) be the commutative R_{0}-algebra generated by M, \overline{E} ⊂ M_{n}(R_{0}/I) be the image of E, J ⊂ E the kernel of the induced map E → \overline{E}. We have J^{2} = 0, so we can apply proposition 2.5.13(i) to lift \overline{M} to some idempotent matrix in E. □

Proposition 5.4.21. Let t ∈ R be a non-zero-divisor, I ⊂ R an ideal, R^{∧} := lim_{n∈ℕ}R/t^{n}I the

(t, I)-adic completion of R, I^{∧} the topological closure of I in R^{∧}, and suppose that the pair (R, tI) is henselian. Let X be a smooth quasi-projective R[t^{−1}]-scheme, and endow X(R[t^{−1}]) (resp. X(R^{∧}[t^{−1}])) with its (t, I)-adic (resp. (t, I^{∧})-adic) topology. Then the natural map X(R[t^{−1}]) → X(R^{∧}[t^{−1}]) has dense image.

Proof. We begin with the following special case: Claim 5.4.22. The proposition holds if X is affine. Proof of the claim: Say that X = Spec S, where S is some finitely presented smooth R[t^{−1}]- algebra, and let σ : S → R^{∧}[t^{−1}] be any element of X(R^{∧}[t^{−1}]). We have to show that there are elements of X(R[t^{−1}]) in every (t, I)-adic neighborhood of σ. To this aim, we pick a finitely presented R-algebra S_{0} such that S_{0}[t^{−1}] ≃ S; after clearing some denominators we can assume that σ extends to a map S_{0} → R^{∧}. Let

(5.4.23)  S_{0} = R[X_{1}, ..., X_{N}]/J

be a finite presentation of S_{0}, and set H := H_{R}(R[X_{1}, ..., X_{N}], J) (notation of definition 5.4.1). By assumption we have

(5.4.24)  t^{h} ∈ H + J

for h ∈ ℕ large enough. The presentation (5.4.23) defines a closed imbedding X ⊂ 𝔸_{R}^{N}, and we can then find a section σ_{0} : Spec R → 𝔸_{R}^{N} that is (t, I)-adically close to σ, so that the restrictions of σ and σ_{0} agree on Spec R/t^{n}I. Let 𝔭 := Ker σ_{0}^{♯} ⊂ R[X_{1}, ..., X_{N}] be the ideal corresponding to σ_{0}; it then follows that

(5.4.25)  J ⊂ 𝔭 + t^{n}IR[X_{1}, ..., X_{N}].

From (5.4.24) and (5.4.25) we deduce that t^{h} ∈ H + t^{n}IR[X_{1}, ..., X_{N}] + 𝔭; up to enlarging n (which is harmless) we can assume that n > 2h. Let \overline{H} := (H + 𝔭)/𝔭 ⊂ R; we derive: t^{h} ∈ \overline{H} + t^{n}I, whence t^{h}(1 + at^{k}) ∈ \overline{H} for some a ∈ I and k > 0; since tI ⊂ rad(R), it follows:

(5.4.26)  t^{h} ∈ H + 𝔭.

By assumption we have Ann_{R}(t) = 0, hence from (5.4.24), (5.4.26) and proposition 5.4.13 we deduce the contention.

Claim 5.4.27. Suppose that X = ℙ_{R[t^{−1}]}^{r}. Then every section σ : Spec R^{∧}[t^{−1}] → X factors through an open imbedding U ⊂ X, where U is an affine R-scheme. Proof of the claim: By [27, Ch.II, Th.4.2.4], σ corresponds to a rank one projective quotient L of (R^{∧}[t^{−1}])^{r+1}; we can then find an idempotent (r+1)×(r+1) matrix e such that Coker(e) = L. By lemma 5.4.20, the scheme U that represents the (r + 1) × (r + 1) idempotent matrices is smooth; clearly U decomposes as a disjoint union of open and closed subschemes U = ⋃_{n=0}^{r+1} U_{n}, where U_{n} represents the subfunctor that classifies all (r + 1) × (r + 1) idempotent matrices of rank n, for every n = 0, ..., r + 1. Therefore each U_{n} is smooth and affine over Spec ℤ and then by claim 5.4.22 it follows that e can be approximated closely by an idempotent matrix e_{0} ∈ M_{r+1}(R[t^{−1}]) whose rank equals the rank of e. We have e_{0} · e = e_{0} · (I_{r+1} − e_{0} + e) and if e_{0} is sufficiently (t, I)-adically close to e, the matrix I_{r+1}−e_{0}+e is invertible in M_{r+1}(R), hence we can assume that:

(5.4.28)  Im(e_{0} · e) = Im(e_{0}) ⊂ R^{∧}[t^{−1}]^{r+1}.

The projection

π : R[t^{−1}]^{r+1} → Im e_{0} : x ↦ e_{0}(x)

determines a closed imbedding ℙ(Im e_{0}) → ℙ_{R[t^{−1}]}^{r} representing the transformation of functors that, to every R[t^{−1}]-algebra S and every rank one projective quotient of S⊗_{R[t^{−1}]}Im(e_{0}) assigns the same module seen as a quotient of S ⊗_{R[t^{−1}]} R[t^{−1}]^{r+1} via π. By (5.4.28), the restriction of 1_{R^{∧}} ⊗_{R} π to R^{∧} ⊗_{R} Im(e) is an isomorphism, hence it remains such after every base change R^{∧}[t^{−1}] → S; this means that the image of σ lands in the complement of ℙ(Im e_{0}), which implies the contention.

Claim 5.4.29. Let X be any quasi-projective R[t^{−1}]-scheme, σ : Spec R^{∧}[t^{−1}] → X any section. Then σ factors through an open imbedding U ⊂ X, where U is an affine R-scheme. Proof of the claim: Due to claim 5.4.27 we can assume that X is an open subscheme of an affine R[t^{−1}]-scheme of finite type. Thus, we can write X = Y \ V (J), where J ⊂ S := 𝒪_{Y} (Y), and a section σ ∈ X(R^{∧}[t^{−1}]) induces a map ϕ : S → R^{∧}[t^{−1}] such that ϕ(J)R^{∧}[t^{−1}] = R^{∧}[t^{−1}]. Let a_{i} ∈ R^{∧}[t^{−1}], f_{i} ∈ J such that ∑_{i} a_{i} · ϕ(f_{i}) = 1; we choose b_{i} ∈ R[t^{−1}], (t, I)-adically close to a_{i} (i = 1, ..., n), so that ∑_{i} ϕ(f_{i})·(a_{i} −b_{i}) ∈ tIR^{∧}. Since tIR^{∧} ⊂ rad(R^{∧}), we deduce easily that ∑_{i} ϕ(f_{i})b_{i} ∈ R^{∧} is invertible in R^{∧}. Set f := ∑_{i} f_{i}b_{i}; it follows that the image of σ is contained in the affine open subset Spec S[f^{−1}] ⊂ X.

The proposition follows easily from claims (5.4.22)and (5.4.29). □

Proposition 5.4.30. Resume the assumptions of proposition 5.4.21 and let ϕ : X → Y be a morphism of quasi-projective R[t^{−1}]-schemes; then we have:

(i) The map ϕ_{∗} : X(R[t^{−1}]) → Y (R[t^{−1}]) is continuous for the (t, I)-adic topologies. (ii) If ϕ is smooth, ϕ_{∗} is an open map.

Proof. (i): we can factor ϕ as a closed imbedding X → X × Y followed by a projection X × Y → Y, so it suffices to prove the claim for the latter maps. The case of a projection is immediate, and the case of a closed imbedding follows straightforwardly from lemma 5.4.16(i).

(ii): due to lemmata 5.4.16(iv) and 5.4.17 we can assume that both X and Y are affine schemes, say X = Spec S, Y = Spec T where S and T are finitely generated R[t^{−1}]-algebras and S is smooth over T, especially S is a finitely presented T -algebra. Let σ : SpecR[t^{−1}] → X be an element of X(R[t^{−1}]); after choosing presentations for S and T and clearing some denominators, we can find morphisms of finitely presented R-algebras f : T_{0} → S_{0} and g : S_{0} → R such that f[t^{−1}] = ϕ^{♯} : T → S and g[t^{−1}] = σ^{♯} : S → R[t^{−1}]. Let X_{0} := Spec S_{0}, Y_{0} := Spec T_{0}, σ_{0} := Spec(g). We have to show that, for every section ξ : Spec R[t^{−1}] → Y sufficiently close to σ, there is ξ′ ∈ X(R[t^{−1}]) such that ϕ_{∗}(ξ′) = ξ. Claim 5.4.31. There is an open neighborhood U ⊂ Y (R[t^{−1}]) of ϕ_{∗}(σ) such that every ξ ∈ U extends to a section ξ_{0} : Spec R → Y_{0}. Proof of the claim: By construction, ϕ_{∗}(σ) is induced by the map g ∘ f : S_{0} → R; any other section ξ ∈ Y (R[t^{−1}]) is determined by a map ξ^{♯} : S → R[t^{−1}]; we have S = S_{0}[t^{−1}] and S_{0} is of finite type over R, say S_{0} = R[x_{1}, ..., x_{N}]/J. Therefore a_{i} := g ∘f(x_{i}) ∈ R for every i ≤ N. The claim follows by observing that the set of sections ξ ∈ Y (R[t^{−1}]) such that ξ^{♯}(x_{i}) − a_{i} ∈ R for every i ≤ N forms an open neighborhood of ϕ_{∗}(σ).

Next, choose a presentation

(5.4.32)  S_{0} ≃ T_{0}[x_{1}, ..., x_{N}]/J

and let H := H_{T_{0}}(T_{0}[x_{1}, ..., x_{N}], J) (notation of definition 5.4.1). Since S is smooth over T, we have

(5.4.33)  t^{h} ∈ H + J

for h ∈ ℕ large enough. Let ξ ∈ Y (R[t^{−1}]); by claim 5.4.31 we can suppose that ξ extends to a section ξ_{0} : Spec R → Y_{0}. Define X_{0}(ξ_{0}) as the fibre product in the cartesian diagram

\xymatrix{X_{0}(ξ_{0}) \ar[r] \ar[d] & X_{0} \ar[d]^{ϕ_{0}} \\ Spec R \ar[r]^{ξ_{0}} & Y_{0}.}

The presentation (5.4.32) induces a closed imbedding X_{0}(ξ_{0}) ⊂ 𝔸_{R}^{N} whose defining ideal J(ξ_{0}) ⊂ R[x_{1}, ..., x_{N}] is the image of J; define similarly H(ξ_{0}) ⊂ R[x_{1}, ..., x_{N}] as the image of H. From (5.4.33)we deduce that

(5.4.34)  t^{h} ∈ H(ξ_{0}) + J(ξ_{0}).

Suppose now that ξ is sufficiently close to σ; this means that the restrictions of ξ_{0} and ϕ_{0∗}(σ_{0}) agree on some closed subset Spec R/t^{n}I ⊂ Spec R. Hence, let \overline{σ}_{0} : Spec R/t^{n}I → X_{0}(ξ) be the restriction of σ_{0}, and choose any extension of \overline{σ}_{0} to a morphism σ_{1} : Spec R → 𝔸_{R}^{N}. Finally, let 𝔭 := Ker σ_{1}^{♯} ⊂ R[x_{1}, ..., x_{N}] be the ideal corresponding to σ_{1}. By construction we have

(5.4.35)  J(ξ_{0}) ⊂ 𝔭 + t^{n}IR[x_{1}, ..., x_{N}].

Now, arguing as in the proof of claim 5.4.22 we see that (5.4.34) and (5.4.35) imply t^{h} ∈ H(ξ_{0}) + 𝔭, at least if n > 2h, which can always be arranged. Finally, proposition 5.4.13 shows that \overline{σ}_{0} can be extended to a section ξ′ : Spec R → X_{0}(ξ_{0}), as required. □

5.4.36. Resume the assumptions of proposition 5.4.21 and let (X_{0}, X_{1}, s, t, c, ι) be a groupoid of quasi-projective R[t^{−1}]-schemes (see (4.5.1)for our general notations concerning groupoids in a category). We have a natural map

(5.4.37)  π_{0}(X_{0}(R[t^{−1}])) → π_{0}(X_{0}(R^{∧}[t^{−1}]))

where, for any groupoid of sets G := (G_{0}, G_{1}, s_{G}, t_{G}, c_{G}, ι_{G}), we denote by π_{0}(G) the set of isomorphism classes of elements of G_{0}; this is the same as the set of connected components of the geometric realization of the simplicial set associated to the groupoid G. Theorem 5.4.38. Keep the notation of (5.4.36), and suppose that X_{0} is smooth over R[t^{−1}] and that the morphism (s, t) : X_{1} → X_{0} ×_{R[t^{−1}]} X_{0} is smooth. Then (5.4.37)is a bijection.

Proof. Let σ ∈ X_{0}(R^{∧}[t^{−1}]); since X_{0} is smooth over R[t^{−1}] we can find sections σ_{0} ∈ X_{0}(R^{∧}[t^{−1}]) arbitrarily (t, I)-adically close to σ (proposition 5.4.21). Then (σ, σ_{0}) can be made arbitrarily close to (σ, σ) ∈ X_{0} ×_{R[t^{−1}]} X_{0}(R^{∧}[t^{−1}]); since the latter lies in the image of X_{1}(R^{∧}[t^{−1}]) under the morphism (s, t), it follows that the same holds for (σ, σ_{0}), provided σ_{0} is sufficiently close to σ (proposition 5.4.30(ii)). This shows that (5.4.37) is onto. Next, suppose that σ, τ ∈ X_{0}(R[t^{−1}]) and that their images σ^{∧}, τ^{∧} in X_{0}(R^{∧}[t^{−1}]) lie in the same homotopy class. By definition, this means that (σ^{∧}, τ^{∧}) = (s, t)(α) for some α ∈ X_{1}(R^{∧}[t^{−1}]); since X_{1} is smooth over R[t^{−1}] it follows that there exist sections α_{0} ∈ X_{1}(R[t^{−1}]) arbitrarily close to α (again proposition 5.4.21). Since (s, t)_{∗} is continuous (proposition 5.4.30(i)), it follows that (σ_{0}, τ_{0}) := (s, t)_{∗}(α_{0}) can be made arbitrarily close to (σ, τ) in X_{0} ×_{R[t^{−1}]} X_{0}(R[t^{−1}]). This means that the pair (σ, σ_{0}) can be made arbitrarily close to (σ, σ), hence (σ, σ_{0}) is in the image of (s, t)_{∗} provided σ and σ_{0} are sufficiently close. So σ is in the same homotopy class as σ_{0}. Likewise we can arrange that τ and τ_{0} are in the same homotopy class, which shows that (5.4.37)is injective as well. □

5.4.39. Let n ∈ ℕ and define U_{n} as in lemma 5.4.20; sometimes we identify U_{n} to the functor which it represents. We define a functor from F_{n} : ℤ-Alg → Set as follows. Given a ring R, we let F_{n}(R) be the set of all data of the form (S, T, ϕ, ψ), where S, T ∈ M_{n}(R) are two idempotent matrices and ϕ, ψ ∈ M_{n}(R) are two other matrices submitted to the following conditions:

(a) ϕ · S = 0 = T · ϕ.

(b) ψ · T = 0 = S · ψ.
(c) (I_{n} − S) · (ψ · ϕ − I_{n}) = 0 = (I_{n} − T) · (ϕ · ψ − I_{n}).

The meaning of (a) is that ϕ induces a map \overline{ϕ} : Coker(S) → Coker(T); likewise, (b) means that ψ induces a map \overline{ψ} : Coker(T) → Coker(S). Finally (c) means that \overline{ϕ} ∘ \overline{ψ} is the identity of Coker(T), and likewise for \overline{ψ} ∘ \overline{ϕ}. It is easy to see from this description that the functor F_{n} is representable by an affine ℤ-scheme of finite type, which we shall denote by the same name. Moreover, the rule (S, T, ϕ, ψ) ↦ (S, T) defines a transformation of functors F_{n} → U_{n} × U_{n}, whence a natural morphism of schemes:

(5.4.40)  F_{n} → U_{n} ×_{ℤ} U_{n}.

Lemma 5.4.41. The morphism (5.4.40)is smooth.

Proof. Since (5.4.40) is clearly of finite presentation, it suffices to verify that it is formally smooth. Hence, let R → R_{0} be a surjective ring homomorphism with nilpotent kernel J, let (S, T) ∈ U_{n} ×_{ℤ} U_{n}(R) and (S_{0}, T_{0}, ϕ_{0}, ψ_{0}) ∈ F_{n}(R_{0}) such that (S_{0}, T_{0}) coincides with the image of (S, T) in U_{n} ×_{ℤ}U_{n}(R_{0}). We need to show that there exist ϕ, ψ ∈ M_{n}(R) lifting ϕ_{0} and ψ_{0}, such that (S, T, ϕ, ψ) ∈ F_{n}(R). However, since S and T are idempotent, P := Coker(S) and Q := Coker(T) are finitely generated projective R-modules. According to (5.4.39), the induced map \overline{ϕ}_{0} : P_{0} =: P ⊗_{R} R_{0} → Q_{0} := Q ⊗_{R} R_{0} is an isomorphism with inverse \overline{ψ}_{0}. Let π_{P} : P → P_{0}, π_{Q} : Q → Q_{0} be the projections; since \overline{ψ}_{0} ∘ π_{Q} : Q → P_{0} is surjective, we can find a map \overline{ϕ} : P → Q such that \overline{ψ}_{0} ∘ π_{Q} ∘ \overline{ϕ} = π_{P}. Using Nakayama’s lemma one checks easily that \overline{ϕ} is an isomorphism that lifts \overline{ϕ}_{0}. Let α : R^{n} → Q and β : R^{n} → P be the natural projections. We set ϕ := (I_{n} − T) ∘ \overline{ϕ} ∘ β and ψ := (I_{n} − S) ∘ \overline{ϕ}^{−1} ∘ α and leave to the reader the verification of the identities (a)-(c) of (5.4.39). □

Corollary 5.4.42. Resume the assumptions of proposition 5.4.21. Then the base change functor R[t^{−1}]-Mod → R^{∧}[t^{−1}]-Mod : M ↦ M ⊗_{R} R^{∧} induces a bijection from the set of isomorphism classes of finitely generated projective R[t^{−1}]-modules to the set of isomorphism classes of finitely generated projective R^{∧}[t^{−1}]-modules.

Proof. Resume the notation of (5.4.39) and let s, t : F_{n} → U_{n} be the morphisms obtained by composing (5.4.40) with the two projections onto U_{n}. The datum (U_{n}, F_{n}, s, t) can be completed to a groupoid of schemes, by letting ι : U_{n} → F_{n} be the morphism representing the transformation of functors: S ↦ (S, S, I_{n}, I_{n}), and c : F_{n} ×_{U_{n}} F_{n} → F_{n} the morphism representing the transformation: ((S, T, ϕ_{1}, ψ_{1}), (S, T, ϕ_{2}, ψ_{2})) ↦ (S, T, ϕ_{2} ·ϕ_{1}, ψ_{2} ·ψ_{1}). Since every finitely generated projective module can be realized as the cokernel of an idempotent endomorphism of a free module of finite rank, the assertion is a straightforward consequence of theorem 5.4.38 and lemmata 5.4.20, 5.4.41. □

5.4.43. Let S be an R[t^{−1}]-scheme, 𝒫 a coherent 𝒪_{S}-module. An S-algebra structure on 𝒫 is a datum (μ, \underline{1}) consisting of a map μ : 𝒫 ⊗_{𝒪_{S}} 𝒫 → 𝒫 and a global section \underline{1} ∈ 𝒫(S), such that (𝒫, μ, \underline{1}) is an 𝒪_{S}-algebra, i.e. such that μ and \underline{1} satisfy the following conditions for every open subset U ⊂ S and every local sections x, y, z ∈ 𝒫(U) :

(a) μ(μ(x ⊗ y) ⊗ z) = μ(x ⊗ μ(y ⊗ z)).
(b) μ(x ⊗ y) = μ(y ⊗ x).
(c) μ(\underline{1} ⊗ x) = x.

We say that an S-algebra structure on 𝒫 is étale if the datum (𝒫, μ, \underline{1}) is an étale 𝒪_{S}-algebra. We denote by Alg_{S}(𝒫) (resp. Et_{S}(𝒫)) the set of all S-algebra structures (resp. étale S-algebra structures) on 𝒫. If now P is a finitely presented R[t^{−1}]-module, we obtain a functor:

(5.4.44)  R[t^{−1}]-Scheme → Set  S ↦ Alg_{S}(𝒪_{S} ⊗_{R[t^{−1}]} P).

Lemma 5.4.45. Suppose that P is a finitely generated projective R[t^{−1}]-module. Then the functor (5.4.44)is representable by a finitely presented R[t^{−1}]-algebra.

Proof. Clearly (5.4.44)is a sheaf on the fppf topology of Spec R[t^{−1}], hence it suffices to show that we can cover Spec R[t^{−1}] by finitely many affine Zariski open subsets U_{i}, such that the restriction of (5.4.44)to U_{i} is representable and finitely presented. However, P is locally free of finite rank on the Zariski topology of Spec R[t^{−1}], hence we can assume that P is a free R[t^{−1}]- module. Let e_{1}, ..., e_{n} be a basis of P; then a multiplication law μ is determined by its values a_{ij} ∈ P on e_{i} ⊗ e_{j}; by writing a_{ij} = ∑_{k} a_{ijk}e_{k} we obtain n^{3} elements of R[t^{−1}]; likewise, \underline{1} is represented by elements b_{1}, ..., b_{n} ∈ R[t^{−1}], and conditions (a),(b) and (c) of (5.4.43) translate as a finite system of polynomial identities for the a_{ijk} and the b_{l}; in other words, our functor is represented by a quotient of the free polynomial algebra R[t^{−1}, X_{ijk}, Y_{l} | i, j, k, l = 1, ..., n] by a finitely generated ideal, which is the contention. □

Lemma 5.4.46. Keep the assumptions of lemma 5.4.45. Let X_{P} be an affine R[t^{−1}]-scheme representing the functor (5.4.44). Then the functor

(5.4.47)  R[t^{−1}]-Scheme → Set  S ↦ Et_{S}(𝒪_{S} ⊗_{R[t^{−1}]} P)

is represented by an affine open subset U_{P} ⊂ X_{P}. Moreover, U_{P} is smooth over Spec R[t^{−1}].

Proof. Let us show first that the functor (5.4.47) is formally smooth. Indeed, suppose that Z is an affine R[t^{−1}]-scheme and Z_{0} ⊂ Z is a closed subset defined by a nilpotent ideal. Let (𝒪_{Z} ⊗_{R[t^{−1}]} P, μ_{0}) be an étale Z_{0}-algebra structure; we can lift it to some étale 𝒪_{Z}-algebra (𝒬, μ) and then 𝒬 is necessarily a locally free sheaf of finite rank (for instance by lemma 3.2.25(i),(iii)); from proposition 3.2.30 we deduce that 𝒬 ≃ 𝒪_{Z} ⊗_{R[t^{−1}]} P, whence the claim. Next, let (𝒫, μ, \underline{1}) be the universal 𝒪_{X_{P}} -algebra structure on P ⊗_{R[t^{−1}]} 𝒪_{X_{P}}; let also δ ∈ 𝒪_{X_{P}} (X_{P}) be the discriminant of the trace form of 𝒫. By theorem 4.1.14, a point x ∈ X_{P} is in the support of δ if and only if 𝒫_{x} is not étale over 𝒪_{X_{P} ,x}; therefore, the subset U_{P} over which 𝒫 is étale is indeed open and affine. To conclude, it suffices to show that U_{P} represents the functor (5.4.47). This amounts to showing that, for every morphism f : S → X_{P} of R[t^{−1}]- schemes, the algebra f^{∗}𝒫 is étale over 𝒪_{S} if and only if the image of f lands in U_{P}. However, the latter statement follows easily from [30, Ch.IV, Cor.17.6.2]. □

5.4.48. Let S be an R[t^{−1}]-scheme, 𝒫 a coherent 𝒪_{S}-module. We denote by Aut_{𝒪_{S}} (𝒫) the group of 𝒪_{S}-linear automorphisms of 𝒫. Then, for a given finitely presented R[t^{−1}]-module P we obtain a group-valued functor

(5.4.49)  R[t^{−1}]-Scheme → Grp  S ↦ Aut_{𝒪_{S}} (𝒪_{S} ⊗_{R[t^{−1}]} P).

Lemma 5.4.50. Keep the assumptions of lemma 5.4.45. Then the functor (5.4.49) is representable by a finitely presented R[t^{−1}]-group scheme.

Proof. It is analogous to the proof of lemma 5.4.45 : up to restricting to a Zariski open subset, we can assume that P is free of some rank n. Then the group scheme representing our functor is just GL_{n,R[t^{−1}]}. □

5.4.51. Let now P be a finitely generated projective R[t^{−1}]-module. For any R[t^{−1}]-scheme, let 𝒫_{S} := 𝒪_{S} ⊗_{R[t^{−1}]} P; we wish to define an action of Aut_{𝒪_{S}} (𝒫_{S}) on the set Et_{S}(𝒫_{S}). Indeed, if μ : 𝒫_{S} ⊗_{𝒪_{S}} 𝒫_{S} → 𝒫_{S} is any étale S-algebra structure and g ∈ Aut_{𝒪_{S}} (𝒫_{S}), let μ^{g} be the unique S-algebra structure on 𝒫_{S} such that g is an isomorphism of étale 𝒪_{S}-algebras: g : (𝒫_{S}, μ) →^{∼} (𝒫_{S}, μ^{g}). It is obvious that the rule (g, μ) ↦ μ^{g} is a functorial group action. Let U_{P} be as in lemma 5.4.46, and let Aut_{P} be a R[t^{−1}]-scheme representing the functor 5.4.49; the functorial map (g, μ) ↦ (μ^{g}, μ) is represented by a morphism of schemes:

(5.4.52)  Aut_{P} ×_{R[t^{−1}]} U_{P} → U_{P} ×_{R[t^{−1}]} U_{P} .

Lemma 5.4.53. The morphism (5.4.52)is étale.

Proof. The map is clearly of finite presentation, hence it suffices to show that it is formally étale. Therefore, let ϕ : Z → U_{P} ×_{R[t^{−1}]} U_{P} be a morphism of R[t^{−1}]-schemes, Z_{0} ⊂ Z a closed subscheme defined by a nilpotent ideal, and suppose that the restriction of ϕ to Z_{0} lifts to a morphism ψ_{0} : Z_{0} → Aut_{P} ×_{R[t^{−1}]} U_{P}. We need to show that ϕ lifts uniquely to a morphism ψ that extends ψ_{0}. However, the datum of ϕ is equivalent to the datum consisting of a pair of Zalgebra structures (𝒫_{Z}, μ_{1}) and (𝒫_{Z}, μ_{2}). The datum of ψ_{0} is equivalent to the datum of a Z_{0}- algebra structure μ_{0} on 𝒫_{Z_{0}}, and of an automorphism g_{0} of 𝒫_{Z_{0}}. Finally, the fact that ψ_{0} lifts the restriction of ϕ means that μ_{0} = μ_{2} ⊗_{Z} 1_{Z_{0}}, and g_{0} : (𝒫_{Z_{0}}, μ_{1} ⊗_{Z} 1_{Z_{0}}) →^{∼} (𝒫_{Z_{0}}, μ_{2} ⊗_{Z} 1_{Z_{0}}) is an isomorphism of étale 𝒪_{Z_{0}}-algebras. By theorem 3.2.18(iii), such an isomorphism extends uniquely to an isomorphism of étale 𝒪_{Z}-algebras g : (𝒫_{Z}, μ_{1}) →^{∼} (𝒫_{Z}, μ_{2}). The datum (g, μ_{2}) is equivalent to the sought map ψ. □

Proposition 5.4.54. Resume the assumptions of proposition 5.4.21. Then the base change functor R[t^{−1}]-Alg → R^{∧}[t^{−1}]-Alg induces an equivalence of categories from the category of finite étale R[t^{−1}]-algebras to the category of finite étale R^{∧}[t^{−1}]-algebras.

Proof. Let (P^{∧}, μ^{∧}) be a finite étale R^{∧}[t^{−1}]-algebra; in particular, P^{∧} is a finitely generated projective R^{∧}[t^{−1}]-module, hence by corollary 5.4.42 we can find a finitely generated projective R[t^{−1}]-module P such that R^{∧}[t^{−1}] ⊗_{R[t^{−1}]} P ≃ P^{∧}. The functorial action (5.4.52)of Aut_{P} on the set of étale algebra structures on P defines a groupoid of quasi-projective schemes. It then follows from theorem 5.4.38 and lemmata 5.4.46, 5.4.53 that the base change map from the set of isomorphism classes of R[t^{−1}]-algebra structures on P to the set of isomorphism classes of R^{∧}[t^{−1}]-algebra structures on P^{∧} is bijective. This shows that the base change functor is essentially surjective on finite étale R^{∧}[t^{−1}]-algebras. To prove full faithfulness, let Y_{1}, Y_{2} be any two finite étale schemes over X := R[t^{−1}]; we let H be the functor that assigns to every R[t^{−1}]-scheme Z the set Hom_{Z}(Z ×_{X} Y_{1}, Z ×_{X} Y_{2}). Since the functors represented by Y_{1} and Y_{2} are locally constant sheaves, say in the fppf topology of X := Spec R[t^{−1}], the same holds for the functor H, hence the latter is represented by a finite étale X-scheme (proposition 8.2.23), which we denote by the same name. We can view H as a trivial groupoid (i.e. such that for every X-scheme Z, the only morphisms of the groupoid H(Z) are the identity morphisms of its objects). In this case, the associated morphism (s, t) : H → H ×_{X} H is none other than the diagonal morphism; especially, the latter is an open imbedding, hence theorem 5.4.38 applies and yields the sought assertion. □

5.5. Lifting theorems for henselian pairs. For the considerations that follow, it will be useful to generalize a little our usual setup : we wish to work with sheaves of almost modules (or almost algebras) on a scheme. This is just a matter of introducing the relevant language, so we will proceed somewhat briskily.

5.5.1. Let X be a scheme over Spec(V). For every open subset U ⊂ X, Γ(U, 𝒪_{X}) is a V - algebra, hence we obtain a sheaf of V^{a}-algebras 𝒪_{X}^{a} on X by setting Γ(U, 𝒪_{X}^{a}) := Γ(U, 𝒪_{X})^{a} for every open U ⊂ X. We refer to [26, Ch.0, §3.1] for generalities on sheaves with values in arbitrary categories; in particular the sheaves of V^{a}-modules on X form an abelian tensor category, and hence we can define a sheaf ℱ of 𝒪_{X}^{a}-modules on X (briefly: a 𝒪_{X}^{a}-module) as a sheaf of almost modules endowed with a scalar multiplication 𝒪_{X}^{a} ⊗_{V^{a}} ℱ → ℱ. Those gadgets form a category that we denote by 𝒪_{X}^{a}-Mod. There is a functor

(5.5.2)  Γ(X, 𝒪_{U}^{a})-Mod → 𝒪_{X}^{a}-Mod  M ↦ M^{∼}

defined as one expects. We say that ℱ is quasi-coherent if we can cover X by affine open subsets U_{i}, such that ℱ_{|U_{i}} is in the essential image of a functor (5.5.2). We denote by 𝒪_{X}^{a}-Mod_{qcoh} the full subcategory of quasi-coherent 𝒪_{X}^{a}-modules. Similarly, we denote by 𝒪_{X}^{a}-Alg (resp. 𝒪_{X}^{a}-Alg_{qcoh}) the category of 𝒪_{X}^{a}-algebras (resp. quasi-coherent 𝒪_{X}^{a}-algebras) defined as one expects.

5.5.3. Since the functors M → M_{!} and M ↦ M_{∗} from A-modules to A_{∗}-modules are right exact, we can globalize them to the situation of (5.5.1). Thus, for every V -scheme X there are functors

𝒪_{X}^{a}-Mod → 𝒪_{X}-Mod ℱ ↦ ℱ_{!} (resp. ℱ ↦ ℱ_{∗})

which are left (resp. right) adjoint to the localization functor 𝒪_{X}-Mod → 𝒪_{X}^{a}-Mod. The functor ℱ ↦ ℱ_{!} is exact and preserves quasi-coherence (as can be easily deduced from proposition 2.4.33), hence it provides a left adjoint to the localization functor 𝒪_{X}-Mod_{qcoh} → 𝒪_{X}^{a}-Mod_{qcoh}. The functor ℱ ↦ ℱ_{∗} does not preserve quasi-coherence, in general.

5.5.4. Let R be a V -algebra and set X := Spec(R). Using the full faithfulness of the functor ℱ ↦ ℱ_{!} one can easily verify that the functor M ↦ M^{∼} from R^{a}-modules to quasi-coherent 𝒪_{X}^{a}-modules is an equivalence, whose quasi-inverse is given by the global section functor.

After these preliminaries, we are ready to state the following descent result which will be crucial for the proof of theorem 5.5.6. Proposition 5.5.5. (i) Let R be a V -algebra, J ⊂ R a finitely generated ideal and R → R′ a flat morphism inducing an isomorphism R/J → R′/JR′. Let X := Spec(R), X′ := Spec(R′), U := X \ V (J) and U′ := U ×_{X} X′. Then the natural commutative diagrams of functors:

\xymatrix{𝒪_{X}^{a}-Mod_{qcoh} \ar[r] \ar[d] & 𝒪_{X′}^{a}-Mod_{qcoh} \ar[d] & 𝒪_{X}^{a}-Alg_{qcoh} \ar[r] \ar[d] & 𝒪_{X′}^{a}-Alg_{qcoh} \ar[d] \\ 𝒪_{U}^{a}-Mod_{qcoh} \ar[r] & 𝒪_{U′}^{a}-Mod_{qcoh} & 𝒪_{U}^{a}-Alg_{qcoh} \ar[r] & 𝒪_{U′}^{a}-Alg_{qcoh}}

are 2-cartesian (that is, cartesian in the category of 2-categories).

(ii) Let A be a V^{a}-algebra, f ∈ A_{∗} a non-zero-divisor, A^{∧} the f-adic completion of A. Denote by A-Mod_{f} (resp. A-Alg_{f}) the full subcategory of f-torsion free A-modules (resp. A-algebras), and similarly define A^{∧}-Mod_{f} (resp. A^{∧}-Alg_{f}). Then the natural commutative diagrams of functors

\xymatrix{A-Mod_{f} \ar[r] \ar[d] & A^{∧}-Mod_{f} \ar[d] & A-Alg_{f} \ar[r] \ar[d] & A^{∧}-Alg_{f} \ar[d] \\ A[f^{−1}]-Mod \ar[r] & A^{∧}[f^{−1}]-Mod & A[f^{−1}]-Alg \ar[r] & A^{∧}[f^{−1}]-Alg}

are 2-cartesian.

Proof. (i): for the functors on 𝒪_{X}^{a}-modules, one applies the functor ℱ ↦ ℱ_{!}, thereby reducing to the corresponding assertion for quasi-coherent 𝒪_{X}-modules. Under assumption (a), the latter is proved in [36, Prop.4.2] (actually, in loc.cit. one assumes that X′ is faithfully flat over X, but one can reduce to such case after replacing X′ by X′ ⨿ (U_{1} ⨿ ... ⨿ U_{n}), where (U_{i} | i ≤ n) is a finite cover of U by affine open subsets; notice also that loc.cit. omits the assumption that J is finitely generated, but the proof works only under such assumption). Since all the functors involved commute with tensor products, the assertion about 𝒪_{X}^{a}-algebras follows formally.

(ii): for modules one argues as in the proof of (i), except that instead of invoking [36], one uses [9, Theorem]. For algebras, one has to proceed a little more carefully, since the tensor product of two f-torsion free modules may fail to be f-torsion free. Hence, let (B_{1}, B_{2}, β) the datum consisting of an A[f^{−1}]-algebra B_{1}, an A^{∧}-algebra B_{2} and an isomorphism β : B_{1} ⊗_{A} A^{∧} →^{∼} B_{2}[f^{−1}] of A^{∧}[f^{−1}]-algebras. Let I := ⋃_{n>0} Ann_{B_{2}⊗_{A^{∧}}B_{2}}(f^{n}) and set C := B_{2}⊗_{A^{∧}}B_{2}/I; β induces an isomorphism γ : B_{1}⊗_{A}B_{1}⊗_{A}A^{∧} →^{∼} C of A^{∧}-modules, so by the foregoing there exists an A-module D such that D[f^{−1}] ≃ B_{1} ⊗_{A} B_{1} and D ⊗_{A} A^{∧} ≃ C. Furthermore, the multiplication morphism μ_{B_{2}/A^{∧}} factors through a morphism μ̃ : D → B_{2}, and consequently the datum (μ_{B_{1}/A[f^{−1}]}, μ̃, γ, β) determines a unique morphism D → B. Let I′ := ⋃_{n>0} Ann_{B⊗_{A}B}(f^{n}); one verifies easily that (B ⊗_{A} B/J) ⊗_{A} A^{∧} ≃ C, so again the same sort of arguments show that D ≃ B ⊗_{A} B/J, hence we obtain a morphism μ_{B/A} : B ⊗_{A} B → B that lifts μ_{B_{1}/A[f^{−1}]} and μ_{B_{2}/A^{∧}}. Arguing along the same lines one can now verify easily that (B, μ_{B/A}) is really a B-algebra : we leave the details to the reader. □

Theorem 5.5.6. Let (A, I) be a tight henselian pair, \overline{P} an almost finitely generated projective A/I-module, 𝔪_{1} ⊂ 𝔪 a finitely generated subideal. We have:

(i) If 𝔪̃ has homological dimension ≤ 1, then there exists an almost finitely generated projective A-module such that P ⊗_{A} (A/I) ≃ \overline{P}.

(ii) If P_{1} and P_{2} are two liftings of P as in (i) and if there exists an isomorphism \overline{β} : P_{1} ⊗_{A} (A/𝔪_{1}I) →^{∼} P_{2} ⊗_{A} (A/𝔪_{1}I), then there exists an isomorphism β : P_{1} → P_{2} such that β ⊗_{A} 1_{A/I} = \overline{β} ⊗_{A} 1_{A/I}.

(iii) With the notation of (5.3.21), the natural functor A-Ét_{afp} → (A/I)-Ét_{afp} is an equivalence of categories.

Proof. We begin by showing (ii): indeed, the obstruction to the existence of a morphism α : P_{1} → P_{2} such that α⊗_{A}1_{(A/𝔪_{1}I)} = β is a class ω ∈ Ext_{A}^{1}(P_{1}, 𝔪_{1}IP_{2}). The same argument used in the proof of claim 5.1.19 shows that the natural map Ext_{A}^{1}(P_{1}, 𝔪_{1}IP_{2}) → Ext_{A}^{1}(P_{1}, IP_{2}) vanishes identically, and proves the assertion.

Next we wish to show that the functor of (iii) is fully faithful. Therefore, let B, C be two almost finitely presented étale A-algebras, and \overline{ϕ} : B/IB → C/IC a morphism. According to lemma 5.2.20, \overline{ϕ} is characterized by its associated idempotent, call it \overline{e} ∈ (B ⊗_{A} C/IC)_{∗}. Set D := B ⊗_{A} C; according to lemma 5.1.12(iii), the pair (D, ID) is tight henselian. Then proposition 5.1.17 says that \overline{e} lifts uniquely to an idempotent e ∈ D_{∗}. Claim 5.5.7. The associated morphism Γ(e) is an isomorphism (notation of (5.2.19)). Proof of the claim: Indeed, by naturality of Γ, we have Γ(e) ⊗_{A} 1_{A/I} = Γ(\overline{e}), so the assertion follows from corollary 5.1.7.

By claim 5.5.7 and lemma 5.2.20, e corresponds to a unique morphism ϕ : B → C which is the sought lifting of \overline{ϕ}. The remaining steps to complete the proof of (iii) will apply as well to the proof of (i). Pick an integer n > 0 and a finitely generated subideal 𝔪_{0} ⊂ 𝔪 such that I^{n} ⊂ 𝔪_{0}A; we notice that assertions (i) and (iii) also hold when I is nilpotent, since in this case they reduce to theorem 3.2.28(i.b),(ii). It follows easily that it suffices to prove the assertions for the pair (A, I^{n}), hence we can and do assume throughout that I ⊂ 𝔪_{0}A.

Claim 5.5.8. Assertions (i) and (iii) hold if I is generated by a non-zero-divisor of A_{∗}. Proof of the claim: Say that I = fA, for some non-zero-divisor f ∈ 𝔪_{0}A_{∗}, and let A^{∧} be the f-adic completion of A. By theorem 3.2.28(i), \overline{P} lifts to a compatible system (P_{n} | n ∈ ℕ) of almost finitely generated projective A/I^{n+1}-modules; by theorem 5.3.24, the latter compatible system gives rise to a unique almost finitely generated projective A^{∧}-module P^{∧}. Notice that f is regular on A^{∧}, hence also on P^{∧}. Since A^{∧}[f^{−1}] is a (usual) V -algebra, the A^{∧}[f^{−1}]-module P^{∧}[f^{−1}] is finitely generated projective; it follows from corollary 5.4.42 that there exists a finitely generated projective A[f^{−1}]-module Q with an isomorphism β : Q ⊗_{A} A^{∧} ≃ P^{∧}[f^{−1}]. By proposition 5.5.5(ii) the datum (P^{∧}, Q, β) determines a unique f-torsion-free A-module P which lifts \overline{P}. Since f is regular on both P and A, we have Tor_{i}^{A}(P, A/fA) = 0 for i = 1, 2, hence P is A-flat, by virtue of lemma 5.2.1. Next, set C := A[f^{−1}] × A^{∧}; from lemma 5.2.3(i) we deduce that Ann_{C}(P ⊗_{A} C)^{2} ⊂ Ann_{A}(P), and since Tor_{1}^{A}(C, P) = 0, remark 3.2.26(i) implies that P is almost finitely presented, therefore almost projective over A, which shows that (i) holds. Likewise, let \overline{B} be an almost finitely presented étale A/I-algebra; by theorem 5.3.27, \overline{B} admits a unique lifting to an almost finitely presented étale A^{∧}-algebra B^{∧}. Then B^{∧}[f^{−1}] is a finite étale A^{∧}[f^{−1}]-algebra, hence by proposition 5.4.54 there exists a unique finite étale A[f^{−1}]-algebra B_{0} with an isomorphism β : B_{0} ⊗_{B} B^{∧} →^{∼} B^{∧}[f^{−1}]. By proposition 5.5.5(ii), the datum (B^{∧}, B_{0}, β) determines a unique f-torsion-free A-algebra B; the foregoing proof of assertion (i) applies to the A-module underlying B and shows that B is an almost finitely generated projective A-algebra. By construction B/fB is unramified over A/fA, so theorem 5.2.12(iv) applies and shows that B is unramified over A. Thus, we have shown that the functor of (iii) is essentially surjective under the present assumptions; since it is already known in general that this functor is fully faithful, assertion (iii) is completely proved in this case.

Claim 5.5.9. Assertions (i) and (iii) hold if I is a principal ideal. Proof of the claim: Say that I = fA for some f ∈ 𝔪_{0}A_{∗}. Let J := ⋃_{n>0} Ann_{A}(f^{n}); we have a cartesian diagram

\xymatrix{A/(J ∩ I) \ar[r] \ar[d] & A/I \ar[d] \\ A/J \ar[r] & A/(I + J).}

Let \overline{P} be as in (i) and let \overline{B} be a finitely presented étale A/I-algebra; by claim 5.5.8 we can find an almost finitely generated projective A/J-module \overline{P}_{1} with an isomorphism β : \overline{P}_{1}/I\overline{P}_{1} →^{∼} \overline{P} /J\overline{P} and a finitely presentedétale A/J-algebra \overline{B}_{1} that lifts \overline{B}/J\overline{B}. By proposition 3.4.21, the datum (\overline{P} , \overline{P}_{1}, β) determines a unique almost finitely generated projective A/(I ∩ J)-module \overline{P}_{2}; likewise, using corollary 3.4.22 we obtain an étale almost finitely presented A/(I ∩ J)- algebra \overline{B}_{2} that lifts \overline{B}. Next, let K := ⋂_{n>0} f^{n}A and set N := K ∩ J ∩ I; we have a cartesian diagram

\xymatrix{A/N \ar[r] \ar[d] & A/(I ∩ J) \ar[d] \\ A^{∧} \ar[r] & A^{∧}/(I ∩ J)A^{∧}.}

Due to theorem 3.2.28(i.b) we can lift \overline{P}_{2} ⊗_{A} A^{∧} to an almost finitely generated projective A^{∧}/(I ∩ J)^{2}A^{∧}-module \overline{P} _{2}^{∧}. For the same reason, \overline{P} _{2}^{∧} ⊗_{A} A/f^{2}A can be lifted to a compatible family (\overline{Q}_{n} | n ∈ ℕ), where \overline{Q}_{n} is almost finitely generated projective over A/f^{n+2}A for every n ∈ ℕ. Finally, by theorem 5.3.24, the projective limit Q of the system (\overline{Q}_{n} | n ∈ ℕ) is an almost finitely generated projective A^{∧}-module. By construction, there is an isomorphism \overline{β} : Q/f^{2}Q →^{∼} \overline{P} _{2}^{∧} ⊗_{A} A/f^{2}A; by assertion (ii) it follows that there exists an isomorphism β : Q/(I ∩ J)^{2}Q →^{∼} \overline{P} _{2}^{∧} that lifts \overline{β} ⊗_{A} 1_{A/fA}. By proposition 3.4.21, the datum (\overline{P}_{2}, Q, β ⊗_{A} 1_{A/(I∩J)}) determines a unique almost finitely generated projective A/N-module P_{1}; since N^{2} = 0, P_{1} can be further lifted to an almost finitely generated projective A-module P, so assertion (i) holds in this case. The proof of assertion (iii) is analogous, but easier : we need to show that \overline{B}_{2} lifts to an almost finitely presented étale A-algebra B; to this aim, it suffices to show that \overline{B}_{2} lifts to an almost finitely presented étale A/N-algebra B_{1}, since in that case B_{1} can be lifted to an almost finitely presented étale A-algebra B, by theorem 3.2.28(ii). To obtain B_{1} it suffices to find an almost finitely presented étale A^{∧}-algebra B^{∧} with an isomorphism β : B^{∧} ⊗_{A} A/(I ∩ J) →^{∼} \overline{B}_{2} ⊗_{A} A^{∧}; indeed, in this case the datum (B^{∧}, \overline{B}_{2}, β) determines a unique étale almost finitely presented A/N-algebra in view of corollary 3.4.22. Finally, we consider the natural functors

A^{∧}-Ét_{afp} → A^{∧}/(I ∩ J)A^{∧}-Ét_{afp} → A/I-Ét_{afp}.

By theorem 5.3.27, the composition of these two functors is an equivalences of categories and the rightmost functor is fully faithful by (ii), so the leftmost functor is an equivalence, thus B_{1} as sought can be found, which concludes the proof of (iii) in this case.

Claim 5.5.10. Assertions (i) and (iii) hold if I is a finitely generated ideal. Proof of the claim: We proceed by induction on the number n of generators of I, the case n = 1 being covered by claim 5.5.8. So suppose n > 1 and let f_{1}, ..., f_{n} ∈ I_{∗} be a finite set of generators of I. Let \overline{P} be as in (i) and \overline{B} any almost finitely presented étale A/I-algebra. We let A′ := A/f_{1}A and J := Im(I → A′). By lemma 5.1.12(iii) the pair (A′, J) is again tight henselian, so by inductive assumption we can find lift \overline{P} (resp. \overline{B}) to an almost finitely generated projective A′-module (resp. to an almost finitely presented étale A′-algebra) P′ (resp. B′). Thence we apply claim 5.5.8 to further lift P′ to a module P (resp. to an algebra B) as stated.

Let now (A, I) be a general tight henselian pair; we can find a henselian pair (R, J) such that R^{a} = A, J^{a} = I and J ⊂ 𝔪_{0}. Denote by (R^{h}, 𝔪_{0}R^{h}) the henselization of the pair (R, 𝔪_{0}R), and let \overline{R}^{h} := R^{h}/JR^{h}. Let us write J = colim_{λ∈Λ} J_{λ}, where J_{λ} runs over the filtered

family of all finitely generated subideals of J; set R_{λ} := R/J_{λ} and R_{λ}^{h} := R^{h} ⊗_{R} R_{λ} for every λ ∈ Λ. Furthermore, let X := Spec R, X_{λ} := Spec R_{λ}, X^{h} := Spec R^{h}, X_{λ}^{h} := X^{h} ×_{X} X_{λ}, U := X \ V (𝔪_{0}), ϕ^{h} : U^{h} := U ×_{X} X^{h} → U, ϕ_{λ}^{h} : U_{λ}^{h} := U ×_{X} X_{λ}^{h} → U_{λ} := U ×_{X} X_{λ} the natural morphisms of schemes. Now, let \overline{P} be an almost finitely generated A/I-module and \overline{B} an almost finitely presented étale A/I-algebra; \overline{P} induces a quasi-coherent 𝒪_{X}^{a}-module \overline{P}^{∼}, and by restriction we obtain a quasi-coherent 𝒪_{U}-module \overline{P} _{|U}^{∼}. Furthermore, since \overline{P} is almost finitely presented, we see that \overline{P} _{|U}^{∼} is finitely presented; by [29, Ch.IV, Th.8.5.2(ii)] it follows that for some λ_{0} ∈ Λ there exists a quasi-coherent finitely presented module 𝒫 on U_{λ_{0}} whose restriction to the closed subset U agrees with \overline{P} _{|U}^{∼}. By restricting further, we can even achieve that 𝒫 be locally free of finite rank ([29, Ch.IV, Prop.8.5.5]). Similarly, we can find a locally free 𝒪_{λ_{0}}-algebra ℬ such that ℬ_{|U} ≃ \overline{B}_{|U}^{∼}.

Claim 5.5.11. For every almost finitely generated projective (\overline{R}^{h})^{a}-module \overline{Q} there exists an almost finitely generated projective (R^{h})^{a}-module Q such that Q ⊗_{(R^{h})^{a}} (\overline{R}^{h})^{a} ≃ \overline{Q}.

Proof of the claim: By theorem 3.2.28(i.b) we know that \overline{Q} lifts to an almost finitely generated projective module Q_{1} over (R^{h}/J^{2}R^{h})^{a}. Then Q_{2} := Q_{1} ⊗_{A} (A/𝔪_{0}^{2}A) is an almost finitely generated projective (R^{h}/𝔪_{0}^{2}R^{h})^{a}-module, therefore by claim 5.5.10 we can lift Q_{2} to an almost finitely generated projective R^{h}-module Q (notice that (R^{h}, 𝔪_{0}^{2}) is still a henselian pair). It remains only to show that Q is a lifting of \overline{Q}; however, by construction we have Q_{1} ⊗_{(R^{h})^{a}} (R^{h}/𝔪_{0}^{2}R^{h})^{a} ≃ Q ⊗_{(R^{h})^{a}} (R^{h}/𝔪_{0}^{2}R^{h})^{a}, so it follows from (ii) that Q_{1} ≃ Q/I^{2}Q, whence the claim.

Claim 5.5.12. The natural functor (R^{h})^{a}-Ét_{afp} → (\overline{R}^{h})^{a}-Ét_{afp} is an equivalence of categories.

Proof of the claim: By claim 5.5.10, the natural functors (\overline{R}^{h})^{a}-Ét_{afp} → (R^{h}/𝔪_{0}R^{h})^{a}-Ét_{afp} and (R^{h})^{a}-Ét_{afp} → (R^{h}/𝔪_{0}R^{h})^{a}-Ét_{afp} are equivalences of categories. The claim is a formal consequence.

We apply claim 5.5.11 with \overline{Q} := \overline{P} ⊗_{A} (R^{h})^{a}; let Q^{∼} be the quasi-coherent 𝒪_{X^{h}}^{a}-module associated to Q. By construction, the restriction Q_{|U^{h}}^{∼} is a quasi-coherent 𝒪_{U^{h}}-module of finite presentation, and we have an isomorphism \overline{β} : Q_{|U^{h}}^{∼} →^{∼} ϕ^{h∗}(\overline{P} _{|U}^{∼}). It then follows by [29, Ch.IV, Cor.8.5.2.5] that there exists some μ ∈ Λ with X_{μ} ⊂ X_{λ_{0}}, such that the isomorphism \overline{β} extends to an isomorphism β_{μ} : Q_{|U_{μ}^{h}}^{∼} →^{∼} ϕ_{μ}^{h∗}(𝒫_{|U_{μ}}). Similarly, by claim 5.5.12, we can find an

almost finitely presented étale (R^{h})^{a}-algebra C with an isomorphism γ_{μ} : C_{|U_{μ}^{h}}^{∼} ≃ ϕ^{h∗}(\overline{B}_{|U_{μ}^{h}}^{∼}). According to (5.5.4), the global section functors:

𝒪_{X_{λ}}^{a}-Mod_{qcoh} → R_{λ}^{a}-Mod 𝒪_{X_{λ}^{h}}^{a}-Mod_{qcoh} → (R_{λ}^{h})^{a}-Mod

are equivalences. Clearly the localization functors:

𝒪_{U_{λ}}-Mod_{qcoh} → 𝒪_{U_{λ}}^{a}-Mod_{qcoh} 𝒪_{U_{λ}^{h}}-Mod_{qcoh} → 𝒪_{U_{λ}^{h}}^{a}-Mod_{qcoh}

are equivalences as well, and similarly for the corresponding categories of algebras. By proposition 5.5.5(i) it follows that the natural diagrams:

\xymatrix{R_{λ}^{a}-Mod \ar[r] \ar[d] & (R_{λ}^{h})^{a}-Mod \ar[d] & R_{λ}^{a}-Alg \ar[r] \ar[d] & (R_{λ}^{h})^{a}-Alg \ar[d] \\ 𝒪_{U_{λ}}-Mod_{qcoh} \ar[r]^{ϕ_{λ}^{h∗}} & 𝒪_{U_{λ}^{h}}-Mod_{qcoh} & 𝒪_{U_{λ}}-Alg_{qcoh} \ar[r]^{ϕ_{λ}^{h∗}} & 𝒪_{U_{λ}^{h}}-Alg_{qcoh}}

are 2-cartesian for every λ ∈ Λ. Hence, the datum (𝒫_{|U_{μ}}, Q ⊗_{R^{a}} R_{μ}^{a}, β_{μ}) determines uniquely a R_{μ}^{a}-module P_{μ} that lifts \overline{P}, and the datum (ℬ_{|U_{μ}}, C ⊗_{R^{a}} R_{μ}^{a}, γ_{μ}) determines a R_{μ}^{a}-algebra B_{μ} that lifts \overline{B}. Furthermore, since the natural morphism U_{μ} ⨿ X_{μ}^{h} → X_{μ} is faithfully flat, it follows from remark 3.2.26(ii) that P_{μ} and B_{μ} are almost finitely generated projective over R_{μ}^{a}. For the same reasons, B_{μ} is unramified, hence étale over R_{μ}^{a}. Finally, we apply claim 5.5.10 to the henselian pair (R^{a}, J_{μ}^{a}) to lift P_{μ} and B_{μ} all the way to A, thereby concluding the proof of the theorem. □

Lemma 5.5.13. Suppose that 𝔪̃ has homological dimension ≤ 1, and let (A, I) be a tight henselian pair, \overline{A} := A/I, \overline{Q} an almost finitely generated projective \overline{A}-module, M an A-module and \overline{ϕ} : \overline{Q} → M/IM an \overline{A}-linear epimorphism. Then there exists an almost finitely generated projective A-module Q and a morphism ϕ : Q → M such that ϕ ⊗_{A} 1\overline{A} = \overline{ϕ}.

Proof. We begin with the following special case:

Claim 5.5.14. The lemma holds if I^{2} = 0. Furthermore, in this case ϕ is an epimorphism.

Proof of the claim: First of all, notice that M/IM is almost finitely generated, hence the same holds for M, in view of lemma 3.2.25(ii). If now Q is an almost finitely generated projective Amodule and ϕ : Q → M is a morphism that lifts \overline{ϕ}, we have Coker(ϕ) ⊗_{A} A/I ≃ Coker \overline{ϕ} = 0, whence Coker ϕ = 0 by lemm 5.1.6. In other words, the second assertion follows from the first.

Define the A-module N as the fibre product in the cartesian diagram of A-modules:

\xymatrix{N \ar[r]^{α} \ar[d]_{β} & \overline{Q} \ar[d]^{\overline{ϕ}} \\ M \ar[r]^{π} & M/IM}

(where π is the natural projection). Notice that IN = Ker α; indeed, clearly α(IN) = 0, and on the other hand β(IN) = IM = Ker π ≃ Ker α. We derive an isomorphism \overline{ψ} : \overline{Q} →^{∼} N/IN, and clearly it suffices to find a morphism Q → N that lifts \overline{ψ}. Under our current assumptions, theorem 5.5.6(i) provides an almost finitely generated projective A-module Q_{1} such that Q_{1} ⊗_{A} \overline{A} ≃ \overline{Q}, which in turns determines an extension of A-modules \underline{E} := (0 → I ⊗_{A} \overline{Q} → Q_{1} → \overline{Q} → 0). Furthermore, \overline{ψ} induces an epimorphism χ : I ⊗_{A} \overline{Q} → IN, whence an extension χ ∗ \underline{E} := (0 → IN → Q_{2} → \overline{Q} → 0). Another such extension is defined by \underline{F} := (0 → IN → N →^{α} \overline{Q} → 0). However, any extension \underline{X} of \overline{Q} by IN induces a morphism u(\underline{X}) : I ⊗\overline{A} \overline{Q} → IN, defined as in (3.2.19). Directly on the definition one can check that u(\underline{X}) depends only on the class of \underline{X} in Ext_{A}^{1}(\overline{Q}, IN), and moreover, if \underline{Y} is any other such extension, then u(\underline{X} + \underline{Y}) = u(\underline{X}) + u(\underline{Y}) (where \underline{X} + \underline{Y} denotes the Baer sum of the two extensions). We can therefore compute: u(χ ∗ \underline{E} − \underline{F}) = χ ∘ u(\underline{E}) − u(\underline{F}); but the definition of \underline{E} is such that u(\underline{E}) = 1_{I⊗\overline{A}\overline{Q}} and by inspecting the construction of \underline{F} we get u(\underline{F}) = χ. So finally u(χ ∗ \underline{E} − \underline{F}) = 0; this means that χ ∗ \underline{E} − \underline{F} is an extension of \overline{A}-modules, that is, its class is contained in the subgroup Ext1 ; \overline{A}(\overline{Q}, IN) ⊂ Ext_{A}^{1}(\overline{Q}, IN). Notice now that Ext2 ; \overline{A}(\overline{Q}, Ker χ) = 0, due to lemma 2.4.14(i),(ii); since χ is an epimorphism, it then follows that the induced map Ext1 ; \overline{A}(\overline{Q}, χ) is surjective. Hence there exists an extension \underline{X} := (0 → I ⊗_{A} \overline{Q} → Q_{3} → \overline{Q} → 0) of \overline{A}-modules, such that χ ∗ \underline{X} = χ ∗ \underline{E} − \underline{F}, i.e. \underline{F} = χ ∗ (\underline{E} − \underline{X}). Say \underline{E}′ := \underline{E} − \underline{X} = (0 → I ⊗_{A} \overline{Q} → Q → \overline{Q} → 0); by construction, u(\underline{E}′) = u(\underline{E}), so Q is a flat A-module (see (3.2.19)) that lifts \overline{Q}. Finally, Q is almost finitely generated projective by lemma 3.2.25(i),(ii). The push-out \underline{E}′ → χ ∗ \underline{E}′ delivers the promised morphism Q → N.

Next, since the pair (A, I^{n+1}) is still tight henselian for every n ∈ ℕ, an easy induction shows that the lemma holds when I is a nilpotent ideal. For the general case, pick n > 0 and a finitely generated subideal 𝔪_{0} ⊂ 𝔪 with I^{n} ⊂ 𝔪_{0}A; by the foregoing we can find an almost finitely generated projective A/I^{n+1}-module Q_{n+1} and an epimorphism ϕ_{n+1} : Q_{n+1} → M/I^{n+1}M. By theorem 5.5.6(i), we can lift Q_{n+1} to an almost finitely generated projective A-module Q; the obstruction to lifting the induced morphism Q → M/I^{n+1}M to a morphism Q → M is a class ω ∈ Ext_{A}^{1}(Q, I^{n+1}M); by the argument of claim 5.1.19 we see that the image of ω in Ext_{A}^{1}(Q, IM) vanishes, whence the claim. □

Corollary 5.5.15. Let A be a V^{a}-algebra, I ⊂ rad(A) a tight ideal and set \overline{A} := A/I; let P an almost finitely generated A-module, such that \overline{P} := P ⊗_{A} \overline{A} is an almost projective \overline{A}-module. Then the following conditions are equivalent:

(i) P is an almost projective A-module.

(ii) P is a flat A-module.

(iii) P is an almost finitely presented A-module and Tor_{1}^{A}(P, \overline{A}) = 0.

(iv) The natural morphism P^{∗} → (\overline{P})^{∗} is an epimorphism.

Proof. Clearly (i) implies all the other assertions, so it suffices to show that each of the assertions (ii)-(iv) implies (i). Let us first remark that, in view of lemmata 2.3.13, 2.4.13 and theorem 2.1.12, we can assume that the homological dimension of 𝔪̃ is ≤ 1. Furthermore, let (A^{h}, I^{h}) be the henselization of the pair (A, I); according to (5.1.13), the morphism A → A^{h} is faithfully flat. In view of remark 3.2.26(ii) and lemma 2.4.31(i) we deduce that each of the statements (i)- (iii) on P and \overline{P} is equivalent to the corresponding statement (i)^{h}-(iii)^{h} made on the A^{h}-module P ⊗_{A} A^{h} and the (\overline{A} ⊗_{A} A^{h})-module \overline{P} ⊗_{A} A^{h}. Moreover, one checks easily that (iv)⇒(iv)^{h}. Thus, up to replacing (A, I) by (A^{h}, I^{h}) we can assume that (A, I) is a tight henselian pair (notice as well that I^{h} = IA^{h}). By lemma 5.5.13 we can find an almost finitely generated projective A-module Q and a morphism ϕ : Q → P such that ϕ ⊗_{A} 1_{A/I} is an isomorphism. By lemma 5.1.6 we deduce easily that ϕ is an epimorphism, Suppose now that (ii) holds; then to deduce (i) it remains only to prove the following :

Claim 5.5.16. Ker ϕ = 0.

Proof of the claim: In view of proposition 2.4.28(v), it suffices to show that ℰ := ℰ_{Q/A}(x) = 0 for every x ∈ Ker ϕ_{∗} (see definition 2.4.23(iv)). However, since Q is A-flat, we can compute:

0 = Tor_{1}^{A}(A/ℰ , P) ≃ Ker(Ker(ϕ) ⊗_{A} (A/ℰ) → Q/ℰ Q) ≃ ((Ker ϕ) ∩ ℰ Q)/(ℰ · Ker ϕ)

that is, (Ker ϕ) ∩ ℰ Q = ℰ · Ker ϕ. By proposition 2.4.28(v) we have x ∈ (ℰ Q)_{∗}; on the other hand we also know that Ker ϕ ⊂ IQ, whence x ∈ (ℰ IQ)_{∗}. We apply once again proposition 2.4.28(v) to derive ℰ = ℰ I, so finally ℰ = 0 in view of lemma 5.1.6.

Next, assume (iii); we compute: 0 = Tor_{1}^{A}(P, \overline{A}) ≃ Ker(Ker(ϕ) ⊗_{A} \overline{A} → Ker(ϕ ⊗_{A} 1\overline{A})) = Ker(ϕ) ⊗_{A} \overline{A}. Since P is almost finitely presented, Ker ϕ is almost finitely generated by lemma 2.3.17, whence Ker ϕ = 0 by lemma 5.1.6, so (i) holds. Finally, let ϕ^{∗} : P^{∗} → Q^{∗} be the transposed of the morphism ϕ; by lemma 2.4.31(i), the natural morphism ψ : Q^{∗} → (Q/IQ)^{∗} is an epimorphism. The composition ψ ∘ϕ^{∗} factors through the transposed morphism (ϕ⊗_{A} 1\overline{A})^{∗} : (\overline{P})^{∗} → (Q/IQ)^{∗}, so it is an epimorphism when (iv) holds; then lemma 5.1.6 implies easily that ϕ^{∗} is an epimorphism; since it is obviously a monomorphism, we deduce that P^{∗} →^{∼} Q^{∗}. so (ϕ^{∗})^{∗} : Q ≃ (Q^{∗})^{∗} → (P^{∗})^{∗} is an isomorphism; since the latter factors through the natural morphism P → (P^{∗})^{∗}, we see that ϕ is a monomorphism and (i) follows. □

5.6. Smooth locus of an affine almost scheme. Thoughout this section we fix a V^{a}-algebra A and set S := Spec A. Let X be an affine S-scheme. We often identify X with the functor it represents:

X : A-Alg → Set T ↦ X(T^{o}) := Hom_{A-Alg^{o}}(T^{o}, X).

The usual argument from faithfully flat descent shows that X is a sheaf for the fpqc topology of A-Alg^{o}. In this section we aim to study, for every such X, the smooth locus of X over S, which will be a certain natural subsheaf of X. The starting point is the following: Definition 5.6.1. Let S and X be as in (5.6). Given an affine S-scheme T and σ ∈ X(T), we say that σ lies in the smooth locus of X over S if the following two conditions hold:

(a) H_{1}(Lσ^{∗}𝕃_{X/S}^{a}) = 0 and

(b) H_{0}(Lσ^{∗}𝕃_{X/S}^{a}) is an almost finitely generated projective 𝒪_{T} -module.

We denote by X_{sm}(T) ⊂ X(T) the subset of all the T -sections of X that lie in the smooth locus of X over S.

5.6.2. Using remark 3.2.26(iii) one sees that X_{sm} is a subsheaf of X. Just as for usual schemes, in order to get a handle on the smooth locus X_{sm}, one often needs to assume that the almost scheme X satisfies some finiteness conditions. For our purposes, the following will do: Definition 5.6.3. We say that the affine almost S-scheme X is almost finitely presented if there exists an almost finitely generated projective 𝒪_{S}-module, and an almost finitely generated ideal J of S_{P} := Sym_{𝒪_{S}}^{•} (P), such that X ≃ Spec S_{P} /J.

Lemma 5.6.4. Let X = Spec S_{F} /J, where S_{F} := Sym_{𝒪_{S}}^{•} (F) for some flat 𝒪_{S}-module F, and J is any ideal. Then there is a natural isomorphism in D(𝒪_{X}-Mod):

τ_{[−1}𝕃_{X/S}^{a} ≃ (0 → J/J^{2} → 𝒪_{X} ⊗_{𝒪_{S}} P → 0).

Proof. Let us remark the following: Claim 5.6.5. With the notation of the lemma, there is a natural isomorphism 𝕃_{Spec S_{F} /S}^{a} ≃ S_{F} ⊗_{𝒪_{S}} F [0] in D(S_{F} -Mod). Proof of the claim: By a theorem of Lazard ([51, Ch.I, Th.1.2]), every flat 𝒪_{S∗}-module is the filtered colimit of a family of free modules of finite rank; in particular this holds for F_{!}; since both functors Sym^{•} and 𝕃 commute with filtered colimits (proposition 2.5.33), we can then reduce to the case where S_{F} ≃ 𝒪_{S}[T_{1}, ..., T_{n}] for some n ∈ ℕ. In this case, we have natural isomorphisms 𝕃_{Spec S_{F} /S}^{a} ≃ 𝕃_{𝒪_{S∗}[T_{1},...,T_{n}]/𝒪_{S∗}}^{a} in light of proposition 8.1.7(ii). The claim follows.

Using claim 5.6.5, the assertion can be shown as in the proof of [46, Ch.III, Cor.1.2.9.1]. □

5.6.6. Let X be an almost finitely presented S-scheme and t ∈ 𝔪 any element. Then 𝒪_{S}[t^{−1}] is a (usual) V [t^{−1}]-algebra, and we let S_{t} := Spec 𝒪_{S}[t^{−1}], X_{t} := X ×_{S} S_{t}. Both S_{t} and X_{t} are represented by (usual) affine schemes over Spec V [t^{−1}], and obviously X_{t} is finitely presented over S_{t}. Using lemma 5.6.4 it is also easy to see that the subfunctor X_{sm,t} := X_{sm} ∩ X_{t} of the functor X is represented by the smooth locus of X_{t} over S_{t}, which is an open subscheme of the latter scheme.

5.6.7. In the situation of (5.6.6), suppose moreover that I ⊂ 𝒪_{S} is a given ideal, and that t is regular in 𝒪_{S}. Let R := 𝒪_{S∗}; then t is a non-zero-divisor in R, and we have a well defined (t, I_{∗})-adic topology on X_{t}(R[t^{−1}]) = X_{t}(S_{t}). Furthermore, it is clear that the restriction map:

(5.6.8)  X(S) → X_{t}(S_{t})  σ ↦ σ_{∗}[t^{−1}]

is injective. Consequently we can endow X(S) with the (t, I)-adic topology, defined as the topology induced by the (t, I_{∗})-adic topology of X_{t}(S_{t}). Lemma 5.6.9. In the situation of (5.6.7), the map (5.6.8) is an open imbedding for the respective (t, I)-adic and (t, I_{∗})-adic topologies.

Proof. Let us write X = Spec S_{P} /J, where S_{P} is the symmetric algebra of an almost finitely generated projective 𝒪_{S}-module P, and set 𝒫 := Spec S_{P}. Then X is a closed subscheme of 𝒫 and X_{t} is a closed subscheme of 𝒫_{t}, which is a vector bundle of finite rank over S_{t}. The (t, I_{∗})-adic topology of X_{t}(S_{t}) is induced by the (t, I_{∗})-adic topology of 𝒫_{t}(S_{t}), and consequently the (t, I)-adic topology of X(S) is induced by the (t, I)-adic topology of 𝒫(S). Since the commutative diagram of sets

(5.6.10)  \xymatrix{X(S) \ar[r] \ar[d] & X_{t}(S_{t}) \ar[d] \\ 𝒫(S) \ar[r] & 𝒫_{t}(S_{t})}

is cartesian, we reduce to showing that the restriction map 𝒫(S) → 𝒫_{t}(S_{t}) is open. To this aim, we set P [t^{−1}]^{∗} := Hom_{𝒪_{S}} (P, 𝒪_{S}[t^{−1}]) and we consider the diagram

\xymatrix{Hom_{𝒪_{S}} (P, 𝒪_{S}) \ar[r]^{∼} \ar[d] & 𝒫(S) \ar[d] \\ P [t^{−1}]^{∗} \ar[r]^{∼} & 𝒫_{t}(S_{t})}

whose horizontal arrows are given by the rule: ϕ ↦ Sym_{𝒪_{S}}^{•} ϕ. Since P [t^{−1}]^{∗} is a finitely generated 𝒪_{S}[t^{−1}]-module, it is endowed with a well defined (t, I_{∗})-preadic topology. We define a linear (t, I)-adic topology on Hom_{𝒪_{S}} (P, 𝒪_{S}), by declaring that the system of submodules (Hom_{𝒪_{S}} (P, t^{n}I) | n ∈ ℕ) forms a cofinal family of open neighborhoods of zero. Claim 5.6.11. With these (t, I)-adic and (t, I_{∗})-adic topologies, (5.6.10)is a diagram of continuous maps, and the horizontal arrows are homeomorphisms. Proof of the claim: In case P [t^{−1}] ≃ R[t^{−1}]^{n} for some n ∈ ℕ, we have 𝒫_{t}(S_{t}) ≃ R[t^{−1}]^{n}, and then the bottom arrow of (5.6.10) is a homeomorphism, essentially by definition. The general case can be reduced to the case of a free module, by writing R[t^{−1}]^{n} = P [t^{−1}] ⊕ Q for some projective R[t^{−1}]-module Q, and remarking that the (t, I_{∗})-adic topologies are compatible with cartesian products. To prove that the top arrow is a homeomorphism, it suffices therefore to show that the (t, I)-adic topology of Hom_{𝒪_{S}} (P, 𝒪_{S}) is induced by the (t, I_{∗})-adic topology of P [t^{−1}]^{∗}. To this aim, pick a finitely generated R-module P_{0} ⊂ P_{∗} with tP_{∗} ⊂ P_{0}. Clearly Hom_{R}(P_{0}, 𝒪_{S}[t^{−1}]) = P [t^{−1}]^{∗}, and again by reducing to the case where P [t^{−1}] is free, one verifies that the (t, I_{∗})-adic topology on P [t^{−1}]^{∗} admits the system (Hom_{R}(P_{0}, t^{n}I_{∗}) | n ∈ ℕ) as a cofinal family of open neighborhoods of zero. For every n ∈ ℕ set U_{n} := {ϕ : P → 𝒪_{S} | ϕ(P_{0}) ⊂ t^{n}I_{∗}}; we have

(5.6.12)  Hom_{𝒪_{S}} (P, t^{n}I) ⊂ U_{n} ⊂ Hom_{𝒪_{S}} (P, t^{n−1}I)  for every n ∈ ℕ

which implies the claim.

In view of claim 5.6.11, we are reduced to showing that the map Hom_{𝒪_{S}} (P, 𝒪_{S}) → P [t^{−1}]^{∗} is open. But this is again a direct consequence of (5.6.12). □

Proposition 5.6.13. Let X, T be affine S-schemes and σ ∈ X(T) a T -section, I ⊂ rad(𝒪_{T}) an ideal, and set T_{0} := Spec 𝒪_{T} /I; suppose that the restriction σ_{0} ∈ X(T_{0}) of σ lies in the smooth locus of X. Suppose moreover that either:

(a) I is nilpotent, or

(b) I is tight and X is almost finitely presented over S. Then σ ∈ X_{sm}(T).

Proof. Suppose that (a) holds; for any quasi-coherent 𝒪_{T} -module ℱ, let us denote by Fil_{I}^{•}ℱ the I-adic filtration on ℱ. We can write τ_{[−1}Lσ^{∗}𝕃_{X/S}^{a} ≃ (0 → N →^{ϕ} P → 0) for two 𝒪_{T} - modules N and P, and we can assume that P is almost projective over 𝒪_{T}, so that the natural morphism

τ_{[−1}(𝒪_{T_{0}} ⊗^{L}_{𝒪_{T}} Lσ^{∗}𝕃_{X/S}^{a}) → (0 → N/IN ⟶^{gr_{I}^{0}ϕ} P/IP → 0)

is an isomorphism in D(𝒪_{T_{0}}-Mod). Hence, the assumption on σ means that gr_{I}^{0}ϕ is a monomorphism with almost finitely generated projective cokernel over 𝒪_{T_{0}}. We consider, for every integer i ∈ ℕ the commutative diagram:

\xymatrix{gr_{I}^{i}N \ar[r]^{gr_{I}^{i}ϕ} & gr_{I}^{i}P \\ gr_{I}^{0}N ⊗_{𝒪_{T}} gr_{I}^{i}𝒪_{T} \ar[u]^{α_{i}} \ar[r]^{gr_{I}^{0}ϕ⊗_{𝒪_{T}} 1_{gr^{i}_{I} 𝒪_{T}}} & gr_{I}^{0}P ⊗_{𝒪_{T}} gr_{I}^{i}𝒪_{T} \ar[u]_{β_{i}}}

Since P is almost projective (especially, flat) β_{i} is an isomorphism. Moreover, since gr_{I}^{0}ϕ is a monomorphism with almost projective cokernel, the long exact Tor sequence shows that gr_{I}^{0}ϕ ⊗_{𝒪_{T}} 1_{gr_{I}^{i}𝒪_{T}} is a monomorphism. It follows that α_{i} is a monomorphism for every i ∈ ℕ, and since it is obviously an epimorphism, we deduce that α_{i} is an isomorphism and gr_{I}^{•}ϕ is a monomorphism, therefore the same holds for ϕ. Let C := Coker(N → P); we deduce easily that Tor_{1}^{𝒪_{T}} (𝒪_{T_{0}}, C) = 0, and then it follows from the local flatness criterion (see [54, Ch.8,Th.22.3]) that C is a flat 𝒪_{T} -module. Finally lemma 3.2.25(i),(ii) says that C is almost finitely generated projective, whence the claim, in case (a).

Next, suppose that assumption (b) holds; by lemma 5.6.4 there is an isomorphism :

τ_{[−1}𝕃_{X/S}^{a} ≃ (0 → N →^{ϕ} Q → 0)

where N is almost finitely generated and Q is almost finitely generated projective over 𝒪_{X}.

Since Q is almost projective, we have τ_{[−1}(𝒪_{T_{0}} ⊗^{L}_{𝒪_{T}} Lσ^{∗}𝕃_{X/S}^{a}) ≃ (0 → N/IN →^{ϕ_{0}} Q/IQ → 0), and by assumption Ker ϕ_{0} = 0 and Coker ϕ_{0} is an almost finitely generated projective 𝒪_{T_{0}}- module. Using the long exact Ext sequence we deduce that N/IN is almost projective. Thus, the bottom arrow of the natural commutative diagram

\xymatrix{Q^{∗} \ar[r]^{ϕ^{∗}} \ar[d]_{α} & N^{∗} \ar[d]^{β} \\ (Q/IQ)^{∗} \ar[r]^{ϕ_{0}^{∗}} & (N/IN)^{∗}}

is an epimorphism. Invoking twice corollary 5.5.15 we find first that α is an epimorphism (whence so is β), and then that N is almost projective. It then follows that ϕ^{∗} ⊗_{𝒪_{S}} 1_{𝒪_{S_{0}}} is an epimorphism as well, hence the same holds for ϕ^{∗}, by applying Nakayama’s lemma 5.1.6 to Coker ϕ^{∗}. The long exact Ext sequence then shows that Ker ϕ^{∗} is almost projective, and the latter is almost finitely generated as well, since Q is. Dualizing, we see that ϕ is a monomorphism and Coker ϕ ≃ (Ker ϕ^{∗})^{∗} is almost finitely generated projective, which is the claim. □

Lemma 5.6.14. Let X be an affine S-scheme, and suppose we are given a cartesian diagram of 𝒪_{S}-algebras as (3.4.8). There follows a cartesian diagram of sets:

\xymatrix{X_{sm}(Spec A_{0}) \ar[r] \ar[d] & X_{sm}(Spec A_{2}) \ar[d] \\ X_{sm}(Spec A_{1}) \ar[r] & X_{sm}(Spec A_{3}).}

Proof. We have to check that every section σ ∈ X(Spec A_{0}) whose restrictions to Spec A_{1} and Spec A_{2} lie in the smooth locus of X over S, lies itself in the smooth locus. Hence, let τ_{[−1}Lσ^{∗}𝕃_{X/S} ≃ (0 → N →^{ϕ} P → 0), for some A_{0}-modules N and P, chosen so that P is

almost projective. Hence A_{i} ⊗^{L}_{A_{0}} τ_{[−1}Lσ^{∗}𝕃_{X/S} ≃ (0 → N_{i} →^{ϕ_{i}} P_{i} → 0) (where N_{i} := A_{i} ⊗_{A_{0}} N and likewise for P_{i}, i = 1, 2). By assumption, Ker ϕ_{i} = 0 and Coker ϕ_{i} is almost finitely generated projective over A_{i} (i = 1, 2). It follows that N_{i} is almost projective for i = 1, 2. Using proposition 3.4.21 we deduce that Coker ϕ is almost finitely generated projective over A_{0} and N is almost projective. In particular, N is flat and consequently N ⊂ N_{1} ⊕ N_{2}, so Ker ϕ = 0, and the assertion follows. □

Lemma 5.6.15. Let X and Y be two affine S-schemes, and suppose that Tor_{i}^{𝒪_{S}} (𝒪_{X}, 𝒪_{Y}) = 0 for every i > 0. Then we have a natural isomorphism of sheaves:

(X ×_{S} Y)_{sm} ≃ X_{sm} × Y_{sm}.

Proof. Let π_{X} : X ×_{S} Y → X be the natural projection, and define likewise π_{Y}. By theorem 2.5.35, our assumptions imply that the natural morphism π_{X}^{∗}𝕃_{X/S}^{a} ⊕ π_{Y}^{∗} 𝕃_{Y/S}^{a} → 𝕃_{X×_{S}Y/S}^{a} is a quasi-isomorphism. Let now T be an affine S-scheme and (σ, τ) ∈ X ×_{S} Y (T) = X(T) × Y (T); we derive a natural isomorphism:

L(σ, τ)^{∗}𝕃_{X×_{S}Y/S} ≃ Lσ^{∗}𝕃_{X/S} ⊕ Lτ^{∗}𝕃_{Y/S}

from which the claim follows straightforwardly. □

Theorem 5.6.16. Assume that hom.dim_{V} 𝔪̃ ≤ 1. Let X be an almost finitely presented affine S-scheme, I ⊂ 𝒪_{S} an ideal such that the pair (𝒪_{S}, I) is henselian, 𝔪_{0} ⊂ 𝔪 a finitely generated subideal, and set S_{n} := Spec 𝒪_{S}/𝔪_{0}^{n}I for every n ∈ ℕ. Then we have:

(i) Im(X_{sm}(S_{1}) → X_{sm}(S_{0})) = Im(X_{sm}(S) → X_{sm}(S_{0})).

(ii) Set X_{sm}^{∧}(S) := lim_{n∈ℕ}X_{sm}(S_{n}), and endow X_{sm}^{∧}(S) with the corresponding pro-discrete

topology. Then the natural map X_{sm}(S) → X_{sm}^{∧}(S) has dense image.

Proof. We begin with an easy reduction: Claim 5.6.17. In order to prove (i), we can assume that I is a tight ideal. Proof of the claim: Indeed, let 𝔪_{0} ⊂ 𝔪 be any subideal, and choose 𝔪_{1} ⊂ 𝔪 such that 𝔪_{0} ⊂ 𝔪_{1}^{2}. Set I′ := 𝔪_{1}I, S′_{0} := Spec 𝒪_{S}/I′, S′_{1} := Spec 𝒪_{S}/𝔪_{1}I′. Notice that I′ is tight, and suppose that the assertion is known for this ideal. By a simple chase on the commutative diagram:

[figure: X_{sm}(S); X_{sm}(S_{0}); X_{sm}(S_{1}); X_{sm}(S); X_{sm}(S′_{0}); X_{sm}(S′_{1})]

the assertion can then be deduced for I as well.

Let σ_{0} ∈ X_{sm}(S_{0}) that admits an extension σ̃ ∈ X_{sm}(S_{1}), pick finitely many generators ε_{1}, ..., ε_{k} for 𝔪_{0}, and define a map ϕ : I^{⊕k} → 𝔪_{0}I by the rule: (a_{1}, ..., a_{k}) ↦ ∑_{i}^{k} ε_{i} · a_{i}. Claim 5.6.18. Suppose that I^{2} = 0 (so that I is an 𝒪_{S_{0}}-module); then 𝔼xt_{𝒪_{S_{0}}}^{1} (Lσ_{0}^{∗}𝕃_{X/S}^{a}, ϕ) is onto and 𝔼xt_{𝒪_{S_{0}}}^{1} (Lσ_{0}^{∗}𝕃_{X/S}^{a}, I)^{a} = 0.

Proof of the claim: For the first assertion we use the spectral sequence

E_{2}^{pq} := Ext_{𝒪_{S_{0}}}^{p} (H_{q}(Lσ_{0}^{∗}𝕃_{X/S}^{a}), I^{⊕k}) ⇒ 𝔼xt_{𝒪_{S_{0}}}^{p+q}(Lσ_{0}^{∗}𝕃_{X/S}^{a}, I^{⊕k})

and a similar one for 𝔪_{0}I. Since σ_{0} lies in the smooth locus of X, the E_{2}^{01} terms vanish, hence we are reduced to verifying that the map Ext_{𝒪_{S_{0}}}^{1} (H_{0}(Lσ_{0}^{∗}𝕃_{X/S}^{a}), ϕ) is surjective. However, the cokernel of this map is a submodule of Ext_{𝒪_{S_{0}}}^{2} (H_{0}(Lσ_{0}^{∗}𝕃_{X/S}^{a}), Ker ϕ); again the assumption that σ_{0} lies in the smooth locus of X, and hom.dim_{V} 𝔪̃ ≤ 1, imply that the latter Ext group vanishes by lemma 2.4.14(ii). The same spectral sequence argument also proves the second assertion.

Claim 5.6.19. Assertion (i) holds if I^{2} = 0. Proof of the claim: We need to show that there exists a morphism σ : S → X extending σ_{0}; then σ ∈ X_{sm}(S) in view of proposition 5.6.13. In other words, we have to find σ^{♯} that fits into a morphism of extensions of 𝒪_{S}-algebras:

[figure: 0; 0; 𝒪_{X}; σ^{♯}; 𝒪_{X}; σ_{0}^{♯}; 0; 0; I; 𝒪_{S}; 𝒪_{S_{0}}; 0.]

By proposition 3.2.16, the obstruction to the existence of σ^{♯} is a class

ω ∈ 𝔼xt_{𝒪_{X}}^{1} (𝕃_{X/S}^{a}, I) ≃ 𝔼xt_{𝒪_{S_{0}}}^{1} (Lσ_{0}^{∗}𝕃_{X/S}^{a}, I).

Likewise, the obstruction to extending σ̃ is a class ω̃ ∈ 𝔼xt_{𝒪_{S_{0}}}^{1} (Lσ_{0}^{∗}𝕃_{X/S}^{a}, 𝔪_{0}I), and ω is the image of ω̃ under the map 𝔼xt_{𝒪_{S_{0}}}^{1} (Lσ_{0}^{∗}𝕃_{X/S}^{a}, j) (where we have denoted by j : 𝔪_{0}I → I the inclusion). From claim 5.6.18 one deduces easily first: that 𝔼xt_{𝒪_{S_{0}}}^{1} (Lσ_{0}^{∗}𝕃_{X/S}^{a}, j ∘ ϕ) vanishes, and therefore, second: that 𝔼xt_{𝒪_{S_{0}}}^{1} (Lσ_{0}^{∗}𝕃_{X/S}^{a}, j) must already vanish. Thus ω = 0, and the assertion holds.

Claim 5.6.20. Choose a finitely generated subideal 𝔪_{1} ⊂ 𝔪 such that 𝔪_{0} ⊂ 𝔪_{1}^{2}. The section σ_{0} can be lifted to an element of lim_{n∈ℕ}X_{sm}(Spec 𝒪_{S}/𝔪_{1}I^{n}).

Proof of the claim: For every n > 0, let T_{n} := Spec 𝒪_{S}/𝔪_{1}^{2}I^{n} and j_{n} : Spec 𝒪_{S}/𝔪_{1}I^{n} → T_{n} the natural morphism; we construct by induction on n ∈ ℕ a sequence of section σ_{n} ∈ X(T_{n}), such that the family (σ_{i}∘j_{i} | i > 0) defines an element of lim_{n∈ℕ}X_{sm}(Spec 𝒪_{S}/𝔪_{1}I^{n}). To this aim,

we take σ_{1} equal to the restriction of σ̃; suppose then that n > 1 and that σ_{n−1} is already given. Notice that the image J ⊂ 𝒪_{S_{n}} of 𝔪_{1}I^{n−1} satisfies J^{2} = 0. By the claim 5.6.19, it follows that σ_{n−1} ∘ j_{n−1} extends to a section in X_{sm}(T_{n}), and this we call σ_{n}.

Let us now show how to deduce assertion (i) from (ii). By claim 5.6.17 we can suppose that I^{m} ⊂ 𝔪_{0}𝒪_{S} for some m ≥ 0 and a finitely generated subideal 𝔪_{0} ⊂ 𝔪; let 𝔪_{1} ⊂ 𝔪 be as in claim 5.6.20; clearly for every n ∈ ℕ there exists n ≥ 0 such that 𝔪_{1}I^{m} ⊂ 𝔪_{0}^{n}I, hence claim 5.6.20 shows that σ_{0} can be lifted to an element of X_{sm}^{∧}(S) and then (ii) yields (i) trivially. Hence, it remains only to show (ii). Claim 5.6.21. In order to prove (ii) we can assume that 𝔪_{0} is a principal ideal. Proof of the claim: We argue by induction on the number of generators of 𝔪_{0}. Thus, let ε_{1}, ..., ε_{k} be a finite system of generators for 𝔪_{0}, and suppose that the assertion is known for all ideals generated by less than k elements. Let 𝔪_{0}^{(n)} be the ideal generated by ε_{1}^{n}, ..., ε_{k}^{n}, and set S_{(n)} := Spec 𝒪_{S}/𝔪_{0}^{(n)}I; for every n ∈ ℕ there exists N ∈ ℕ such that 𝔪_{0}^{N} ⊂ 𝔪_{0}^{(n)} ⊂ 𝔪_{0}^{n}, whence an isomorphism of pro-discrete spaces: X_{sm}^{∧}(S) ≃ lim_{n∈ℕ}X_{sm}(S_{(n)}). Thus, we can suppose that we

are given a compatible system of sections σ_{n} ∈ X_{sm}(S_{(n)}), and wish to show that, for every n ∈ ℕ, there exists σ ∈ X_{sm}(S) whose restriction to S_{(n)} agrees with σ_{n}. Fix N > 0, set T := Spec 𝒪_{S}/ε_{1}^{N}I and let 𝔪_{1} ⊂ 𝔪 be the ideal generated by ε_{2}, ..., ε_{k}. Let also T_{(n)} := T ×_{S} S_{(n)} for every n ∈ ℕ, and denote by σ_{n|T} ∈ X_{sm}(T_{(n)}) the restriction of σ_{n}. Clearly T_{(n)} ≃ Spec 𝒪_{T} /𝔪_{1}^{(n)}𝒪_{T} for every n ≥ N, and the pair (𝒪_{T} , I𝒪_{T}) is henselian (cp. remark 5.1.9(v)). Hence, by inductive assumption, for every n ≥ N we can find σ_{T} ∈ X(T) whose restriction to T_{(n)} agrees with σ_{n|T}. We can then apply claim 5.6.20, in order to find a compatible system of sections (σ′_{n} ∈ X(Spec 𝒪_{S}/ε_{1}^{n}I) | n ≥ N), whose restriction to Spec 𝒪_{S}/ε_{1}^{N−1}I agrees with the restriction of σ_{T}. Finally, if we assume that (ii) is known whenever 𝔪_{0} is principal, we can find a section σ ∈ X_{sm}(S) whose restriction to Spec 𝒪_{S}/ε_{1}^{N−1}𝒪_{S} agrees with σ′_{N}, whence the claim.

For a given ε ∈ 𝔪, set K(ε) := ⋃_{n∈ℕ} Ann_{𝒪_{S}} (ε^{n}). Claim 5.6.22. K(ε)_{∗} = ⋃_{n∈ℕ} Ann_{𝒪_{S∗}} (ε^{n}). Proof of the claim: Clearly we have only to show the inclusion K(ε)_{∗} ⊂ ⋃_{n∈ℕ} Ann_{𝒪_{S∗}} (ε^{n}). By applying the left exact functor M ↦ M_{∗} to the left exact sequence 0 → Ann_{𝒪_{S}} (ε^{n}) → 𝒪_{S} →^{ε^{n}} 𝒪_{S} we deduce that Ann_{𝒪_{S}} (ε^{n})_{∗} = Ann_{𝒪_{S∗}} (ε^{n}). However, ε · K(ε)_{∗} ⊂ ⋃_{n∈ℕ} Ann_{𝒪_{S}} (ε^{n})_{∗}, so the claim follows easily.

Claim 5.6.23. (K(ε) ∩ ε^{n}I)_{∗} ⊂ nil(𝒪_{S∗}) for every n > 0. Proof of the claim: We have (K(ε) ∩ ε^{n}I)_{∗} = K(ε)_{∗} ∩ (ε^{n}I)_{∗}. Let x ∈ (ε^{n}I)_{∗}; according to lemma 5.1.14, we have x^{k} ∈ ε^{n}𝒪_{S∗} for every sufficiently large k ∈ ℕ. On the other hand, it is easy to check that Ann_{𝒪_{S∗}} (ε^{m}) ∩ ε^{n}𝒪_{S∗} ⊂ nil(𝒪_{S∗}) for every n, m > 0. In view of claim 5.6.22, the assertion follows.

Claim 5.6.24. If n > 0, every almost finitely generated subideal of K(ε) ∩ ε^{n}I is nilpotent.

Proof of the claim: Let ℐ be such an ideal; we can find a finitely generated ideal ℐ_{0} ⊂ (K(ε)∩ ε^{n}I)_{∗} such that εℐ_{∗} ⊂ ℐ_{0} ⊂ ℐ_{∗}. Since ℐ ⊂ ε^{n}I, lemma 5.1.14 says that there exists N ∈ ℕ such that (ℐ_{∗})^{N} ⊂ ε^{n}𝒪_{S∗}, hence (ℐ_{∗})^{N+1} ⊂ ℐ_{0}. From claim 5.6.23 we deduce that ℐ_{0} is a nilpotent ideal, so the same holds for ℐ.

Claim 5.6.25. In order to prove (ii) we can assume that 𝔪_{0} = εV, where ε ∈ 𝔪 is an 𝒪_{S}-regular element.

Proof of the claim: Let us write X = Spec S_{P} /J, where S_{P} := Sym_{𝒪_{S}}^{•} P for some almost finitely projective 𝒪_{S}-module P, and J ⊂ S_{P} is an almost finitely generated ideal. By claim 5.6.21 we can assume that 𝔪_{0} = εV for some ε ∈ 𝔪. Set \overline{S} := Spec 𝒪_{S}/K(ε) and \overline{S}_{n} := \overline{S} ×_{S} S_{n} for every n ∈ ℕ. Let σ^{∧} ∈ X_{sm}^{∧}(S); by definition σ^{∧} is a compatible family of sections σ_{n} : S_{n} → X lying in the smooth locus of X over S. In turns, σ_{n} can be viewed as the datum of an 𝒪_{S}-linear morphism τ_{n} : P → 𝒪_{S}/ε^{n}I, such that the induced morphism of 𝒪_{S}-algebras Sym_{𝒪_{S}}^{•} τ : Sym_{𝒪_{S}}^{•} P → 𝒪_{S}/ε^{n}I satisfies the condition: Sym_{𝒪_{S}}^{•} τ(J) = 0. For every n ∈ ℕ we have a cartesian diagram of 𝒪_{S}-algebras:

\xymatrix{𝒪_{S}/(K(ε) ∩ ε^{n}I) \ar[r] \ar[d] & 𝒪\overline{S} \ar[d]^{π_{n}} \\ 𝒪_{S_{n}} \ar[r]^{p_{n}} & 𝒪_{\overline{S}_{n}}.}

Notice that ε is regular in 𝒪\overline{S} and suppose that assertion (ii) holds for all almost finitely presented \overline{S}-schemes, especially for \overline{X} := X ×_{S} \overline{S}. Let \overline{σ}_{n} : \overline{S}_{n} → \overline{X} be the restriction of σ_{n}, for every n ∈ ℕ. It then follows that, for every n ∈ ℕ, \overline{σ}_{n} extends to a section \overline{σ} ∈ \overline{X}_{sm}(\overline{S}). The datum of \overline{σ} is equivalent to the datum of an 𝒪\overline{S}-linear morphism \overline{τ} : P → 𝒪\overline{S} such that Sym_{𝒪_{S}}^{•} \overline{τ}(J) = 0 and such that π_{n} ∘ \overline{τ} = p_{n} ∘ τ_{n}. The pair (τ_{n}, \overline{τ}) determines a morphism ω_{n} : P → 𝒪_{S}/(K(ε) ∩ ε^{n}I), and by construction we have: Sym_{𝒪_{S}}^{•} ω_{n}(J) = 0, i.e. ω_{n} induces a section σ′ : Spec 𝒪_{S}/(K(ε) ∩ ε^{n}I) → X, and then σ′ must lie in the smooth locus of X over S, in view of lemma 5.6.14. The obstruction to the existence of a lifting ω : P → 𝒪_{S} of ω_{n}, is a class α_{n} ∈ Ext_{𝒪_{S}}^{1} (P, K(ε) ∩ ε^{n}I). A simple verification shows that

(5.6.26)  K(ε) ∩ ε^{n}I = ε · (K(ε) ∩ ε^{n−1}I)  for all n > 0.

From (5.6.26), an argument as in the proof of claim 5.1.19 allows to conclude that the image of α_{n} in Ext_{𝒪_{S}}^{1} (P, K(ε) ∩ ε^{n−1}I) vanishes; however, this image is none other than α_{n−1}, so actually α_{n} = 0, and the sought lifting can be found for every n > 0. By construction we have \overline{J} := Sym_{𝒪_{S}}^{•} ω(J) ⊂ K(ε) ∩ ε^{n}I. Now, ω induces a section σ′′ : Spec 𝒪_{S}/\overline{J} → X, which by construction extends σ′; moreover, the pair (𝒪_{S}/\overline{J}, I/\overline{J}) is henselian (cp. remark 5.1.9(v)), hence ε^{n}I/\overline{J} is a tight radical ideal. Thus, using proposition 5.6.13(ii) we derive that σ′′ lies as well in the smooth locus of X over S. Since J is almost finitely generated, \overline{J} is nilpotent in view of claim 5.6.24. Moreover, by (5.6.26) we can write \overline{J} = ε𝒥 for some ideal 𝒥 ⊂ K(ε) ∩ ε^{n−1}I; 𝒥 is nilpotent if n ≥ 2, since in that case 𝒥^{2} ⊂ \overline{J}. Hence we can apply claim 5.6.20, to deduce that the restriction of σ′′ to Spec 𝒪_{S}/𝒥 extends to a section σ ∈ X_{sm}(S); by construction σ extends σ_{n−1}. Since n can be taken to be arbitrarily large, the claim follows.

So finally we suppose that 𝔪_{0} = tV, with t an 𝒪_{S}-regular element. Set 𝒪_{S}^{∧} := lim_{n∈ℕ}𝒪_{S_{n}} and

S^{∧} := Spec 𝒪_{S}^{∧}. We have a natural bijection

(5.6.27)  X(S^{∧}) ≃ X^{∧}(S) := lim_{n∈ℕ}X(S_{n}).

Let I_{∗}^{∧} ⊂ 𝒪_{S∗}^{∧} ≃ lim_{n∈ℕ}𝒪_{S_{n}∗} be the topological closure of I_{∗}; since t^{n}I_{∗}^{∧} is the topological closure

of t^{n}I_{∗}, one verifies easily that (5.6.27) identifies the pro-discrete topology of X^{∧}(S) with the (t, I_{∗}^{∧})-adic topology of X(S^{∧}) (see (5.6.7)and the proof of lemma 5.6.9). Claim 5.6.28. The homeomorphism (5.6.27)induces a bijection: X_{sm}(S^{∧}) ≃ X_{sm}^{∧}(S). Proof of the claim: Let (σ_{n} | n ∈ ℕ) be an element of X_{sm}(S^{∧}), and σ^{∧} ∈ X(S^{∧}) the corresponding section. We have to show that σ^{∧} lies in the smooth locus of X. Thus, let I^{∧} := Ker(𝒪_{S}^{∧} → 𝒪_{S}/I); we have a natural isomorphism: 𝒪_{S}^{∧}/tI^{∧} ≃ 𝒪_{S}/tI, and by assumption, the restriction of σ^{∧} to Spec 𝒪_{S}^{∧}/tI^{∧} lies in the smooth locus of X. The assertion then follows from proposition 5.6.13.

Under the standing assumptions, we have a cartesian diagram of almost algebras:

\xymatrix{𝒪_{S} \ar[r] \ar[d] & 𝒪_{S}[t^{−1}] \ar[d] \\ 𝒪_{S}^{∧} \ar[r] & 𝒪_{S}^{∧}[t^{−1}]}

whence a cartesian diagram of sets:

\xymatrix{X(S) \ar[r] \ar[d] & X_{t}(S_{t}) \ar[d]^{α} \\ X(S^{∧}) \ar[r]^{β} & X_{t}(S_{t}^{∧}).}

Now, let (σ_{n} | n ∈ ℕ) ∈ X_{sm}^{∧}(S); by claim 5.6.28, the corresponding section σ^{∧} : S^{∧} → X lies in the smooth locus of X. Let σ_{t}^{∧} ∈ X_{sm,t}(S_{t}^{∧}) be the restriction of σ^{∧}. By corollary 5.1.16(ii), the pair (𝒪_{S∗}, tI_{∗}) is henselian; then by (5.6.6)and proposition 5.4.21, the restriction α_{sm} : X_{sm,t}(S_{t}) → X_{sm,t}(S_{t}^{∧}) has dense image for the (t, I_{∗}^{∧})-adic topology. Hence, we can approximate σ_{t}^{∧} arbitrarily (t, I_{∗}^{∧})-adically close by a section of the form τ_{t}^{∧} := α(τ_{t}), where τ_{t} ∈ X_{sm,t}(S_{t}). Furthermore, β is an open imbedding, by lemma 5.6.9. Hence, if τ_{t}^{∧} is close enough to σ^{∧}, we can find τ^{∧} ∈ X(S^{∧}) such that β(τ^{∧}) = τ_{t}^{∧}. In view of proposition 5.6.13 we can also achieve that τ^{∧} lies in the smooth locus of X. The pair (τ_{t}, τ^{∧}) determines a unique section τ ∈ X(S), and by construction τ can be obtained as (t, I)-adically close to (σ_{n} | n ∈ ℕ) as desired. Especially, we can achieve that τ lies in the smooth locus of X, which concludes the proof of the theorem. □

5.7. Quasi-projective almost schemes. Let R be a V -algebra; we denote by (R-Alg)_{fpqc}^{o} the large fpqc site of affine R-schemes, and similarly for the site (R^{a}-Alg)_{fpqc}^{o}. The localization functor R-Alg → R^{a}-Alg defines a morphism of sites:

j: (R^{a}-Alg)_{fpqc}^{o} → (R-Alg)_{fpqc}^{o}.

If ℱ is any sheaf on (R-Alg)_{fpqc}^{o}, then ℱ^{a} := j^{∗}ℱ can be described as the sheaf associated to the presheaf B ↦ ℱ (B_{∗}). Especially, we can regard any R-scheme X as a sheaf on (R-Alg)_{fpqc}^{o}, and hence we obtain the almost scheme X^{a} associated to X. If X is affine, this notation agrees with that of (3.3.3). For a general X, pick a Zariski hypercovering Z_{•} → X, where each Z_{i} is a disjoint union of affine R-schemes; then X ≃ colim_{Δ^{o}} Z_{•} as fpqc-sheaves, and thus X^{a} ≃ colim_{Δ^{o}} Z_{•}^{a} in the topos of sheaves on (R^{a}-Alg)_{fpqc}^{o}. Furthermore, we have

j_{∗}Spec B = Spec B_{!!} ⊗_{R_{!!}^{a}} R for every R^{a}-algebra B.

If G is an R-group scheme, then G^{a} is clearly an R^{a}-group scheme, and if Y → X is any G-torsor over an R-scheme X (for the fpqc topology of X), then Y^{a} → X^{a} is a G^{a}-torsor.

5.7.1. Let now A be an R^{a}-algebra, J ⊂ A_{∗} a finitely generated ideal. We define the quasiaffine R^{a}-scheme Spec A \ V (J) as the almost scheme j^{∗}(Spec A_{∗} \ V (J)). Let f_{1}, ..., f_{k} be a finite set of generators of J; by (5.7) this can be realized as the subsheaf ⋃_{i=1}^{k} Spec A[f_{i}^{−1}] ⊂ Spec A, where the union takes place in the category of sheaves on (R^{a}-Alg)_{fpqc}^{o}.

Lemma 5.7.2. Let B be a V^{a}-algebra and g_{1}, ..., g_{k} ∈ B_{∗}. Then ∑_{i=1}^{k} g_{i}B = B if and only if the natural morphism B → ∏_{i=1}^{k} B[g_{i}^{−1}] is faithfully flat.

Proof. Supppose that ∑_{i=1}^{k} g_{i}B = B; we have to show that ∏_{i=1}^{k} C[g_{i}^{−1}] ≠ 0 for every nonzero quotient C of B. Replacing B by C, we are reduced to showing that, if B[g_{i}^{−1}] = 0 for every i ≤ k, then B = 0. However, the condition B[g_{i}^{−1}] = 0 implies that for every ε ∈ 𝔪 there exists n_{i} ∈ ℕ such that ε · g_{i}^{n_{i}} = 0; set n := max(n_{i} | i ≤ k). We have ∑_{i=1}^{k} g_{i}^{n}B = B, hence ε · B = 0. Since ε is arbitrary, the claim follows. Conversely, let B′ := ∏_{i=1}^{k} B[g_{i}^{−1}] and I := ∑_{i=1}^{k} g_{i}B; clearly IB′ = B′, therefore I = B provided B′ is faithfully flat over B. □

Lemma 5.7.3. With the notation of (5.7.1), the almost scheme Spec A \ V (J) is the subfunctor of Spec A, whose T -section are the morphisms σ : T → Spec A such that σ^{♯}(J)𝒪_{T} = 𝒪_{T}, for every affine R^{a}-scheme T.

Proof. Let σ be a T -section of Spec A \ V (J); by definition, this is the same as saying that the natural morphism ⨿_{i=1}^{k}T ×_{Spec A} Spec A[f_{i}^{−1}] → T is a fpqc-covering of T. Hence the claim is just a restatement of lemma 5.7.2. □

5.7.4. Alternatively, one can regard a quasi-affine R^{a}-scheme as a difference of two affine R^{a}-schemes. This viewpoint is elaborated in the following: Definition 5.7.5. Let ϕ : ℱ → 𝒢 be a morphisms of sheaves on (R^{a}-Alg)_{fpqc}^{o}.

(i) We say that ϕ is a closed imbedding if, for every affine R^{a}-scheme T and every section T → 𝒢, the induced morphism T ×_{𝒢} ℱ → T is a closed imbedding of affine R^{a}- schemes.

(ii) The difference 𝒢 \ ℱ is the subsheaf of 𝒢 whose T -sections are all the T -sections T → 𝒢 of 𝒢 such that T ×_{𝒢} ℱ = ∅, for every affine R^{a}-scheme T. Here ∅ denotes the initial object of the topos of sheaves on (R^{a}-Alg)_{fpqc}^{o}.

Using lemma 5.7.3 one checks easily that the sheaf-theoretic difference Spec A \ Spec A/J is the same as the almost scheme considered in (5.7.1). Lemma 5.7.6. Let ϕ : ℱ → 𝒢 be a morphism of sheaves on (R-Alg)_{fpqc}^{o}. Then:

j^{∗}(𝒢 \ ℱ) ⊂ j^{∗}𝒢 \ j^{∗}ℱ .

Proof. 𝒢 \ ℱ is also the largest subobject of of 𝒢 that has empty intersection with Im(ϕ); in particular this is defined for every topos. Then the statement holds for any morphism of topoi (in this case j), and it follows easily from the facts that the pull-back functor (here j^{∗}) is left exact and sends empty objects to empty objects. □

Example 5.7.7. Let E be an R^{a}-module.

(i) We define the presheaf ℙ_{R^{a}}(E) on (R^{a}-Alg)_{fpqc}^{o} as follows. For every affine R^{a}- scheme T, the T -sections of ℙ(E) are the strictly invertible quotients of E ⊗_{R^{a}} 𝒪_{T}, i.e. the equivalence classes of epimorphisms E ⊗_{R^{a}} 𝒪_{T} → L where L is a strictly invertible 𝒪_{T} -module (see definition (4.4.32)).

(ii) More generally, for every integer r ∈ ℕ we can define the presheaf Grass_{R^{a}}^{r}(E); its

T -sections are the rank r almost 𝒪_{T} -projective quotients E ⊗_{R^{a}} 𝒪_{T} → P. By faithfully flat descent it is easy to verify that both these presheaves are in fact fpqc-sheaves.

5.7.8. Let E be an R-module, B an R^{a}-algebra; a section σ ∈ Grass_{R}^{r}(E)(Spec B_{∗}) is a quotient map σ : E ⊗_{R} B_{∗} → P, onto a projective B_{∗}-module P of rank r. To σ we associate the quotient σ^{a} : E^{a} ⊗_{R} B → P^{a}; after passing to the associated sheaves we obtain a morphism:

(5.7.9)  j^{∗}Grass_{R}^{r}(E) → Grass_{R^{a}}^{r}(E^{a})  σ ↦ σ^{a}.

Since P ≃ P_{∗}^{a}, it is clear that the rule σ ↦ σ^{a} defines an injective map of presheaves, hence (5.7.9)is a monomorphism. Lemma 5.7.10. Suppose that E is a finitely generated R-module. Then (5.7.9) is an isomorphism.

Proof. We have to show that (5.7.9) is an epimorphism of fpqc-sheaves; hence, let T be any affine R^{a}-scheme, σ : E^{a} ⊗_{R^{a}} 𝒪_{T} → P a quotient morphism, with P almost projective over 𝒪_{T} of rank r. The contention is that there exists a faithfully flat morphism f : U → T such that f^{∗}σ is in the image of Grass_{R}^{r}(E)(U_{∗}) (where U_{∗} := Spec 𝒪_{U∗}). By theorem 4.4.24 we can then reduce to the case when P = 𝒪_{T}^{r}. We deduce an epimorphism Λ_{𝒪_{T}}^{r} σ : Λ_{𝒪_{T}}^{r} (E^{a} ⊗_{R^{a}} 𝒪_{T}) → Λ_{𝒪_{T}}^{r} (𝒪_{T}^{r}) ≃ 𝒪_{T}. Let e_{1}, ..., e_{k} be a finite set of generators of E, and for every subset J := {j_{1}, ..., j_{r}} ⊂ {1, ..., k} of cardinality r, let e_{J} := e_{j_{1}} ∧ ... ∧ e_{j_{r}} ∈ Λ_{𝒪_{T}}^{r} (E^{a} ⊗_{R^{a}} 𝒪_{T}); it follows that the set (t_{J} := Λ_{𝒪_{T}}^{r} σ(e_{J}) | J ⊂ {1, ..., k}) generates 𝒪_{T}. Set U_{J} := Spec 𝒪_{T} [t_{J}^{−1}] for every J ⊂ {1, ..., k} as above; by construction σ induces surjective maps E ⊗_{R} 𝒪_{T ∗}[t_{J}^{−1}] → 𝒪_{T ∗}^{r}[t_{J}^{−1}]; after tensoring by 𝒪_{T} [t_{J}^{−1}]_{∗} we obtain surjections E ⊗_{R} 𝒪_{T} [t_{J}^{−1}]_{∗} → 𝒪_{T} [t_{J}^{−1}]_{∗}^{r}. In other words, the restriction σ_{|U_{J}} lies in the image of Grass_{R}^{r}(E)(U_{J∗}) for every such J; on the other hand, lemma 5.7.2 says that the natural morphism ⨿_{J}U_{J} → T is a fpqc covering, whence the claim. □

Lemma 5.7.10 says in particular that Grass_{R^{a}}^{r}(E^{a}) is an almost scheme associated to a scheme whenever E is finitely generated. Next we wish to define the projective R^{a}-scheme associated a graded R^{a}-algebra. Definition 5.7.11. Let A := ⊕_{i∈ℕ}A_{i} be a graded R^{a}-algebra which is generated by A_{1} over A_{0} := R^{a}, in the sense that the natural morphism of graded R^{a}-algebras Sym_{R^{a}}^{•}A_{1} → A is an epimorphism. We let Proj(A) be the sheaf on (R^{a}-Alg)_{fpqc}^{o} defined as follows. For every affine R^{a}-scheme T, the T -points of Proj(A) are the strictly invertible quotients A_{1} ⊗_{R^{a}} 𝒪_{T} → L of A_{1} ⊗_{R^{a}} 𝒪_{T}, inducing morphisms of graded R^{a}-algebras A → Sym_{𝒪_{T}}^{•} L. Lemma 5.7.12. With the notation of (5.7.11), the natural map of sheaves Proj(A) → ℙ_{R^{a}}(A_{1}) is a closed imbedding.

Proof. Let T be any affine R^{a}-scheme and σ : T → ℙ_{R^{a}}(A_{1}) a morphism of fpqc-sheaves. We have to show that T′ := T ×_{ℙ_{R^{a}}(A_{1})} Proj(A) is a closed subscheme of T. By definition, σ : A_{1} ⊗_{R^{a}} 𝒪_{T} → L is a strictly invertible quotient; we deduce via Sym_{𝒪_{T}}^{•} σ a natural structure of Sym_{R^{a}}^{•}A_{1}-module on Sym_{𝒪_{T}}^{•} L. Let U be an affine R^{a}-scheme and U → T′ a morphism of fpqc-sheaves; set J_{n} := Ker (Sym_{R^{a}}^{n}A_{1} → A) for every n ∈ ℕ. By definition:

J_{n} · Sym_{𝒪_{T}}^{n} L ⊂ Ker (Sym_{𝒪_{T}}^{n} L → 𝒪_{U} ⊗_{𝒪_{T}} Sym_{𝒪_{T}}^{n} L). for every n ∈ ℕ

Since L is flat, this is the same as:

(5.7.13)  J_{n} · Sym_{𝒪_{T}}^{n} L ⊂ Ker (𝒪_{T} → 𝒪_{U}) · Sym_{𝒪_{T}}^{n} L.  for every n ∈ ℕ

For every n ∈ ℕ denote by ev_{n} : Sym_{𝒪_{T}}^{n} L ⊗_{𝒪_{T}} Sym_{𝒪_{T}}^{n} L^{∗} → 𝒪_{T} the evaluation morphism of the 𝒪_{T} -module Sym_{𝒪_{T}}^{n} L. Then (5.7.13)is equivalent to:

∑_{n∈ℕ}ev_{n}(J_{n} · Sym_{𝒪_{T}}^{n} L ⊗_{𝒪_{T}} Sym_{𝒪_{T}}^{n} L^{∗}) ⊂ Ker (𝒪_{T} → 𝒪_{U}).

Conversely, let

(5.7.14)  I := ∑_{n∈ℕ}ev_{n}(J_{n} · Sym_{𝒪_{T}}^{n} L ⊗_{𝒪_{T}} Sym_{𝒪_{T}}^{n} L^{∗}).

Then the restriction of σ to Spec 𝒪_{T} /I is a section of Proj(A). All in all, this shows that T′ represents the closed subscheme Spec 𝒪_{T} /I of T, whence the claim. □

Lemma 5.7.15. Let S := ⊕_{i∈ℕ}S_{i} be a graded R-algebra, with S_{0} = R; suppose that S is generated by S_{1} and that S_{1} is a finitely generated R-module. Then the natural map

j^{∗}Proj(S) → Proj(S^{a})

is an isomorphism.

Proof. Let B be an R^{a}-algebra, and σ : S_{1}^{a} ⊗_{R^{a}} B → L a strictly invertible B-module quotient. By lemma 5.7.10 we know that there is a faithfully flat B-algebra C such that the induced map σ′ : S_{1} ⊗_{R} C_{∗} → (L ⊗_{B} C)_{∗} is a rank one projective quotient. Suppose now that σ is a section of Proj(S^{a}) and let J := Ker(Sym_{R}^{•}(S_{1}) → S); by definition we have J^{a} ⊂ Ker Sym_{B}^{•}σ, and since Sym_{C_{∗}}^{•}(L ⊗_{B} C)_{∗} does not contain 𝔪-torsion, it follows that J ⊂ Ker Sym_{C_{∗}}^{•}σ′. □

5.7.16. Next we consider quasi-projective almost schemes. Hence, let A be as in definition 5.7.11 and 𝒥 := ⊕_{i∈ℕ}𝒥_{i} ⊂ A a graded ideal. We set X := Proj(A) \ Proj(A/𝒥). Lemma 5.7.17. Keep the assumptions of (5.7.16), and suppose moreover that A = S^{a} and 𝒥 = J^{a} for a graded R-algebra S and a finitely generated graded ideal J ⊂ S. Suppose moreover that S_{1} is a finitely generated R-module and that S_{1} generates S. Then

X ≃ j^{∗}(Proj(S) \ Proj(S/J)).

Proof. Set Y := Proj(S), Y_{0} := Proj(S/J), X′ := Y \ Y_{0}; in view of lemmata 5.7.6 and 5.7.15, we have only to show that

j^{∗}Y \ j^{∗}Y_{0} ⊂ j^{∗}X′.

Hence, let B be an R^{a}-algebra and σ ∈ j^{∗}Y (Spec B); by lemma 5.7.15 we can find a faithfully flat morphism T → Spec B and a projective 𝒪_{T ∗}-module L_{∗} of rank one such that the restriction of σ to T is induced by a surjective map τ : S_{1}⊗_{R}𝒪_{T ∗} → L_{∗}. Suppose now that σ is a section of j^{∗}Y \j^{∗}Y_{0}; it follows that τ_{0} := τ ×_{Y} Y_{0} = ∅. From lemma 5.7.12 and its proof we deduce easily that τ_{0} is represented the R^{a}-scheme Spec 𝒪_{T} /I, where I is defined as in (5.7.14); consequently I = 𝒪_{T}. Since Sym_{𝒪_{T}}^{•} L^{∗} is generated by (L_{∗})^{∗}, which is a finitely generated 𝒪_{T ∗}-module, and since J is finitely generated by assumption, we see that I is generated by the images x_{1}, ..., x_{k} of finitely many elements in J · Sym_{𝒪_{T ∗}}^{•} L_{∗} ⊗_{𝒪_{T}} Sym_{𝒪_{T ∗}}^{•} (L_{∗})^{∗}. It then follows from lemma 5.7.2 that there exists a covering morphism U → T such that ∑_{i=1}^{k} x_{i}𝒪_{U∗} = 𝒪_{U∗}, which means that the restriction of σ to U is a section of j^{∗}X′. □

5.7.18. In the situation of (5.7.16), let Y := Spec A \ Spec (A/𝒥 A_{+}), where A_{+} := ⊕_{i>0}A_{i}; then Y represents the functor that assigns to every R^{a}-scheme T the set of all maps of R^{a}- algebras ϕ : A → 𝒪_{T} such that

(5.7.19)  ϕ(𝒥 A_{+}) · 𝒪_{T} = 𝒪_{T} .

Moreover, we have a map of R^{a}-schemes π : Y → X, representing the natural transformation of functors that assigns to any T -section ϕ : A → 𝒪_{T} of Y the induced map of 𝒪_{T} -modules ϕ_{1} : A_{1}⊗_{R}𝒪_{T} → 𝒪_{T}. Condition (5.7.19)ensures that ϕ_{1} is surjective and ϕ_{1} ∉ Proj(A/𝒥)(T) whenever T is non-empty, hence this rule yields a well defined T -section of X. It is a standard fact that π is a 𝔾_{m}-torsor. The action of 𝔾_{m}(T) on Y (T) can be described explicitly as follows. To a given pair (ϕ, u) ∈ Y (T) × 𝒪_{T}^{×} one assigns the unique R-algebra map ϕ_{u} : A → 𝒪_{T} such that ϕ_{u}(x) = uϕ(x) for every x ∈ A_{1}.

Lemma 5.7.20. Keep the notation of (5.7.18). The subsheaf

Y_{sm} := (Spec A)_{sm} \ Spec (A/𝒥 A_{+}) ⊂ Y

is a subtorsor of Y for the natural 𝔾_{m}-action on Y.

Proof. Clearly 𝔾_{m} = (𝔾_{m})_{sm}; hence, by lemma 5.6.15, we have a natural identification:

(𝔾_{m} ×_{R^{a}} Y)_{sm} ≃ 𝔾_{m} × Y_{sm}.

However, the natural transformation of functors 𝔾_{m} ×_{R^{a}} Y → Y ×_{X} Y defined by the rule:

(u, ϕ) ↦ (ϕ, ϕ_{u}) for every local section u of 𝔾_{m} and ϕ of Y

is an isomorphism, therefore (𝔾_{m} ×_{R^{a}} Y)_{sm} ≃ (Y ×_{X^{a}} Y)_{sm}. Let T be an R^{a}-scheme, ϕ ∈ Y (T) and u ∈ 𝔾_{m}(T); it follows that (ϕ, ϕ_{u}) ∈ (Y ×_{X} Y)_{sm}(T) if and only if ϕ ∈ Y_{sm}(T). Furthermore, it is clear that (ϕ, ϕ_{u}) is in the smooth locus of Y ×_{X} Y if and only if (ϕ_{u}, ϕ) is (use the X-automorphism of Y ×_{X} Y that swaps the two factors). However, the pair (ϕ_{u}, ϕ) can be written in the form (ψ, ψ_{v}) for ψ = ϕ_{u} and v = u^{−1}, so that ψ is in the smooth locus if and only if (ψ, ψ_{v}) is. Ergo, ϕ is in the smooth locus if and only if ϕ_{u} is, as required. □

Definition 5.7.21. Let X and Y be as in (5.7.18). We define the smooth locus of X as the quotient (in the category of sheaves on (R^{a}-Alg)_{fpqc}^{o}):

X_{sm} := Y_{sm}/𝔾_{m}.

The notation X_{sm} remains somewhat ambiguous, until one shows that the smooth locus of X does not depend on the choice of presentation of X as quotient of a quasi-affine almost scheme Y. While we do not completely elucidate this issue, we want at least to make the following remark: Lemma 5.7.22. In the situation of (5.7.18), suppose that ℒ is a strictly invertible R^{a}-module, and set B := ⊕_{r∈ℕ}A_{r} ⊗_{R^{a}} ℒ^{⊗r}, 𝒥_{B} := ⊕_{r∈ℕ}𝒥_{r} ⊗_{R^{a}} ℒ^{⊗r}, so B is a graded R^{a}-algebra and 𝒥_{B} ⊂ B a graded ideal. Then there are natural isomorphisms

X ≃ X_{B} := Proj(B) \ Proj(B/𝒥_{B}) X_{sm} ≃ X_{B,sm}.

Proof. We define as follows a natural isomorphism of functors X →^{∼} X_{B}. To every R^{a}-scheme T and every T -section A_{1}⊗_{R^{a}} 𝒪_{T} → L, we assign the strictly invertible quotient A_{1}⊗_{R^{a}} ℒ ⊗_{R^{a}} 𝒪_{T} → ℒ ⊗_{R^{a}} L. Set Y_{B} := Spec B \ Spec (B/𝒥_{B}B_{+}); it follows that Y_{B}/𝔾_{m} ≃ X_{B}. Hence, the identity X_{sm} ≃ X_{B,sm} can be checked locally on the fpqc topology. However, ℒ is locally free of rank one on (R^{a}-Alg)_{fpqc}^{o}, and the claim is obvious in case ℒ is free. □

Theorem 5.7.23. Suppose 𝔪̃ has homological dimension ≤ 1, and that (R^{a}, I) is a henselian pair. Let P be an almost finitely generated projective R^{a}-module and ℐ , 𝒥 ⊂ S_{P} := Sym_{R^{a}}^{•}P two graded ideals, with 𝒥 almost finitely generated. Set X := Proj(S_{P} /𝒥) \ V (ℐ). Then, for every finitely generated subideal 𝔪_{0} ⊂ 𝔪, we have:

Im(X_{sm}(Spec R^{a}/𝔪_{0}I) → X_{sm}(Spec R^{a}/I)) = Im(X_{sm}(Spec R^{a}) → X_{sm}(Spec R^{a}/I)).

Proof. First of all, arguing as in the proof of claim 5.6.17 we reduce to the case where I is a tight ideal. Next, set 𝒥 := ⊕_{i∈ℕ}𝒥_{i}, ℐ := ⊕_{i∈ℕ}ℐ_{i} and let \overline{σ} : (P/𝒥_{1}) ⊗_{R^{a}} R^{a}/𝔪_{0}I → \overline{L} be an R^{a}/𝔪_{0}I-section of X. By theorem 5.5.6 we can lift \overline{L} to an almost finitely generated projective R^{a}-module L. We have Λ_{R^{a}}^{2}L ⊗_{R^{a}} R^{a}/𝔪_{0}I ≃ Λ_{R^{a}}^{2}\overline{L} = 0, whence Λ_{R^{a}}^{2}L = 0 by Nakayama’s lemma 5.1.6, so L is strictly invertible. We set S′ := Sym_{R^{a}}^{•}(P ⊗_{R^{a}} L^{∗}), ℐ′ := ⊕_{r∈ℕ}ℐ_{r} ⊗_{R^{a}} L^{∗⊗r}, 𝒥′ := ⊕_{r∈ℕ}𝒥_{r} ⊗_{R^{a}} L^{∗⊗r}, B := S′/𝒥′, Y′ := Spec B \ Spec (B/ℐ′B_{+}) and X′ := Proj(B) \ Proj(B/ℐ′). By lemma 5.7.22 we have an isomorphism β : X′ ≃ X preserving the smooth loci; by inspecting the definition we find that β^{−1}(\overline{σ}) lies in the image of the projection Y′_{sm}(Spec R^{a}/𝔪_{0}I) → X′_{sm}(Spec R^{a}/𝔪_{0}I). Hence we can replace P by P ⊗_{R^{a}} L^{∗} and assume from start that \overline{σ} lies in the image of the map Y_{sm}(Spec R^{a}/𝔪_{0}I) → X_{sm}(Spec R^{a}/𝔪_{0}I). Let \overline{τ} ∈ Y_{sm}(Spec R^{a}/𝔪_{0}I) that maps to \overline{σ}. By theorem 5.6.16(i), the image τ_{0} of \overline{τ} in (Spec S_{P} /𝒥)_{sm} lifts to a section τ ∈ (Spec S_{P} /𝒥)_{sm}(Spec R^{a}). To conclude, it suffices to show the following: Claim 5.7.24. Suppose that I is tight. Then τ ∈ Y (Spec R^{a}). Proof of the claim: Let A := S_{P} /𝒥, so that τ : A → R^{a} is a morphism of R^{a}-algebras, and set J := τ(A_{+}ℐ); by assumption τ_{0} ∈ Y (Spec R^{a}/I), which means that J · (R^{a}/I) = (R^{a}/I), i.e. J + I = R^{a}. Set M := R^{a}/JR^{a}; it then follows that M = IM, hence M = 0 by lemma 5.1.6; hence JR^{a} = R^{a}, which is the claim. □

5.8. Lifting and descent of torsors. We begin this section with some generalities about group schemes and their linear representations, that are preliminary to our later results about liftings of torsors in an almost setting (theorems 5.8.19 and 5.8.21).

Furthermore, we apply the results of section 5.4 to derive a descent theorem for G-torsors, where G is a group scheme satisfying some fairly general conditions.

5.8.1. Let R be a ring, G an affine group scheme defined over R. We set 𝒪(G) := Γ(G, 𝒪_{G}); so 𝒪(G) is an R-algebra and the multiplication map of G determines a structure of co-algebra on 𝒪(G). In (3.3.7)we have defined the notion of a (left or right) G-action on a quasi-coherent 𝒪_{X}-module M, where X is a scheme acted on by G. We specialize now to the case where X = Spec R, and G acts trivially on X. In such situation, a G-action on M is the same as a map of fpqc-sheaves, from the sheaf represented by G to the sheaf of automorphisms of M; i.e. the datum, for every R-scheme T, of a functorial group homomorphism:

(5.8.2)  G(T) → Aut_{𝒪_{T}} (M ⊗_{R} 𝒪_{T}).

Explicitly, given a map as in (5.8.2), take T := G; then the identity map G → G determines an 𝒪(G)-linear automorphism β of M ⊗_{R} 𝒪(G), and one verifies easily that β fulfills the conditions of (3.3.7). Conversely, an automorphism β as in (3.3.7) extends uniquely to a well defined map of sheaves (5.8.2). Furthermore, the G-action on M can be prescribed by choosing, instead of a β as above, an R-linear map:

γ : M → M ⊗_{R} 𝒪(G)

satisfying certain identities analogous to (3.3.9)and (3.3.10); of course one can then recover the corresponding β by extension of scalars. One says that the pair (M, γ) is an 𝒪(G)-comodule; more precisely, one defines right and left 𝒪(G)-comodules, and the bijection just sketched sets up an equivalence from the category of R-modules with left G-actions to the category of right 𝒪(G)-comodules. One says that an 𝒪(G)-comodule is finitely generated (resp. projective) if its underlying R-module has the same property. Lemma 5.8.3. Suppose that R is a Dedekind domain and that 𝒪(G) is flat over R. Then:

(i) Every 𝒪(G)-comodule is the filtered union of its finitely generated 𝒪(G)-comodules. (ii) Every finitely generated 𝒪(G)-comodule is quotient of a projective 𝒪(G)-comodule.

Proof. (i) is [63, §1.5, Cor.] and (ii) is [63, §2.2, Prop.3]. □

5.8.4. Let R be a ring, G be a flat group scheme of finite presentation over R, and suppose that G admits a closed imbedding as a subgroup scheme G ⊂ GL_{n}. We suppose moreover that the quotient X_{G} := GL_{n}/G (for the right action of G on GL_{n}) is representable by a quasi-projective R-scheme, and that X_{G} admits an ample GL_{n}-equivariant line bundle. Then X_{G} is necessarily of finite presentation by [30, lemme 17.7.5]. Furthermore, X_{G} is smooth; indeed X_{G} is flat over Spec R, hence smoothness can be checked on the geometric fibres over the points of Spec R, which reduces to the case where R is an algebraically closed field. In such case, smoothness over R is the same as regularity and the latter follows since GL_{n} is regular and the quotient map GL_{n} → X_{G} is faithfully flat. The following lemma 5.8.5 provides plenty of examples of the situation envisaged in this paragraph. Lemma 5.8.5. Suppose that R is a Dedekind domain and G is a flat affine group scheme of finite type over R. Then G fulfills the conditions of (5.8.4).

Proof. The proof is obtained by assembling several references to the existing literature. To start with, let P be a finitely generated subrepresentation of the left (or right) regular representation of G on 𝒪(G) which generates 𝒪(G) as an R-algebra. By [19, 1.4.5] P is a finitely generated projective R-module and the action of G on P is faithful, in the sense that it gives a closed immersion: G → Aut_{R}(P). Taking the direct sum with a trivial representation one gets a closed immersion of R-group schemes: G → GL_{n,R}. By a theorem of M.Artin (see [1, 3.1.1]) the quotient fppf sheaf GL_{n}/G is represented by an algebraic space of finite presentation over Spec R. Then by [1, Th.4.C] this algebraic space is a scheme. Let K be the fraction field of R; by classical results of Chevalley and Chow, GL_{n,K}/G_{K} is quasi-projective over K and by [59, VIII.2] it follows that GL_{n}/G is quasi-projective over Spec R. The ample line bundle ℒ on GL_{n}/G that one gets always has a GL_{n}-linearization. This follows from the proof of [66, lemma 1.2]. For our purpose it suffices to know that the weaker assertion that some tensor multiple of ℒ is GL_{n}-linearizable. We may assume that ℒ is trivial on the identity section S := Spec R → GL_{n,R}/G.

Claim 5.8.6. Ker(Pic GL_{n,R} →^{e^{∗}} PicS) = 0 (where e : S → GL_{n,R} is the identity section). Proof of the claim: GL_{n,K} is open in an affine space, hence it has trivial Picard group, hence every divisor on GL_{n,R} is linearly equivalent to a divisor whose irreducible components do not dominate S, i.e. a pull-back of a divisor on S, which gives the claim.

The assertion now follows from [59, VII, Prop.1.5]. □

Lemma 5.8.7. Under the assumptions of (5.8.4), there is a GL_{n}-equivariant isomorphism of R-schemes:

(5.8.8)  X_{G} →^{∼} Proj((Sym_{R}^{•}𝒫)/ℐ) \ V (𝒥)

where 𝒫 is a projective GL_{n}-comodule, and ℐ , 𝒥 ⊂ Sym_{R}^{•}𝒫 are two finitely generated graded ideals, that are sub-𝒪(G)-comodules of Sym_{R}^{•}𝒫.

Proof. Let ℒ be an ample GL_{n}-equivariant line bundle on X_{G}. Since X_{G} is quasi-compact, we can find n ∈ ℕ large enough, so that ℒ^{⊗n} is very ample. We can then replace ℒ by ℒ^{⊗n} and therefore assume that X_{G} admits a locally closed imbedding in ℙ_{R}(Γ(X_{G}, ℒ)). Again by compactness argument, we can find a finitely generated R-submodule W ⊂ Γ(X_{G}, ℒ) such that X_{G} already imbeds into ℙ_{R}(W). Let f_{1}, ..., f_{k} be a finite set of generators of the R-module W. By lemma 5.8.3(i) we can find a finitely generated ℤ-module W_{0} ⊂ W which is a 𝒪(GL_{n})- comodule containing the (f_{i} | i ≤ k). Up to replacing W by the R-module generated by W_{0}, we can further assume that W is an 𝒪(GL_{n})-comodule. Claim 5.8.9. W is the quotient of a projective 𝒪(G)-comodule of finite type L. Proof of the claim: By the foregoing we can assume that W is generated by a finitely generated ℤ-submodule W_{0} ⊂ W which is an 𝒪(G)-comodule. By lemma 5.8.3(ii) we can write W_{0} as a quotient of a projective 𝒪(G)-comodule L_{0}; hence W is a quotient of L := L_{0} ⊗_{ℤ} R.

It follows that X_{G} is a locally closed subscheme of ℙ(L). Let Y be the schematic closure of the image of X_{G}; we can write Y ≃ Proj((Sym_{R}^{•}L)/I), for some graded ideal I, and I is necessarily an 𝒪(G)-comodule. Furthermore, we have Y \ X_{G} = V (J), where J ⊂ Sym_{R}^{•}L is another graded ideal, also an 𝒪(G)-comodule. Arguing as in the foregoing we can write J = ⋃_{α} J_{α}, where J_{α} runs over the filtered family of the finitely generated 𝒪(G)-sub-comodules of J. By quasi-compactness one shows easily that Y \ X_{G} = V (J_{α}) for J_{α} large enough; we set 𝒥 := J_{α}. Likewise, we can find a finitely generated 𝒪(G)-subcomodule ℐ ⊂ I such that (ℙ(L) \ V (𝒥)) ∩ V (I) = (ℙ(L) \ V (𝒥)) ∩ V (ℐ). □

5.8.10. Under the assumptions of (5.8.4), let T be a GL_{n}-torsor. Then G acts on T via the imbedding G ⊂ GL_{n}, and the functor T/G is representable by a smooth R-scheme.

Lemma 5.8.11. The functor T/G : R-Scheme → Set is naturally isomorphic to the functor that assigns to every R-scheme S the set of all pairs (H, ω), where H is a G-torsor on S_{fpqc} and ω : H ×^{G} GL_{n} →^{∼} T is an isomorphism of GL_{n}-torsors.

Proof. Indeed, let σ : S → T/G be any morphism of R-schemes; set H_{σ} := S ×_{T/G} T, which is a G-torsor on S_{fpqc}. The natural morphism H_{σ} → T induces a well defined isomorphism of GL_{n}-torsors ω_{σ} : H_{σ} ×^{G} GL_{n} →^{∼} T. Conversely, given (H, ω) over S, we deduce a Gequivariant morphism H → T; after taking quotients by the left action of G, we deduce a morphism S ≃ H/G → T/G. It is easy to verify that these two rules define mutually inverse natural transformations of functors, whence the claim. □

5.8.12. The right GL_{n}-torsor T is endowed by a natural left action by the group Aut_{GL_{n}}(T). The latter is a group scheme locally isomorphic to GL_{n} in the Zariski topology; especially it is smooth over R. After taking the quotient of T by the right action of G we deduce a left action of Aut_{GL_{n}}(T) on T/G :

(5.8.13)  Aut_{GL_{n}}(T) ×_{R} T/G → T/G.

Theorem 5.8.14. Resume the assumptions of proposition 5.4.21. Let G be a group scheme over Spec R[t^{−1}] which satisfies (relative to R[t^{−1}]) the conditions of (5.8.4). Then the natural map

H^{1}(Spec R[t^{−1}]_{fpqc}, G) → H^{1}(Spec R^{∧}[t^{−1}]_{fpqc}, G)

is a bijection.

Proof. We begin with the following special case:

Claim 5.8.15. The theorem holds when G = GL_{n}.

Proof of the claim: Indeed, a GL_{n} torsor over a scheme X is the same as a locally free 𝒪_{X}- module of rank n. Hence the assertion is just a restatement of corollary 5.4.42.

For a given GL_{n}-torsor T, let T^{∧} := T ×_{R[t^{−1}]} R^{∧}[t^{−1}] and denote by

H_{T}^{1} ⊂ H^{1}(Spec R[t^{−1}]_{fpqc}, G)

the subset consisting of all classes of G-torsors H such that H ×^{G} GL_{n} ≃ T; define likewise the subset H_{T^{∧}}^{1} ⊂ H^{1}(Spec R^{∧}[t^{−1}]_{fpqc}, G). According to claim 5.8.15 it suffices to show:

Claim 5.8.16. The restriction H_{T}^{1} → H_{T^{∧}}^{1} is a bijection.

Proof of the claim: The morphism (5.8.13)defines a groupoid T/G of quasi-projective R[t^{−1}]- schemes; T/G is locally isomorphic to X_{G} in the Zariski topology, hence it is smooth (see 5.8.4) and we can therefore apply theorem 5.4.38. The assertion follows directly after one has remarked that, for every R[t^{−1}]-scheme S, the set π_{0}(T/G(S)) is in natural bijection with the set of isomorphism classes of G-torsors H on S_{fpqc} such that H ×^{G} GL_{n} ≃ T ×_{Spec R[t^{−1}]} S.

□ 5.8.17. Finally we come to the lifting problems for G^{a}-torsors. Let A be a V^{a}-algebra, set R := A_{∗} and let G a smooth affine group scheme of finite type over Spec R. We suppose that G is a closed subgroup scheme of GL_{n,R}, for some n ∈ ℕ. Clearly G^{a} is a group scheme over S := Spec A. Unless otherwise stated, G^{a} will be acting on the right on any such torsor. Lemma 5.8.18. Keep the notation of (5.8.17). The category of GL_{n}-torsors over S_{fpqc} is naturally equivalent to the category whose objects are the almost projective 𝒪_{S}-modules of constant rank n, and whose morphisms are the linear isomorphisms.

Proof. Let 𝒪_{∗} be the structure sheaf of S_{fpqc}, i.e. the sheaf of rings defined by the rule: T ↦ 𝒪_{T ∗}. To a given GL_{n}-torsor P we assign the 𝒪_{∗}-module F_{P} := P ×^{GL^{n}} 𝒪_{∗}^{n} (for the natural left action of GL_{n} on 𝒪_{∗}). Locally in S_{fpqc} the sheaf F_{P} is a free 𝒪_{∗}-module of rank n, and the assignment P ↦ F_{P} is an equivalence from GL_{n}-torsors to the category of all such locally free 𝒪_{∗}-modules F of rank n, whose inverse is the rule: F ↦ \underline{Iso}_{𝒪_{∗}} (𝒪_{∗}^{n}, F). On the other hand, theorem 4.4.24 says that any almost projective 𝒪_{S}-module M of constant rank n defines a locally free 𝒪_{∗}-module in S_{fpqc}, by the rule T ↦ (𝒪_{T} ⊗_{𝒪_{S}} M)_{∗} and by faithfully flat descent it is also clear that the resulting functor is an equivalence as well. □

Theorem 5.8.19. Assume that hom.dim_{V} 𝔪̃ ≤ 1. Let S and G be as in (5.8.17), and I ⊂ 𝒪_{S} an ideal such that the pair (𝒪_{S}, I) is henselian. Let also 𝔪_{0} ⊂ 𝔪 be a finitely generated subideal, and set S_{n} := Spec 𝒪_{S}/𝔪_{0}^{n}I for n = 0, 1. Let P and Q be two G^{a}-torsors over S, and suppose that \overline{β} : P ×_{S} S_{1} →^{∼} Q ×_{S} S_{1} is an isomorphism of G^{a} ×_{S} S_{1}-torsors. Then β ×_{S} 1_{S_{0}} lifts to an isomorphism β : P →^{∼} Q of G^{a}-torsors.

Proof. Let \underline{Iso}(P, Q) denote the functor that assigns to every S-scheme T the set of isomorphisms of G^{a} ×_{S} T -torsors P ×_{S} T →^{∼} Q ×_{S} T. It is easy to see that \underline{Iso}(P, Q) is a sheaf for the fpqc topology of S. Since, locally in S_{fpqc}, this sheaf is represented by the almost scheme G^{a}, it follows by faitfully flat descent that \underline{Iso}(P, Q) is represented by an affine S-scheme, which we denote by the same name. To the (right) G^{a}-torsor P we associate a left G^{a}-torsor P′, whose underlying S-scheme is the same as P, and whose G^{a}-action is defined by the rule: g·x := xg^{−1} for every S-scheme T, every g ∈ G^{a}(T) and every x ∈ P (T). Furthermore, we let H denote the trivial left and right G^{a}-torsor whose underlying S-scheme is G^{a}, and whose left and right G^{a}-actions are induced by the multiplication map G^{a} ×_{S} G^{a} → G^{a}. Claim 5.8.20. There is a natural isomorphism of sheaves on S_{fpqc}:

ω : \underline{Iso}(P, Q) ≃ Q ×^{G^{a}} H ×^{G^{a}} P′.

Proof of the claim: Recall that the meaning of the right-hand side is as follows. For every Sscheme T, one consider the presheaf whose T -sections of are the equivalence classes of triples (x, h, y) ∈ Q(T)×H(T)×P′(T), modulo the equivalence relation such that (x, g_{1} ·h·g_{2}, y) ∼ (x · g_{1}, h, g_{2} · y) for every such (x, h, y) and every g_{1}, g_{2} ∈ G^{a}(T). Then Q ×^{G^{a}} H ×^{G^{a}} P′ is the sheaf associated to this presheaf. The sought isomorphism is defined as follows. Let T be an S-scheme, and β a T -section of \underline{Iso}(P, Q). We pick a covering morphism U → T in S_{fpqc} such that P (U) ≠ ∅; let y ∈ P (U); we set ω(β, y) := (β(y), 1, y). If now z ∈ P (U) is any other section, we have z = yg for some g ∈ G^{a}(U), therefore: ω(β, y) = (β(y), g · g^{−1}, y) ∼ (β(y) · g, 1, g^{−1}y) = (β(yg), 1, yg) = ω(β, z), i.e. the equivalence class of ω(β, y) does not depend on the choice of y, hence we have a well defined U-section ω(β) of Q ×^{G^{a}} H ×^{G^{a}} P′. Furthermore, let p_{i} : U ×_{T} U → U, i = 1, 2 be the two projections; the sections p_{1}^{∗}ω(β, y) and p_{2}^{∗}ω(β, y) differ by an element of G^{a}(U ×_{T} U) so, by the same argument, they lie in the same equivalence class, which means that actually ω(β) comes from T, as required. We leave to the reader the verification that ω thus defined is an isomorphism.

Composing the closed imbedding G^{a} ⊂ GL_{n,S} with the standard group homomorphism GL_{n,S} ⊂ SL_{n+1,S}, we can view G^{a} as a closed subgroup scheme of SL_{n,S}, whence a closed G^{a}-equivariant imbedding of S-schemes G^{a} ⊂ M_{n,S} := Spec 𝒪_{S}[x_{ij} | i, j ≤ n]. From claim 5.8.20 we derive a closed imbedding of S-schemes:

\underline{Iso}(P, Q) ⊂ X := Q ×^{G^{a}} M_{n,S} ×^{G^{a}} P′

(where G^{a} acts on the left and right on M_{n,S} in the obvious way). X is locally isomorphic to M_{n,S} in S_{fpqc}, hence it is of the form Spec (Sym_{𝒪_{S}}^{•} M), for an almost finitely generated projective 𝒪_{S}-module M; especially, X is almost finitely presented over S, whence the same holds for \underline{Iso}(P, Q). Now, the given \overline{β} gives a section of \underline{Iso}(P, Q) over S_{1}, so the theorem is an immediate consequence of theorem 5.6.16(i). □

Theorem 5.8.21. Keep the notation and assumptions of theorem 5.8.19, and suppose furthermore that I is tight, and that G fulfills the conditions of (5.8.4). Let P_{0} be any G^{a}-torsor over S_{0}. Then there exists a G^{a}-torsor P over S with an isomorphism of G^{a}-torsors: P_{0} ≃ P ×_{S} S_{0}.

Proof. By assumption there exists a finitely generated subideal 𝔪_{0} ⊂ 𝔪 and an integer n > 0 such that I^{n} ⊂ 𝔪_{0}𝒪_{S}; then by theorem 3.3.38(ii) we can lift P_{0} to a G-torsor over S_{1} := 𝒪_{S}/𝔪_{0}I. We form the GL_{n}-torsor \overline{Q} := \overline{P} ×^{G^{a}} GL_{n}. By lemma 5.8.18, \overline{Q} is the same as an almost projective 𝒪_{S_{1}}-module of constant rank n; by theorem 5.5.6(i) we can then find an almost projective 𝒪_{S}-module Q that lifts \overline{Q}, and then a standard application of Nakayama’s lemma 5.1.6 shows that Q has constant rank equal to n (cp. the proof of theorem 5.7.23). Thus, Q is a GL_{n}-torsor with an isomorphism of GL_{n}-torsors ω : \overline{P} ×^{G^{a}} GL_{n} →^{∼} Q ×_{S} S_{1}. By (the almost version of) lemma 5.8.11, the datum of (\overline{P} , ω) is the same as the datum of a morphism \overline{σ} : S_{1} → Q/G^{a} of sheaves on S_{fpqc}. The contention is that \overline{σ} ×_{S} S_{0} lifts to a morphism σ : S → Q/G^{a}. However, we have a natural GL_{n}-equivariant isomorphism

Q/G^{a} ≃ Q ×^{GL^{n}} (GL_{n}/G^{a}) ≃ Q ×^{GL^{n}} (GL_{n}/G)^{a}.

By lemmata 5.7.17 and 5.8.7 it follows that there is a GL_{n}-equivariant isomorphism:

Q/G^{a} ≃ Proj(Sym_{𝒪_{S}}^{•} 𝒫′)/ℐ′) \ V (𝒥′)

with 𝒫′ := Q ×^{GL^{n}} 𝒫, ℐ′ := ℐ ×^{GL^{n}} 𝒫 and 𝒥′ := 𝒥 ×^{GL^{n}} 𝒫, where 𝒫, ℐ and 𝒥 are as in (5.8.8). By lemma 3.2.25(ii),(iii) and remark 3.2.26(i),(ii) we see that 𝒫′ is almost finitely generated projective and ℐ′, 𝒥′ are almost finitely generated. Finally, Q/G^{a} is smooth since X_{G} is (see 5.8.4), so the existence of the section σ follows from theorem 5.7.23. □

6. Valuation theory

This chapter is an extended detour into valuation theory. The first two sections contain nothing new, and are only meant to gather in a single place some useful material that is known to experts, but for which satisfactory references are hard to find. The main theme of sections 6.3 through 6.5 is the study of the cotangent complex of an extension of valuation rings. To give a sample of our results, suppose that k is a perfect field, and let W be a valuation ring containing k; then we show that Ω_{W/k} is a torsion-free W -module. Notice that this assertion would be an easy consequence of the existence of resolution of singularities for k-schemes; our methods enable us to prove it unconditionally, as well as several other statements and variants for logarithmic differentials. Furthermore, consider a finite separable extension K ⊂ L of henselian valued fields of rank one; it is not difficult to see that the corresponding extension of valuation rings K^{+} ⊂ L^{+} is almost finite, hence one can define the different ideal 𝒟_{L^{+}/K^{+}} as in chapter 4. In case the valuation of K is discrete, it is well known that the length of the module of relative differentials Ω_{L^{+}/K^{+}} equals the length of L^{+}/𝒟_{L^{+}/K^{+}}; theorem 6.3.20 generalizes this identity to the case of arbitrary rank one valuations; notice that in this case the usual notion of length won’t do, since the modules under considerations are only almost finitely generated.

Section 6.6 ties up with earlier work of Coates and Greenberg [21], in which the notion of deeply ramified extension of a local field was introduced, and applied to the study of p-divisible groups attached to abelian varieties defined over such p-adic fields. Essentially, section 2 of [21] rediscovers the results of Fresnel and Matignon [37], although via a different route, closer to the original treatment of Tate in [64]. In particular, an algebraic extension E of K is deeply ramified if and only if 𝒟_{E^{+}/K^{+}} = (0) according to the terminology of [37]. We adopt Coates and Greenberg’s terminology for our section 6.6, and we give some complements which were not observed in [21]; notably, proposition 6.6.2, which we regard as the ultimate generalization of the one-dimensional case of the almost purity theorem. Our proof generalizes Faltings’ method, which relied on the above mentioned relationship between differentials and the different ideal; of course, in the present setting we need to appeal to our theorem 6.3.20, rather than estimating lengths the way Faltings did. Finally, we extend the definition of deeply ramified extension to include valued fields of arbitrary rank.

6.1. Ordered groups and valuations. In this section we gather some generalities on valuations and related ordered groups, which will be used in later sections.

6.1.1. As usual, a valued field (K, | · |_{K}) consists of a field K endowed with a surjective group homomorphism | · |_{K} : K^{×} → Γ_{K} onto an ordered abelian group (Γ_{K}, ≤), such that

(6.1.2)  |x + y|_{K} ≤ max(|x|_{K}, |y|_{K})

whenever x + y ≠ 0. We denote by 1 the neutral element of Γ_{K}, and the composition law of Γ_{K} will be denoted by: (x, y) ↦ x · y. It is customary to extend the map | · |_{K} to the whole of K, by adding a new element 0 to the set Γ_{K}, and setting |0| := 0. One can then extend the ordering of Γ_{K} to Γ_{K} ∪ {0} by declaring that 0 is the smallest element of the resulting ordered set. In this way, (6.1.2) holds for every x, y ∈ K. The map | · |_{K} is called the valuation of K and Γ_{K} is its value group.

6.1.3. An extension of valued fields (K, |·|_{K}) ⊂ (E, |·|_{E}) consists of a field extension K ⊂ E, and a valuation | · |_{E} : E → Γ_{E} ∪ {0} together with an imbedding j : Γ_{K} ⊂ Γ_{E}, such that the restriction to K of | · |_{E} equals j ∘ | · |_{K}. Example 6.1.4. Let | · | : K → Γ ∪ {0} be a valuation on the field K.

(i) Given a field extension K ⊂ E, it is known that there always exist valuations on E which extend | · | (cp. [16, Ch.VI, §1, n.3, Cor.3]).

(ii) If the field extension K ⊂ E is algebraic and purely inseparable, then the extension of | · | is unique. (cp. [16, Ch.VI, §8, n.7, Cor.2]).

(iii) We can construct extensions of | · | on the polynomial ring K[X], in the following way. Let Γ′ be an ordered group with an imbedding of ordered groups Γ ⊂ Γ′. For every x_{0} ∈ K, and every ρ ∈ Γ′, we define the Gauss valuation | · |_{(x_{0},ρ)} : K[X] → Γ′ ∪ {0} centered at x_{0} and with radius ρ (cp. [16, Ch.VI, §10, n.1, Lemma 1]) by the rule:

a_{0} + a_{1}(X − x_{0}) + ... + a_{n}(X − x_{0})^{n} ↦ max{|a_{i}| · ρ^{i} | i = 0, 1, ..., n}.

(iv) The construction of (iii) can be iterated : for instance, suppose that we are given a sequence of k elements ρ := (ρ_{1}, ρ_{2}, ..., ρ_{k}) of the ordered abelian group Γ′ of (iv). Then we can define a Gauss valuation | · |_{(0,ρ)} on the fraction field of K[X_{1}, X_{2}, ..., X_{k}], with values in Γ′, by the rule: ∑_{α∈ℕ^{k}} a_{α}X^{α} ↦ max{|a_{α}| · ρ^{α} | α ∈ ℕ}.

(v) Suppose again it is given an ordered group Γ′ with an imbedding of ordered groups Γ ⊂ Γ′. Let T ⊂ Γ′/Γ be a finite torsion subgroup, say T ≃ ℤ/n_{1}ℤ ⊕ ... ⊕ ℤ/n_{k}ℤ. For every i ≤ k, pick an element γ_{i} ∈ Γ′ whose class in Γ′/Γ generates the direct summands ℤ/n_{i}ℤ of T. Let x_{i} ∈ K such that |x_{i}| = n_{i} · γ_{i}. For every i = 1, ..., k pick an element y_{i} in a fixed algebraic closure E^{a} of E, such that y_{i}^{n_{i}} = x_{i}; then the field E := K(y_{1}, ..., y_{k}) has degree over K equal to the order of T, and it admits a unique valuation | · |_{E} extending | · |. Of course, |y_{i}|_{E} = γ_{i} for every i ≤ k.

6.1.5. We want to explain a construction which is a simultaneous generalization of examples 6.1.4(iv),(v). Suppose it is given the datum 𝔊 := (G, j, N, ≤) consisting of:

(a) an abelian group G with an imbedding j : K^{×} ↪ G such that G/j(V^{×}) is torsion-free;

(b) a subgroup N of G/j(V^{×}) such that the natural map:

Γ →^{∼} K^{×}/V^{×} → Γ_{𝔊} := G/(N + j(V^{×}))

is injective;

(c) an ordering ≤ on Γ_{𝔊} such that the injective map: Γ → Γ_{𝔊} is order-preserving. Let us denote by K[G] (resp. K[K^{×}]) the group K-algebra of the abelian group G (resp. K^{×}). Any element of K[G] can be written uniquely as a formal linear combination ∑_{g∈G} a_{g} · [g], where a_{g} ∈ K for every g ∈ G, and a_{g} = 0 for all but a finite number of g ∈ G. We augment K[K^{×}] over K via the K-algebra homomorphism

(6.1.6)  K[K^{×}] → K : [a] ↦ a for every a ∈ K^{×}.

Then we let K[𝔊] := K[G] ⊗_{K[K^{×}]} K, where the K[K^{×}]-algebra structure on K is defined by the augmentation (6.1.6). It is easy to verify that K[𝔊] is the maximal quotient algebra of K[G] that identifies the classes of [g · a] and a · [g], for every g ∈ G and a ∈ K^{×}. Pick, for every class γ ∈ G/K^{×}, a representative g_{γ} ∈ G. It follows that every element of K[𝔊] can be written uniquely as a formal K-linear combination ∑_{γ∈G/K^{×}} a_{γ} · [g_{γ}]. We define a map | · |_{𝔊} : K[𝔊] → Γ_{𝔊} ∪ {0} by the rule:

(6.1.7)  ∑_{γ∈G/K^{×}} a_{γ} · [g_{γ}] ↦ max_{γ∈G/K^{×}} |a_{γ}| · |g_{γ}|

where |g_{γ}| ∈ Γ_{𝔊} denotes the class of g_{γ}. One verifies easily that | · |_{𝔊} does not depend on the choice of representatives g_{γ}. Indeed, if (h_{γ} | γ ∈ G/K^{×}) is another choice then, for every γ ∈ G/K^{×} we have g_{γ} = j(x_{γ}) · h_{γ} for some x_{γ} ∈ K^{×}; therefore [g_{γ}] = x_{γ} · [h_{γ}] and |g_{γ}| = |x_{γ}| · |h_{γ}|. Lemma 6.1.8. K[𝔊] is an integral domain, and | · |_{𝔊} extends to a valuation:

| · |_{𝔊} : K(𝔊) := Frac(K[𝔊]) → Γ_{𝔊} ∪ {0}.

Proof. Let (G_{α} | α ∈ I) be the filtered system of the subgroups G_{α} of G such that K^{×} ⊂ G_{α} and G_{α}/K^{×} is finitely generated. Each G_{α} defines a datum 𝔊_{α} := (G_{α}, j, N ∩(G_{α}/j(V^{×})), ≤), and clearly K[𝔊] = colim_{α∈I} K[𝔊_{α}]. We can therefore reduce to the case where G/K^{×} is finitely

generated. Write G/K^{×} = T ⊕F, where T is a torsion group and F is torsion-free. There exist unique subgroups T̃, F̃ ⊃ K^{×} in G with T̃/K^{×} = T and F̃/K^{×} = F. Let 𝔊_{T} := (T̃, j, {0}, ≤) and 𝔊_{F} := (F̃, j, N, ≤) be the corresponding data. The functor H ↦ K[H] preserves colimits, since it is left adjoint to the forgetful functor from K-algebras to abelian groups; it follows easily that K[𝔊] ≃ K[𝔊_{T}] ⊗_{K} K[𝔊_{F}]. By inspecting (6.1.7), one can easily show that K[𝔊_{T}] is of the type of example 6.1.4(v) and K[𝔊_{F}] is of the type of example 6.1.4(iv). Especially, K[𝔊] is a domain, and |·|_{𝔊} is induced by a Gauss valuation of a free algebra over the finite field extension K[𝔊_{T}] of K. □

The next result shows that an arbitrary valuation is always “close” to some Gauss valuation. Lemma 6.1.9. Let (E, | · |_{E}) be a valued field extension of (K, | · |). Let x ∈ E \ K, and let (a_{i} | i ∈ I) be a net of elements of K (indexed by the directed set (I, ≤)) with the following property. For every b ∈ K there exists i_{0} ∈ I such that |x − a_{i}|_{E} ≤ |x − b|_{E} for every i ≥ i_{0}. Let f(X) ∈ K[X] be a polynomial that splits in K[X] as a product of linear polynomials. Then there exists i_{0} ∈ I such that |f(x)|_{E} = |f(X)|_{(a_{i},|x−a_{i}|_{E})} for every i ≥ i_{0}.

Proof. To prove the claim, it suffices to consider the case when f(X) = X − b for some b ∈ K. However, from the definition of the sequence (a_{i} | i ∈ I) we have max(|x − a_{i}|_{E}, |a_{i} − b|) ≥ |x−b|_{E} ≥ |x−a_{i}|_{E} for every sufficiently large i ∈ I. Therefore, |x−b|_{E} = max(|x−a_{i}|_{E}, |b− a_{i}|) = |X − b|_{(a_{i},|x−a_{i}|_{E})}. □

6.1.10. To see how to apply lemma 6.1.9, let us consider the case where K is algebraically closed and E = K(X), the field of fractions of the free K-algebra in one generator, which we suppose endowed with some valuation | · |_{E} with values in Γ_{E}. We apply lemma 6.1.9 to the element x := X ∈ E. Suppose first that there exists an element a ∈ K that minimizes the function K → Γ_{E} : b ↦ |X − b|_{E}. In this case the trivial net {a} fulfills the condition of the lemma. Since every polynomial of K[X] splits over K, we see that | · |_{E} is the Gauss valuation centered at a and with radius |X − a|_{E}. Suppose, on the other hand, that the function b ↦ |X − b|_{E} does not admit a minimum. It will still be possible to choose a net of elements {a_{i} | i ∈ I} fulfilling the conditions of lemma 6.1.9 (indexed, for instance, by a subset of the partially ordered set Γ). Then | · |_{E} is determined by the identity:

|f(X)|_{E} = lim_{i∈I} |f(X)|_{(a_{i},|X−a_{i}|_{E})} for every f(X) ∈ E.

6.1.11. Given a valuation | · | on a field K, the subset K^{+} := {x ∈ | |x| ≤ 1} is a valuation ring of K, i.e., a subring of K such that, for every x ∈ K \ {0}, either x ∈ K^{+} or x^{−1} ∈ K^{+}. The subset (K^{+})^{×} of units of K^{+} consists precisely of the elements x ∈ K such that |x| = 1. Conversely, let V be a valuation ring of K with maximal ideal 𝔪; V induces a valuation | · | on K whose value group is Γ_{K} := K^{×}/V^{×} (then | · | is just the natural projection). The ordering on Γ_{K} is defined as follows. For given classes \overline{x}, \overline{y} ∈ Γ_{K}, we declare that \overline{x} < \overline{y} if and only if x/y ∈ 𝔪. Remark 6.1.12. (i) It follows easily from (6.1.11) that every finitely generated ideal of a valuation ring is principal. Indeed, if a_{1}, ..., a_{n} is a set of generators for an ideal I, pick i_{0} ≤ n such that |a_{i_{0}}| = max_{i≤n} |a_{i}|; then I = (a_{i_{0}}).

(ii) It is also easy to show that any finitely generated torsion-free K^{+}-module is free and any torsion-free K^{+}-module is flat (cp. [16, Ch.VI, §3, n.6, Lemma 1]).

(iii) Let E be a field extension of the valued field (K, | · |_{K}). Then the integral closure W of K^{+} in E is the intersection of all the valuation rings of E containing K^{+} (cp. [16, Ch.VI, §1, n.3, Cor.3]). In particular, K^{+} is integrally closed.

(iv) Furthermore, if E is an algebraic extension of K, then W is a Prüfer domain, that is, for every prime ideal 𝔭 ⊂ W, the localization W_{𝔭} is a valuation ring. Moreover, the assignment 𝔪 ↦ W_{𝔪} establishes a bijection between the set of maximal ideals of W and the set of valuation rings V of E whose associated valuation | · |_{V} extends | · |_{K} (cp. [16, Ch.VI, §8, n.6, Prop.6]).

(v) Let R and S be local rings contained in a field K, 𝔪_{R} and 𝔪_{S} their respective maximal ideals. One says that R dominates S if S ⊂ R and 𝔪_{S} = S ∩ 𝔪_{R}. It is clear that the relation of dominance establishes a partial order structure on the set of local subrings of K. Then a local subring of K is a valuation ring of K if and only if it is maximal for the dominance relation (cp. [16, Ch.VI, §1, n.2, Th.1]).

(vi) Let K^{+} be a valuation ring of K with maximal ideal 𝔪, and K^{+h} a henselization of K^{+}. One knows that K^{+h} is an ind-étale local K^{+}-algebra (cp. [60, Ch.VIII, Th.1]), hence it is integral and integrally closed (cp. [60, Ch.VII, §2, Prop.2]). Denote by K^{h} the field of fractions of K^{+h} and W the integral closure of K^{+} in K^{h}. It follows that W ⊂ K^{+h}. Let 𝔪^{h} be the maximal ideal of K^{+h}; since 𝔪^{h} ∩ K^{+} = 𝔪, we deduce that 𝔮 := 𝔪^{h} ∩ W is a maximal ideal of W; then by (iv), W_{𝔮} is a valuation ring of K^{h} dominated by K^{+h}; by (v) it follows that K^{+h} = W_{𝔮}, in particular this shows that the henselization of a valuation ring is again a valuation ring. The same argument works also for strict henselizations.

The following lemma provides a simple method to construct extensions of valuation rings, which is sometimes useful. Lemma 6.1.13. Let (K, |·|) be a valued field, κ the residue field of K^{+}, R a K^{+}-algebra which is finitely generated free as a K^{+}-module, and suppose that R ⊗_{K^{+}} κ is a field. Then R is a valuation ring, and the morphism K^{+} → R induces an isomorphism of value groups Γ_{K} →^{∼} Γ_{R}.

Proof. Let e_{1}, ..., e_{n} be a K^{+}-basis of R. Let us define a map | · |_{R} : R → Γ_{R} ∪ {0} in the following way. Given x ∈ R, write x = ∑_{i=1}^{n} x_{i} · e_{i}; then |x|_{R} := max{|x_{i}| | i = 1, ..., n}. If |x| = 1, then the image \overline{x} of x in R ⊗_{K^{+}} κ is not zero, hence it is invertible by hypothesis. By Nakayama’s lemma it follows easily that x itself is invertible in R. Hence, every element y of R can be written in the form y = u · b, where u ∈ R^{×} and b ∈ K^{+} is an element such that |b| = |y|_{R}. It follows easily that R is an integral domain. Moreover, it is also clear that, given any x ∈ Frac(R) \ {0}, either x ∈ R or x^{−1} ∈ R, so R is indeed a valuation ring and | · |_{R} is its valuation. □

Lemma 6.1.14. Every finitely presented torsion K^{+}-module M is isomorphic to a direct sum of the form

(K^{+}/a_{1}K^{+}) ⊕ ... ⊕ (K^{+}/a_{n}K^{+})

where a_{1}, ..., a_{n} ∈ K^{+}. More precisely, if F →^{ϕ} M is any surjection from a free K^{+}-module F of rank n, then there is a basis e_{1}, ..., e_{n} of F and elements a_{1}, ..., a_{n} ∈ K^{+} \ {0} such that Ker ϕ = (a_{1}K^{+}) ⊕ ... ⊕ (a_{n}K^{+}).

Proof. We proceed by induction on the rank n of F. For n = 1 the claim follows easily from remark 6.1.12(i). Suppose n > 1; first of all, S := Ker(ϕ) is finitely generated by [16, Ch.I, §2, n.8, lemme 9]. Then S is a free K^{+}-module, in light of remark 6.1.12(ii); its rank is necessarily equal to n, since S ⊗_{K^{+}} K = F ⊗_{K^{+}} K.

The image of the evaluation map S ⊗_{K^{+}} F^{∗} → K^{+} given by f ⊗ α ↦ α(f) is a finitely generated ideal I ≠ 0 of K^{+}, hence it is principal, by remark 6.1.12(i). Let ∑_{i=1} f_{i} ⊗ α_{i} be an element whose image generates I; this means that ∑_{i=1} α_{i}(f_{i}) is a generator of I, hence one of the terms in the sum, say α_{1}(f_{1}), is already a generator. The map α_{1} : S → I is surjective onto a free rank one K^{+}-module, therefore it splits, which shows that S = (f_{1}K^{+}) ⊕ (S ∩ Ker α_{1}). In particular, S′ := S ∩ Ker α_{1} is a finitely generated torsion-free, hence free K^{+}-module. Let e_{1}, ..., e_{n} be a basis of F; then f_{1} = ∑_{i=1}^{n} a_{i} · e_{i} for some a_{i} ∈ K^{+}. Consider the projection π_{i} : F → K^{+} such that π_{i}(e_{j}) = δ_{ij} for j = 1, ..., n; clearly π_{i}(f_{1}) = a_{i} ∈ I. This shows that f_{1} = α_{1}(f_{1}) · g for some g ∈ F. It follows that α_{1}(g) = 1, whence F = (gK^{+}) ⊕ Ker α_{1}. Set F′ := Ker α_{1}; we have shown that M ≃ (K^{+}/α_{1}(f_{1})) ⊕ (F′/S′). But F′ is a free K^{+}-module of rank n − 1, hence we conclude by induction. □

6.1.15. In later sections we will be concerned with almost ring theory in the special case where the basic setup (V, 𝔪) (see 2.1.1) consists of a valuation ring V. In preparation for this, we fix the following terminology, which will stand throughout the rest of this work. If V is a valuation ring, then the standard setup attached to V is the pair (V, 𝔪) where 𝔪 := V in case the value group of V is isomorphic to ℤ (i.e. V is a discrete valuation ring), and otherwise 𝔪 is the maximal ideal of V.

6.1.16. Let K → Γ_{K} ∪ {0} be a valuation on the field K, and K^{+} its valuation ring. We consider the category K^{+a}-Mod relative to the standard setup (K^{+}, 𝔪). The topological group Div(K^{+a}) of fractional ideals of K^{+a} is the subspace of ℐ_{K^{+a}}(K^{a}) which consists of all the submodules I ≠ K^{a} of the almost K^{+}-module K^{a}, such that the natural morphism I ⊗_{K^{+a}} K^{a} → K^{a} is an isomorphism. The group structure is induced by the multiplication of fractional ideals. Remark 6.1.17. One verifies easily that Div(K^{+a}) is isomorphic to the group D(K^{+}) defined in [16, Ch.VII, §1, n.1].

The structure of the ideals of K^{+} can be largely read off from the value group Γ. In order to explain this, we are led to introduce some notions for general ordered abelian groups.

6.1.18. We endow an ordered group Γ with the uniform structure defined in the following way. For every γ ∈ Γ such that γ > 1, the subset of Γ × Γ given by E(γ) := {(α, β) | γ^{−1} < α^{−1} · β < γ} is an entourage for the uniform structure, and the subsets of this kind form a fundamental system of entourages. Let Γ^{∧} be the completion of Γ for this uniform structure. Lemma 6.1.19. With the notation of (6.1.16), there exists a natural isomorphism of topological groups: Div(K^{+a}) →^{∼} Γ_{K}^{∧}.

Proof. We only indicate how to construct the morphism, and leave the details to the reader. In light of remark 6.1.12(i), for every ideal I ⊂ K^{+a} we can find a net {J_{i} | i ∈ S} of principal ideals converging to I (for some filtered ordered set (S, ≤)). Let γ_{i} ∈ Γ_{K} be the value of a generator of J_{i}. One verifies that the net {γ_{i} | i ∈ S} converges in Γ_{K}^{∧} to some element γ̂. Then we assign: I ↦ γ̂. One verifies that this rule is well-defined and that it extends uniquely to the whole of Div(K^{+a}). □

Definition 6.1.20. Let Γ be any ordered abelian group with neutral element 1.

(i) We denote by Γ^{+} ⊂ Γ the subset of all the γ ∈ Γ such that γ ≤ 1.

(ii) A subgroup Δ of Γ is said to be convex if it satisfies the following property. If x ∈ Δ^{+} and 1 > y > x, then y ∈ Δ. The set Spec Γ of all the convex subgroups of Γ will be called the spectrum of Γ. We define the convex rank of Γ as the supremum c.rk(Γ) over the lengths r of the chains 0 ⊊ Δ_{1} ⊊ ... ⊊ Δ_{r} := Γ, such that all the Δ_{i} are convex subgroups. In general c.rk(Γ) ∈ ℕ ∪ {∞}, but we will mainly encounter situations for which the convex rank is a positive integer. It is easy to see that the convex rank is always less than or equal to the usual rank, defined as rk(Γ) := dim_{ℚ}(Γ ⊗_{ℤ} ℚ). To keep the two apart, we call rational rank the latter.

Example 6.1.21. (i) If Γ is an ordered abelian group, there exists a unique ordered group structure on Γ ⊗_{ℤ} ℚ such that the natural map Γ → Γ ⊗_{ℤ} ℚ is order-preserving. Indeed, if Γ is the value group of a valuation | · | on a field K, and | · |_{K^{a}} is any extension of | · |_{K} to the algebraic closure K^{a} of K, then it is easy to see (e.g. using example 6.1.4(v)) that Γ_{K^{a}} ≃ Γ ⊗_{ℤ} ℚ.

(ii) Furthermore, let K^{s} ⊂ K^{a} be the separable closure of K; we claim that | · |_{K^{a}} maps K^{s} surjectively onto Γ_{K^{a}}. Indeed, if a ∈ K^{a} is inseparable over K^{s}, then the minimal polynomial m(X) ∈ K^{s}[X] of a is of the form X^{p^{m}} − b for some b ∈ K^{s}. For c ∈ K^{×}, let m_{c}(X) ∈ K^{s}[X] be the polynomial m(X) + cX; if a′ is a root of m_{c}(X), then a′ ∈ K^{s}; moreover, |(a − a′)^{p^{m}}|_{K^{a}} = |c · a′|_{K^{a}}, hence for |c|_{K^{a}} sufficiently small we have |a|_{K^{a}} = |a′|_{K^{a}}.

(iii) For any valued field (K, | · |), and every γ ∈ Γ_{K}, let U_{γ} := {x ∈ K | |x| ≤ γ}. One defines the valuation topology on K as the unique group topology such that the family (U_{γ} | γ ∈ Γ) is a fundamental system of open neighborhoods of 0. The argument in (ii) shows more precisely that K^{s} is dense in K^{a} for the valuation topology of (K^{a}, | · |_{K^{a}}).

(iv) If Δ ⊂ Γ is any subgroup, then c.rk(Γ) ≤ c.rk(Δ) + rk(Γ/Δ) (cp. [16, Ch.VI, §10, n.2, Prop.3]).

(v) A subgroup Δ ⊂ Γ is convex if and only if there is an ordered group structure on Γ/Δ such that the natural map Γ → Γ/Δ is order-preserving. Then the ordered group structure with this property is unique.

(vi) If c.rk(Γ) = 1, we can find an order-preserving imbedding

ρ : (Γ, ·, ≤) ↪ (ℝ, +, ≤).

Indeed, pick an element g ∈ Γ with g > 1. For every h ∈ Γ, and every positive integer n, there exists a largest integer k(n) such that g^{k(n)} < h^{n}. Then (k(n)/n | n ∈ ℕ) is a Cauchy sequence and we let ρ(h) := lim_{n→∞}k(n)/n. One verifies easily that ρ is an order-preserving group homomorphism, and since the convex rank of Γ equals one, it follows that ρ is injective.

6.1.22. There is an inclusion-reversing bijection between the set of convex subgroups of the value group Γ of a valuation | · | and the set of prime ideals of its valuation ring K^{+}. This bijection assigns to a convex subgroup Δ ⊂ Γ, the prime ideal 𝔭_{Δ} := {x ∈ K^{+} | γ > |x| for every γ ∈ Δ}. Conversely, to a prime ideal 𝔭, there corresponds the convex subgroup Δ_{𝔭} := {γ ∈ Γ | γ > |x| for all x ∈ 𝔭}. Then, the value group of the valuation ring K_{𝔭}^{+} is (naturally isomorphic to) Γ/Δ_{𝔭}. Furthermore, K^{+}/𝔭 is a valuation ring of its field of fractions, and its value group is Δ_{𝔭}.

6.1.23. The rank of a valuation is defined as the convex rank of its value group. It is clear from (6.1.22)that this is the same as the Krull dimension of the associated valuation ring.

6.1.24. For any field extension F_{1} ⊂ F_{2}, denote by tr.d(F_{2} : F_{1}) the transcendence degree of F_{2} over F_{1}. Let E be a field extension of the valued field K, and | · |_{E} : E^{×} → Γ_{E} an extension of the valuation | · |_{K} : K^{×} → Γ_{K} of K to E. Let κ (resp. κ(E)) be the residue field of the valuation ring of (K, | · |) (resp. of (E, | · |)). Then we have the inequality: rk(Γ_{E}/Γ_{K}) + tr.d(κ(E) : κ) ≤ tr.d(E : K) (cp. [16, Ch.VI, §10, n.3, Cor.1]).

6.1.25. The image of K^{+} \ {0} in Γ is the monoid Γ^{+}. The submonoids of Γ^{+} are in bijective correspondence with the multiplicative subsets of K^{+}\{0} which contain (K^{+})^{×}. The bijection is exhibited by the following ”short exact sequence” of monoids:

1 → (K^{+})^{×} → K^{+} \ {0} →^{π} Γ^{+} → 1.

Then, to a monoid M ⊂ Γ^{+} one assigns the multiplicative subset π^{−1}(M).

6.1.26. Let us say that a submonoid N of a monoid M is convex if the following holds. If γ, δ ∈ M and γ · δ ∈ N, then γ ∈ N and δ ∈ N. For every submonoid N there is a smallest convex submonoid N^{con} such that N ⊂ N^{con}. One deduces a natural bijection between convex submonoids of Γ^{+} and prime ideals of K^{+}, by assigning on one hand, to a convex monoid M, the ideal 𝔭(M) := K^{+} \ π^{−1}(M), and on the other hand, to a prime ideal 𝔭, the convex monoid M(𝔭) := π(K^{+} \ 𝔭).

6.1.27. The subsets of the form M \ N, where N is a convex submonoid of the monoid M, are the first examples of ideals in a monoid. More generally, one says that a subset I ⊂ M is an ideal of M, if I · M ⊂ I. Then we say that I is a prime ideal if I is an ideal such that, for every x, y ∈ M with x · y ∈ I, we have either x ∈ I or y ∈ I. Equivalently, an ideal I is a prime ideal if and only if M \ I is a submonoid; in this case M \ I is necessarily a convex submonoid. For a monoid M, let us denote by Spec M the set of all the prime ideals of M. Taking into account (6.1.22), we derive bijections

Spec Γ →^{∼} Spec K^{+} →^{∼} Spec Γ^{+} : Δ ↦ 𝔭_{Δ} ↦ π(𝔭_{Δ}) = Γ^{+} \ Δ^{+}.

Furthermore, the bijection Spec K^{+} →^{∼} Spec Γ^{+} extends to an inclusion-preserving bijection between the ideals of K^{+} and the ideals of Γ^{+}.

In the sequel, it will be sometimes convenient to study a monoid via the system of its finitely generated submonoids. In preparation for this, we want to delve a little further into the theory of general commutative monoids. Definition 6.1.28. Let M be a commutative monoid.

(i) We say that M is integral if we have a = b, whenever a, b, c ∈ M and a · c = b · c. (ii) We say that M is free if it isomorphic to ℕ^{(I)} for some index set I. In this case, a

minimal set of generators for M will be called a basis.

6.1.29. Let Mnd be the category of commutative monoids. The natural forgetful functor ℤ-Mod → Mnd admits a left adjoint functor M ↦ M^{gp}. Given a monoid M, the abelian group M^{gp} can be realized as the set of equivalence classes of pairs (a, b) ∈ M × M, where (a, b) ∼ (a′, b′) if there exists c ∈ M such that a · b′ · c = a′ · b · c; the addition is defined termwise, and the unit of the adjunction is the map ϕ : M → M^{gp} : a ↦ (a, 1) for every a ∈ M. It is easy to see that ϕ is injective if and only if M is integral.

6.1.30. The category Mnd admits arbitrary limits and colimits. In particular, it admits direct sums. The functor M ↦ M^{gp} commutes with limits and colimits. Theorem 6.1.31. Let Δ be an ordered abelian group, N ⊂ Δ^{+} a finitely generated submonoid. Then there exists a free finitely generated submonoid N′ ⊂ Δ^{+} such that N ⊂ N′.

Proof. Since N is a submonoid of a group, it is integral, so N ⊂ N^{gp}. The group homomorphism N^{gp} ⊂ Δ induced by the imbedding N ⊂ Δ is injective as well. The verification is straightforward, using the description of N^{gp} in (6.1.29). Then N^{gp} inherits a structure of ordered group from Δ, and we can replace Δ by N^{gp}, thereby reducing to the case where Δ is finitely generated and N spans Δ. Thus, in our situation, the convex rank r of Δ is finite; we will argue by induction on r. Suppose then that r = 1. In this case we will argue by induction on the rank n of Δ. If n = 1, then one has only to observe that ℤ^{+} is a free monoid. Suppose next that n = 2; in this case, let g_{1}, g_{2} ∈ Δ be a basis. We can suppose that g_{1} < g_{2} < 1; indeed, if g_{1} > 1, we can replace it by g_{1}^{−1}; then, since r = 1, we can find an integer k such that g′_{2} := g_{2}·g_{1}^{k} < 1 and g′_{2} > g_{1}; clearly g_{1}, g′_{2} is still a basis of Δ. We define inductively a sequence of elements g_{i} ∈ Δ^{+}, for every i > 2, in the following way. Suppose that i > 2 and that the elements g_{3} < g_{4} < ... < g_{i−1} have already been assigned; let k_{i} := sup{n ∈ ℕ | g_{i−1} · g_{i−2}^{−n} ≤ 1}; notice that, since the convex rank of Δ equals 1, we have k < ∞. We set g_{i} := g_{i−1} · g_{i−2}^{−k_{i}}.

Claim 6.1.32. We have g_{i}^{ℕ} · g_{i+1}^{ℕ} ⊂ g_{i+1}^{ℕ} · g_{i+2}^{ℕ} for every i ≥ 1, and Δ^{+} = ⋃_{i≥1}(g_{i}^{ℕ} · g_{i+1}^{ℕ}). Proof of the claim: The first assertion is obvious. We prove the second assertion. Let g ∈ Δ^{+}; for every i ≥ 1 we can write g = g_{i}^{a_{i}} · g_{i+1}^{b_{i}} for unique a_{i}, b_{i} ∈ ℤ. Notice that a_{i} and b_{i} cannot both be negative. Suppose that either a_{i+1} or b_{i+1} is not in ℕ; we show that in this case

(6.1.33)  |a_{i+1}| + |b_{i+1}| < |a_{i}| + |b_{i}|.

Indeed, we must have either a_{i} < 0 and b_{i} > 0, or a_{i} > 0 and b_{i} < 0. However, a_{i+1} = a_{i} · k_{i+1} + b_{i} and b_{i+1} = a_{i}; thus, if a_{i} < 0, then b_{i+1} < 0, and consequently a_{i+1} > 0, whence

(6.1.34)  |a_{i+1}| < |b_{i}|

and if a_{i} > 0, then a_{i+1} < 0, so again (6.1.34) holds. From (6.1.33) it now follows that eventually a_{i} and b_{i} become both positive.

Since N is finitely generated, claim 6.1.32 shows that N ⊂ g_{i}^{ℕ} · g_{i+1}^{ℕ} for i > 0 sufficiently large, so the claim follows in this case.

Next, suppose that the convex rank r = 1 and n := rk(Δ) > 2. Write Δ = H ⊕ G for two subgroups such that rk(H) = n − 1 and G = gℤ for some g ∈ Δ. Claim 6.1.35. For every δ ∈ Δ^{+} \ {1} we can find a, b ∈ H such that δ < a · g^{−1} < 1 and δ < b^{−1} · g < 1. Proof of the claim: Let ρ : Δ ↪ ℝ be an order-preserving imbedding as in example 6.1.21(vi); since rk(H) > 1, it is easy to see that ρ(H) is dense in ρ(Δ). The claim is an immediate consequence.

Let g_{1}, ..., g_{k} be a set of generators for N. For every i ≤ k we can write g_{i} = h_{i} · g^{n_{i}} for unique h_{i} ∈ H and n_{i} ∈ ℤ. Suppose that n_{i} ≥ 0; it follows easily from claim 6.1.35 that there exists a, b ∈ H such that a < g < b and g_{i} < (b^{−1} · g)^{n_{i}} < 1 (resp. g_{i} < (a^{−1} · g)^{n_{i}} < 1) for every i such that n_{i} ≥ 0 (resp. n_{i} < 0). Then for n_{i} ≥ 0 set h′_{i} := h_{i} · b^{n_{i}} and for n_{i} < 0 set h′_{i} := h_{i} · a^{n_{i}}. Notice that h′_{i} < 1. Set h_{0} := a · b^{−1} and let M be the submonoid of H^{+} spanned by h_{0}, h′_{1}, ..., h′_{k},; we can imbed M in a larger submonoid M′ ⊂ H^{+} such that (M′)^{gp} = H. Then, by inductive assumption, we can imbed M′ in a free submonoid L ⊂ H^{+}. Let l_{1}, ..., l_{n−1} be a basis for L.

We can write h_{0} = ∏_{i=1}^{t} l_{k_{i}} for some integers k_{1}, ..., k_{t} ∈ {1, ..., n − 1}. Notice that h_{0} ≤ b^{−1} · g and let r < t be the largest integer such that ∏_{i=1}^{r} l_{k_{i}} > g · b^{−1}; set l′ := g · b^{−1} · ∏_{i=1}^{r} l_{k_{i}}^{−1} and l′′ := g^{−1} · b · ∏_{i=1}^{r+1} l_{k_{i}}. Claim 6.1.36. The submonoid L′ generated by {l_{1}, ..., l_{n−1}, l′, l′′} \ {l_{k_{r+1}}} contains N. Proof of the claim: Indeed, since l′ · l′′ = l_{k_{r+1}}, it follows that L ⊂ L′; moreover, g · b^{−1} = l′ · ∏_{i=1}^{r} l_{k_{i}} and g^{−1} · a = l′′ · ∏_{i=r+2}^{t} l_{k_{i}} so g · b^{−1}, g^{−1} · a ∈ L′. Now the claim follows by remarking that g_{i} = h′_{i} · (g^{−1} · a)^{−n_{i}} if n_{i} < 0, and g_{i} = h′_{i} · (g · b^{−1})^{n_{i}} if n_{i} ≥ 0.

Now, it is clear that L ⊂ Δ^{+}; since moreover L spans Δ and is generated by n elements, it follows that L is a free monoid, so the proof is concluded in case c.rk(Δ) = 1.

Finally, suppose r > 1 and pick a convex subgroup 0 ≠ Δ_{0} ⊊ Δ; then the ordering on Δ induces a unique ordering on Δ/Δ_{0} such that the projection map π : Δ → Δ/Δ_{0} is orderpreserving. Let N_{0} := π(N). By induction, N_{0} can be imbedded into a finitely generated free submonoid F_{0} of (Δ/Δ_{0})^{+}. By lifting a minimal set of generators of F_{0} to elements f_{1}, ..., f_{n} ∈ Δ^{+}, we obtain a free finitely generated monoid F ⊂ Δ^{+} with π(F) = F_{0}. Now, choose a finite set S of generators for N; we can partition S = S_{1} ∪ S_{2}, where S_{1} = S ∩ Δ_{0} and S_{2} = S \ Δ_{0}. By construction, for every x ∈ S_{2} there exist integers k_{i,x} ≥ 0 (i = 1, ..., n) such that y_{x} := x · ∏_{i=1}^{n} f_{i}^{−k_{i,x}} ∈ Δ_{0}. Let g := max{y_{x} | x ∈ S_{2}}; if g < 1, let e_{i} := f_{i}, otherwise let e_{i} := f_{i} · g for every i ≤ n. Since Δ_{0} is convex, we have in any case: e_{i} < 1 for i ≤ n. Moreover, the elements z_{x} := x · ∏_{i=1}^{n} e_{i}^{−k_{i,x}} are contained in Δ_{0}^{+}. By induction, the submonoid of Δ_{0}^{+} generated by S_{1} ∪ {z_{x} | x ∈ S_{2}} is contained in a free finitely generated monoid F′ ⊂ Δ_{0}^{+}. Using the convexity of Δ_{0} one verifies easily that N′ := F · F′ is a free monoid. Clearly N ⊂ N′, so the assertion follows. □

Remark 6.1.37. Another proof of theorem 6.1.31 can be found in [32, Th.2.2]. Moreover, this theorem can also be deduced from the resolution of singularities of toric varieties ([49, Ch.I, Th.11]).

6.2. Basic ramification theory. This section is a review of some basic ramification theory in the setting of general valuation rings and their algebraic extensions.

6.2.1. Throughout this section we fix a valued field (K, | · |). Its valuation ring will be denoted K^{+} and the residue field of K^{+} will be denoted by κ. If (E, | · |_{E}) is any valued field extension of K, we will denote by E^{+} the valuation ring of E, by κ(E) its residue field and by Γ_{E} its value group. Furthermore, we let K^{a} be an algebraic closure of K, and K^{s} the separable closure of K contained in K^{a}.

6.2.2. Let E ⊂ K^{a} be a finite extension of K. Let W be the integral closure of K^{+} in E; by remark 6.1.12(iv), to every maximal ideal 𝔭 of W we can associate a valuation | · |_{𝔭} : E^{×} → Γ_{𝔭} extending | · |, and (up to isomorphisms of value groups) every extension of | · | to E is obtained in this way. Set κ(𝔭) := W/𝔭; it is known that ∑_{𝔭∈Max(W)}[Γ_{𝔭} : Γ] · [κ(𝔭) : κ] ≤ [E : K] (cp. [16, Ch.VI §8, n.3 Th.1]).

6.2.3. Suppose now that E is a Galois extension of K. Then Gal(E/K) acts transitively on Max(W). For a given 𝔭 ∈ Max(W), the decomposition subgroup D_{𝔭} ⊂ Gal(E/K) of 𝔭 is the stabilizer of 𝔭. Then κ(𝔭) is a normal extension of κ and the natural morphism D_{𝔭} → Aut(κ(𝔭)/κ) is surjective; its kernel I_{𝔭} is the inertia subgroup at 𝔭 (cp. [16, Ch.V, §2, n.2, Th.2] for the case of a finite Galois extension; the general case is obtained by passage to the limit over the family of finite Galois extensions of K contained in E).

6.2.4. If now E is a finite Galois extension of K, then it follows easily from (6.2.2) and (6.2.3)that the integers [Γ_{𝔭} : Γ] and [κ(𝔭) : κ] are independent of 𝔭, and therefore, if W admits n maximal ideals, we have : n · [Γ_{𝔭} : Γ] · [κ(𝔭) : κ] ≤ [E : K]. Lemma 6.2.5. Let K^{+sh} be a strict henselization of K^{+}; then K^{+sh} is a valuation ring and Γ_{K^{+sh}} = Γ.

Proof. It was shown in remark 6.1.12(vi) that K^{+sh} is a valuation ring. To show the second assertion, let R be more generally any integrally closed local domain; the (strict) henselization of R can be constructed as follows (cp. [60, Ch.X, §2, Th.2]). Let F := Frac(R), F^{s} a separable closure of F, 𝔭 any maximal ideal of the integral closure W of R in F^{s}, D and I respectively the decomposition and inertia subgroups of 𝔭; let W^{D} (resp. W^{I}) be the subring of elements of W fixed by D (resp. by I) and set 𝔭^{D} := W^{D} ∩ 𝔭, 𝔭^{I} := W^{I} ∩ 𝔭. Then the localization R^{h} := (W^{D})_{𝔭^{D}} (resp. R^{sh} := (W^{I})_{𝔭^{I}}) is a henselization (resp. strict henselization) of R. Now, let us make R := K^{+}, so F := K and F^{s} := K^{s}; let E ⊂ K^{s} be any finite Galois extension of K; W_{E} := W ∩ E is the integral closure of K^{+} in E; set D_{E} := D ∩ Gal(E/K), I_{E} := I ∩ Gal(E/K), E′ := E^{D_{E}}, E′′ := E^{I_{E}}. Let 𝔭′ := 𝔭 ∩ E′; it then follows from [62, Ch.VI,§12, Th.23] that [Γ_{𝔭′} : Γ] = 1. Clearly the value group Γ_{K^{h}} of K^{+h} is the filtered union of all such Γ_{𝔭′}, so we deduce Γ_{K^{h}} = Γ. Therefore, in order to prove the lemma, we can assume that K = K^{h}. In this case Gal(E/K) coincides with the decomposition subgroup of 𝔭′ and I_{E} is a normal subgroup of Gal(K/E) such that [Gal(K/E) : I_{E}] equals the cardinality n of Aut(κ(E)/κ). By the definition of I_{E} it follows that the natural map : Aut(κ(E)/κ) → Aut(κ(E′′)/κ) is an isomorphism. We derive : [E′′ : K] = n ≤ [κ(E′′) : κ]; then from (6.2.4) we obtain Γ_{E′′} = Γ and the claim follows. □

6.2.6. We suppose now that K^{+} is a henselian valuation ring, with value group Γ. Then, on any algebraic extension E ⊂ K^{a} of K, there is a unique valuation |·|_{E} extending |·|, and thus a unique inertia subgroup, which we denote simply by I. By remark 6.1.12(iv), E^{+} is the integral closure of K^{+} in E. Remark 6.2.7. (i) In the situation of (6.2.6), the inequality of (6.2.2)simplifies to :

[κ(E) : κ] · [Γ_{E} : Γ_{K}] ≤ [E : K].

(ii) Sometimes this inequality is actually an equality; this is for instance the case when the valuation of K is discrete and the extension K ⊂ E is finite and separable (cp. [16, Ch.VI, §8, n.5, Cor.1]).

(iii) However, even when the valuation of K is discrete, it may happen that the inequality (i) is strict, if E is inseparable over K. As an example, let κ be a perfect field of positive characteristic, and choose a power series f(T) ∈ κ[[T]] which is transcendental over the subfield κ[T]. Endow F := Frac(κ(T^{1/p}, f(T))) with the T -adic valuation, and let K be the henselization of F. Then the residue field of K is κ and the valuation of K is discrete. Let E := K[f(T)^{1/p}]. Then [E : K] = p, Γ_{E} = Γ_{K} and κ(E) = κ.

6.2.8. For a field F, we denote by μ(F) the torsion subgroup of F^{×}. Let E be a finite Galois extension of K (with K^{+} still henselian). One defines a pairing

(6.2.9)  I × (Γ_{E}/Γ_{K}) → μ(κ(E))

in the following way. For (σ, γ) ∈ I × Γ_{E}, let x ∈ E^{×} such that |x| = γ; then let (σ, γ) ↦ σ(x)/x ( mod 𝔪_{E}). One verifies easily that this definition is independent of the choice of x; moreover, if x ∈ K^{×}, then σ acts trivially on x, so the definition is seen to depend only on the class of γ in Γ_{E}/Γ_{K}.

6.2.10. Suppose furthermore that κ is separably closed. Then the inertia subgroup coincides with the Galois group Gal(E/K) and moreover μ(κ(E)) = μ(κ). The pairing (6.2.9) induces a group homomorphism

(6.2.11)  Gal(E/K) → Hom_{ℤ}(Γ_{E}/Γ_{K}, μ(κ)).

Let p := char(κ). For a group G, let us denote by G^{(p)} the maximal abelian quotient of G that does not contain p-torsion. Proposition 6.2.12. Under the assumptions of (6.2.10), the map (6.2.11) is surjective and its kernel is a p-group.

Proof. Let n := [E : K]. Notice that μ(κ) does not contain p-torsion, hence every homomorphism Γ_{E}/Γ_{K} → μ(κ) factors through (Γ_{E}/Γ_{K})^{(p)}. Let m be the order of (Γ_{E}/Γ_{K})^{(p)}. Let us recall the definition of the Kummer pairing: one takes the Galois cohomology of the exact sequence of Gal(K^{s}/K)-modules

1 → μ_{m} → (K^{s})^{×} ⟶^{(−)^{m}} (K^{s})^{×} → 1

and applies Hilbert 90, to derive an isomorphism K^{×}/(K^{×})^{m} ≃ H^{1}(Gal(K^{s}/K), μ_{m}). Now, since (m, p) = 1 and κ is separably closed, the equation X^{m} = 1 admits m distinct solutions in κ. Since K^{+} is henselian, these solutions lift to roots of 1 in K, i.e., μ_{m} ⊂ K^{×}, whence H^{1}(Gal(K^{s}/K), μ_{m}) ≃ Hom_{cont}(Gal(K^{s}/K), μ_{m}). By working out the identifications, one checks easily that the resulting group isomorphism

K^{×}/(K^{×})^{m} ≃ Hom_{cont}(Gal(K^{s}/K), μ_{m})

can be described as follows. To a given a ∈ K^{×}, we assign the group homomorphism

Gal(K^{s}/K) → μ_{m} : σ ↦ σ(a^{1/m})/a^{1/m}.

Notice as well that, since κ is separably closed, more generally every equation of the form X^{m} = u admits m distinct solutions in κ, provided u ≠ 0; again by the henselian property we deduce that every unit of K^{+} is an m-th power in K^{×}; therefore K^{×}/(K^{×})^{m} ≃ Γ_{K}/mΓ_{K}.

Dualizing, we obtain an isomorphism

Hom_{ℤ}(Γ/mΓ, μ_{m}) ≃ Hom_{ℤ}(Hom_{cont}(Gal(K^{s}/K), μ_{m}), μ_{m}).

However,

Hom_{cont}(Gal(K^{s}/K), μ_{m}) = colim_{H⊂Gal(K^{s}/K)}Hom_{ℤ}(Gal(K^{s}/K)/H, μ_{m})

where H runs over the cofiltered system of open normal subgroups of Gal(K^{s}/K) such that Gal(K^{s}/K)/H is abelian with exponent dividing m. It follows that

Hom_{ℤ}(Hom_{cont}(Gal(K^{s}/K)/H, μ_{m}), μ_{m}) ≃ lim_{H⊂Gal(K^{s}/K)}Gal(K^{s}/K)/H

where the right-hand side is a quotient of Gal(K^{s}/K)^{(p)}. Hence, we have obtained a surjective group homomorphism

(6.2.13)  Gal(K^{s}/K) → Hom_{ℤ}(m^{−1}Γ_{K}/Γ_{K}, μ_{m}) →^{∼} Hom_{ℤ}(m^{−1}Γ_{K}/Γ_{K}, μ(κ)).

(Since Γ_{K} is torsion-free, we can identify naturally Γ_{K}/mΓ_{K} to the subgroup m^{−1}Γ_{K}/Γ_{K} ⊂ (Γ_{K} ⊗_{ℤ} ℚ)/Γ_{K}). Let j : Γ_{E}/Γ_{K} ↪ m^{−1}Γ_{K}/Γ_{K} be the inclusion map. One verifies directly from the definitions, that the maps (6.2.11)and (6.2.13)fit into a commutative diagram

\xymatrix{Gal(K^{s}/K) \ar[r] \ar[d] & Gal(E/K) \ar[d] \\ Hom_{ℤ}(m^{−1}Γ_{K}/Γ_{K}, μ(κ)) \ar[r]^{ρ} & Hom_{ℤ}(Γ_{E}/Γ_{K}, μ(κ))}

where the top map is the natural surjection, and ρ := Hom_{ℤ}(j, μ(κ)). Finally, an easy application of Zorn’s lemma shows that ρ is surjective, and therefore, so is (6.2.11).

It remains to show that the kernel H of (6.2.11) is a p-group. Suppose that σ ∈ H and nevertheless p does not divide the order l of σ; then we claim that the K-linear map ϕ : E → E given by x ↦ ∑_{i=0}^{l−1} σ^{i}(x) is an isometry. Indeed, ϕ(x) = l · x + ∑_{i=1}^{l−1}(σ^{i}(x) − x); it suffices then to remark that |l · x| = |x| and |σ^{i}(x) − x| < |x|, since σ^{i} ∈ H for i = 0, ..., l − 1. Next, for every x ∈ E we can write 0 = σ^{l}(x) − x = ϕ(x − σ(x)); hence σ(x) = x, that is, σ is the neutral element of Gal(E/K), as asserted. □

Corollary 6.2.14. Keep the assumptions of (6.2.10), and suppose moreover that (p, [E : K]) = 1. Then Γ_{E}/Γ_{K} ≃ Hom_{ℤ}(Gal(E/K), μ(K)). Moreover, if Γ_{E}/Γ_{K} ≃ ℤ/q_{1}ℤ ⊕ ... ⊕ ℤ/q_{k}ℤ, then there exist a_{1}, ..., a_{k} ∈ K with E = K[a_{1}^{1/q_{1}}, ..., a_{k}^{1/q_{k}}].

Proof. To start out, since (p, [E : K]) = 1, proposition 6.2.12 tells us that the map (6.2.11) is an isomorphism. In particular, Gal(E/K) is abelian, and [Γ_{E} : Γ_{K}] ≥ [E : K], whence [Γ_{E} : Γ_{K}] = [E : K] by remark 6.2.7(ii). Therefore Gal(E/K) ≃ ℤ/q_{1}ℤ ⊕ ... ⊕ ℤ/q_{k}ℤ and E is a compositum of cyclic extensions E_{1}, ..., E_{k} of order q_{1}, ..., q_{k}. It follows as well that Γ_{E}/Γ_{K} ≃ Hom_{ℤ}(Gal(E/K), μ(κ)), so the first assertion holds; furthermore the latter holds also for every extension of K contained in E. We deduce : Claim 6.2.15. The Galois correspondence establishes an inclusion preserving bijection between the subgroups of Γ_{E} containing Γ_{K}, and the subfields of E containing K.

To prove the second assertion, we are thus reduced to the case where E is a cyclic extension of prime power order, say Gal(E/K) ≃ ℤ/qℤ, with (q, p) = 1. Let γ ∈ Γ_{E} be an element whose class in Γ_{E}/Γ_{K} is a generator; we can find a ∈ K such that |a| = γ^{q}. Let E′ := K[a^{1/m}] and F := E · E′. Since Γ_{F} is torsion-free, one sees easily that its subgroups Γ_{E} and Γ_{E′} coincide. However, F satisfies again the assumptions of the corollary, therefore claim 6.2.15 applies to F, and yields E = E′. □

Definition 6.2.16. Let (K, |·|) be a valued field. We denote by K^{+sh} be the strict henselisation of K^{+} and set K^{sh} := Frac(K^{+sh}). The maximal tame extension K^{t} of K in its separable closure K^{s} is the union of all the finite Galois extensions E of K^{sh} inside K^{s}, such that ([E : K^{sh}], p) = 1. Notice that, by corollary 6.2.14, every such extension is abelian and the compositum of two such extensions is again of the same type, so the family of all such finite extension is filtered, and therefore their union is their colimit, so the definition makes sense.

6.2.17. Since Γ_{K^{sh}} = Γ_{K}, one verifies easily from the foregoing that there is a natural isomorphism of topological groups Gal(K^{t}/K^{sh}) ≃ Hom_{ℤ}(Γ_{K} ⊗_{ℤ} ℤ_{(p)}/Γ_{K}, μ^{(p)}), where μ denotes the group of roots of 1 in K^{sh} and where we endow the target with the profinite topology.

6.2.18. Let E ⊂ K^{s} be any separable extension of K. Then it is easy to check that E^{t} = E · K^{t}. Indeed, one knows that E^{sh} = E · K^{sh}; then let F be a finite separable extension of E such that ([F : E], p) = 1. By taking roots of elements of K we can find an extension F′ of K such that ([F′ : K], 1) = 1 and (Γ_{F} /Γ_{K})^{(p)} = (Γ_{F′}/Γ_{K})^{(p)} and then E · F′ · K^{sh} ⊃ F.

6.3. Algebraic extensions. In this section we return to almost rings: we suppose it is given a valued field (K, | · |), and then we will study exclusively the almost ring theory relative to the standard setup attached to K^{+} (see (6.1.15)). For an extension E of K, we will use the notation of (6.2.1). Furthermore, we will denote W_{E} the integral closure of K^{+} in E. Lemma 6.3.1. Let R be a ring and 0 → M_{1} → M_{2} → M_{3} → 0 a short exact sequence of finitely generated torsion R-modules, and suppose that the Tor-dimension of M_{3} is ≤ 1. Then F_{0}(M_{2}) = F_{0}(M_{1}) · F_{0}(M_{3}).

Proof. We can find epimorphisms ϕ_{i} : R^{n_{i}} → M_{i} for i ≤ 3, with n_{2} = n_{1} + n_{3}, fitting into a commutative diagram with exact rows:

\xymatrix{0 \ar[r] & R^{n_{1}} \ar[r] \ar[d]_{ϕ_{1}} & R^{n_{2}} \ar[r] \ar[d]^{ϕ_{2}} & R^{n_{3}} \ar[r] \ar[d]^{ϕ_{3}} & 0 \\ 0 \ar[r] & M_{1} \ar[r] & M_{2} \ar[r] & M_{3} \ar[r] & 0.}

Let N_{i} := Kerϕ_{i} (i ≤ 3). By snake lemma we have a short exact sequence: 0 → N_{1} → N_{2} →^{π} N_{3} → 0. Since the Tor-dimension of the M_{3} is ≤ 1, it follows that N_{3} is a flat R-module.

Claim 6.3.2. Λ_{R}^{n_{3}+1}N_{3} = 0.

Proof of the claim: Since N_{3} is flat, the antisymmetrizer operator \overline{a}_{k} : Λ_{R}^{k}N_{3} → N_{3}^{⊗k} is injective for every k ≥ 0 (cp. the proof of proposition 4.3.26). On the other hand, Λ_{R}^{n_{3}+1}R^{n_{3}+1} = 0, thus it suffices to show that the natural map j^{⊗k} : N_{3}^{⊗k} → (R^{n_{3}})^{⊗k} is injective for every k ≥ 0. This is clear for k = 0. Suppose that injectivity is known for j^{⊗k}; we have j^{⊗k+1} = (1_{R^{⊗k}} ⊗_{R} j) ∘ (j^{⊗k} ⊗_{R} 1_{N_{3}}). Since N_{3} is flat, we conclude by induction on k.

Next recall that, for every k ≥ 0 there are exact sequences

(6.3.3)  N_{1} ⊗_{R} Λ^{k}N_{2} → Λ^{k+1}N_{2} ⟶^{π^{∧k+1}} Λ^{k+1}N_{3}.

(To show that such sequences are exact, one uses the universality of Λ_{R}^{k+1}N_{3} for (k + 1)-linear alternating maps to R-modules). From (6.3.3)and claim 6.3.2, a simple argument by induction on k shows that the natural map ψ : Λ_{R}^{n_{1}}N_{1} ⊗ Λ_{R}^{n_{3}}N_{2} → Λ_{R}^{n_{2}}N_{2} is surjective. Finally, by definition, we have F_{0}(N_{i}) = Im(Λ_{R}^{n_{i}}N_{i} ⟶^{j_{i}^{∧n_{i}}} Λ_{R}^{n_{i}}R^{n_{i}} →^{∼} R). To conclude, it suffices therefore to remark that the diagram:

(6.3.4)  \xymatrix{Λ_{R}^{n_{1}}N_{1} ⊗ Λ_{R}^{n_{3}}N_{2} \ar[d]_{1_{Λ{n_{1} \atop R} N_{1}}⊗π^{∧k+1}} \ar[r]^{ψ} & Λ_{R}^{n_{2}}N_{2} \ar[d]^{j_{2}^{∧n_{2}}} \\ Λ_{R}^{n_{1}}N_{1} ⊗ Λ_{R}^{n_{3}}N_{3} \ar[r] & Λ_{R}^{n_{2}}R^{n_{2}}}

commutes. We leave to the reader the task of verifying that the commutativity of (6.3.4) boils down to a well-known identity for determinants of matrices. □

Remark 6.3.5. (i) Lemma 6.3.1 applies especially to a short exact sequence of finitely presented torsion K^{+}-modules, since by lemma 6.1.14, any such module has homological dimension ≤ 1.

(ii) By the usual density arguments (cp. the proof of proposition 2.3.24), it then follows that lemma 6.3.1 holds true verbatim, even when we replace R by K^{+} and the R-modules M_{1}, M_{2}, M_{3} by uniformly almost finitely generated torsion K^{+a}-modules. Proposition 6.3.6. Suppose that K^{+} is a valuation ring of rank one. Let E be a finite separable extension of K. Then W_{E}^{a} and Ω_{W_{E}^{a}/K^{+a}} are uniformly almost finitely generated K^{+a}-modules which admit the uniform bounds [E : K] and respectively [E : K]^{2}. Moreover, W_{E}^{a} is an almost projective K^{+a}-module.

Proof. In view of the presentation (2.5.27), the assertion for Ω_{W_{E}^{a}/K^{+a}} is an immediate consequence of the assertion for W_{E}^{a}. The trace pairing t_{E/K} : E × E → K is perfect since E is separable over K. Let e_{1}, ..., e_{n} be a basis of the K-vector space E and e_{1}^{∗}, ..., e_{n}^{∗} the dual basis under the trace morphism, so that t_{E/K}(e_{i} ⊗ e_{j}^{∗}) = δ_{ij} for every i, j ≤ n. We can assume that e_{i} ∈ W_{E} and we can find a ∈ K^{+} \ {0} such that a · e_{i}^{∗} ∈ W_{E} for every i ≤ n. Let w ∈ W_{E}; we can write w = ∑_{i=1}^{n} a_{i} · e_{i} for some a_{i} ∈ K. We have t_{E/K}(w ⊗ a · e_{j}^{∗}) ∈ K^{+} for every j ≤ n; on the other hand, t_{E/K}(w ⊗ a · e_{j}^{∗}) = a · a_{j}. Thus, if we let ϕ : K^{n} → E be the isomorphism (x_{1}, ..., x_{n}) ↦ ∑_{i=1}^{n} x_{i} · e_{i}, we see that

(6.3.7)  (K^{+})^{n} ⊂ ϕ^{−1}(W_{E}) ⊂ a^{−1} · (K^{+})^{n}.

We can write W_{E} as the colimit of the family 𝒲 of all its finitely generated K^{+}-submodules containing e_{1}, ..., e_{n}; if W_{0} ∈ 𝒲 , then W_{0} is a free K^{+}-module by remark 6.1.12(ii); then it is clear from (6.3.7) that the rank of W_{0} must be equal to n. The proof follows straightforwardly from the following: Claim 6.3.8. Let ε ∈ 𝔪; there exists W_{0} ∈ 𝒲 such that ε · W_{E} ⊂ W_{0}. Proof of the claim: Indeed, suppose that this is not the case. Then we can find an infinite sequence of finitely generated submodules ⊕_{i=1}^{n}e_{i} · K^{+} ⊂ W_{0} ⊂ W_{1} ⊂ W_{2} ⊂ ... ⊂ W_{E} such that ε · W_{i+1} ⊈ W_{i} for every i ≥ 0. From (6.3.7) and lemma 6.3.1 it follows easily that F_{0}((K^{+})^{n}/a · (K^{+})^{n}) ⊂ F_{0}(W_{k+1}/W_{0}) = ∏_{i=0}^{k} F_{0}(W_{i+1}/W_{i}) for every k ≥ 0. However, F_{0}(W_{i+1}/W_{i}) ⊂ Ann_{K^{+}}(W_{i+1}/W_{i}) ⊂ ε · K^{+} for every i ≥ 0. We deduce that |a|^{n} < |ε|^{k} for every k ≥ 0, which is absurd, since the valuation of K has rank one. □

6.3.9. Suppose that the valuation ring of K has rank one. Let K ⊂ E ⊂ F be a tower of finite separable extensions. Let 𝔭 ⊂ W_{E} be any prime ideal; then W_{E,𝔭} is a valuation ring (see remark 6.1.12(iii)), and W_{F,𝔭} is the integral closure of W_{E,𝔭} in F. It then follows from proposition 6.3.6 and remark 6.1.12(ii) that W_{F,𝔭}^{a} is an almost finitely generated projective W_{E,𝔭}^{a}-module; we deduce that W_{F}^{a} is an almost finitely generated projective W_{E}^{a}-module, therefore we can define the different ideal of W_{F}^{a} over W_{E}^{a}. To ease notation, we will denote it by 𝒟_{W_{F} /W_{E}}. If |·|_{F} is a valuation of F extending |·|, then F^{+} = W_{F,𝔭} for some prime ideal 𝔭 ⊂ W_{F}; moreover, if | · |_{E} is the restriction of | · |_{F} to E, then E^{+} = W_{E,𝔮}, where 𝔮 = 𝔭 ∩ W_{E}. For this reason, we are led to define 𝒟_{F^{+}/E^{+}} := (𝒟_{W_{F} /W_{E}})_{𝔭}. Lemma 6.3.10. Let K ⊂ E ⊂ F be a tower of finite separable extensions of K. Then:

(i) The W_{E}^{a}-module (W_{E}^{a})^{∗} is invertible.

(ii) 𝒟_{W_{E}/K^{+}} · 𝒟_{W_{F} /W_{E}} = 𝒟_{W_{F} /K^{+}}.

Proof. In view of proposition 4.1.27, (ii) follows from (i). We show (i): from proposition 6.3.6 we can find, for every ε ∈ 𝔪, a finitely generated K^{+}-submodule M ⊂ W_{E} such that ε · W_{E} ⊂ M. By remark 6.1.12(ii) it follows that M is a free K^{+}-module, so the same holds for M^{∗} := Hom_{K^{+}}(M, K^{+}). The scalar multiplication M^{∗} → M^{∗} : ϕ ↦ ε · ϕ factors through a map M^{∗} → W_{E}^{∗}, and if we let N be the W_{E}-module generated by image of the latter map, then ε · W_{E}^{∗} ⊂ N. Furthermore, for every prime ideal 𝔭 ⊂ W_{E}, the localization N_{𝔭} is a torsion-free W_{E,𝔭}-module; since W_{E,𝔭} is a valuation ring, it follows that N_{𝔭} is free of finite rank, again by remark 6.1.12(ii). Hence, N is a projective W_{E}-module. In particular, this shows that (W_{E}^{a})^{∗} is almost finitely generated projective as a W_{E}^{a}-module. To show that (W_{E}^{a})^{∗} is also invertible, it will suffice to show that the rank of N equals one. However, the rank of N can be computed as dim_{E} N ⊗_{W_{E}} E. We have N ⊗_{W_{E}} E = W_{E}^{∗} ⊗_{K^{+}} K = Hom_{K}(E, K), so the assertion follows by comparing the dimensions of the two sides. □

Proposition 6.3.11. Suppose that K^{+} has rank one. Let K ⊂ E be a finite field extension such that l := [E : K] is a prime. Let p := char(κ). Suppose that either:

(a) l ≠ p and K = K^{sh}, or

(b) l = p and K = K^{t}, or

(c) the valuation of K is discrete and henselian, and E is separable over K, or

(d) the valuation of K is discrete and henselian, Γ_{E} = Γ_{K} and κ(E) = κ. Then :

(i) In case (a), (b) or (d) holds, there exists x ∈ E \ K such that E^{+} is the filtered union of a family of finite K^{+}-subalgebras of the form E_{i}^{+} := K^{+}[a_{i}x + b_{i}], (i ∈ ℕ) where a_{i}, b_{i} ∈ K are elements with |a_{i}x + b_{i}| ≤ 1.

(ii) In case (c) holds, there exists an element x ∈ E^{+} such that E^{+} = K^{+}[x].

(iii) Furthermore, if E is a separable extension of K, then H_{j}(𝕃_{E^{+}/K^{+}}) = 0 for every

j > 0. (iv) If E is an inseparable extension of K, then H_{j}(𝕃_{E^{+}/K^{+}}) = 0 for every j > 1, and

moreover H_{1}(𝕃_{E^{+}/K^{+}}) is a torsion-free E^{+}-module.

Proof. Let us first show how assertions (iii) and (iv) follow from (i) and (ii). Indeed, since the cotangent complex commutes with colimits of algebras, by (i) and (ii) we reduce to dealing with an algebra of the form K^{+}[w] for w ∈ E^{+}. Such an algebra is a complete intersection K^{+}- algebra, quotient of the free algebra K^{+}[X] by the ideal I ⊂ K^{+}[X] generated by the minimal polynomial m(X) of w. In view of [46, Ch.III, Cor.3.2.7], one has a natural isomorphism in D(K^{+}[w]-Mod)

𝕃_{K^{+}[w]/K^{+}} ≃ (0 → I/I^{2} →^{δ} Ω_{K^{+}[X]/K^{+}} ⊗_{K^{+}} K^{+}[w] → 0).

If we identify Ω_{K^{+}[X]/K^{+}} ⊗_{K^{+}} K^{+}[w] to the rank one free K^{+}[w]-module generated by dX, then δ can be given explicitly by the rule: f(X) ↦ f′(w)dX, for every f(X) ∈ (m(X)). However, E is separable over K if and only if m′(w) ≠ 0. It follows that δ is injective if and only if E is separable over K, which proves (iii). If E is inseparable over K, then δ vanishes identically by the same token. This shows (iv).

We prove (ii). Since the valuation is discrete, we must have either e := [Γ_{E} : Γ] = l or f := [κ(E) : κ] = l (see remark 6.2.7(ii)). If e = l, then pick any uniformizer a ∈ E; every element of E can be written as a sum ∑_{i=0}^{l−1} x_{i} ·a^{i} with x_{i} ∈ K for every i < l. Then it is easy to see that such a sum is in E^{+} if and only x_{i} ∈ K^{+} for every i < l. In other words, E^{+} = K^{+}[a]. In case f = l, we can write κ(E) = κ[\overline{u}] for some unit u ∈ (E^{+})^{×}; moreover, 𝔪_{E} = 𝔪E^{+}; then K^{+}[u] + 𝔪E^{+} = E^{+}; since in this case E^{+} is a finite K^{+}-module, we deduce E^{+} = K^{+}[u] by Nakayama’s lemma.

We prove (i). Suppose that (a) holds; then by corollary 6.2.14 it follows that Γ_{E}/Γ_{K} ≃ ℤ/lℤ and E = K[a^{1/l}] for some a ∈ K. Hence:

(6.3.12)  |a^{i/l}| ∉ Γ for every i = 1, ..., l − 1.

We can suppose that the valuation of K is not discrete, otherwise we fall back on case (c); then, for every ε ∈ 𝔪, there exists b_{ε} ∈ K such that |ε| < |b_{ε}^{l} · a| < 1. Let x_{0}, ..., x_{l−1} ∈ K and set w := ∑_{i=0}^{l−1} x_{i} · a^{i/l}. Clearly every element of E can be written in this form. From (6.3.12) we derive that the values |x_{i} · a^{i/l}| such that x_{i} ≠ 0 are all distinct. Hence, |w| = max_{0≤i<l} |x_{i} · a^{i/l}|.

Suppose now that w ∈ E^{+}; it follows that |x_{i} · a^{i/l}| ≤ 1 for i = 0, ..., l − 1, and in fact |x_{i} · a^{i/l}| < 1 for i ≠ 0. Let ε ∈ 𝔪 such that |ε^{l−1}| > |x_{i} · a^{i/l}| for every i ≠ 0. A simple calculation shows that |x_{i} · b_{ε}^{−i}| < 1 for every i ≠ 0, in other words, w ∈ K^{+}[b_{ε} · a^{1/l}], which proves the claim in this case.

In order to deal with cases (b) and (d) we need some preparation. Let x ∈ E \ K be any element, and set:

ρ(x) := inf_{a∈K} |x − a| ∈ Γ_{E}^{∧}.

We consider case (b). Notice that the hypothesis K = K^{t} implies that the valuation of K is not discrete. For any y ∈ E we can write y = f(x) for some f(X) := b_{0} + b_{1}X + ... + b_{d}X^{d} ∈ K[X] with d := deg f(X) < p. The degree of the minimal Galois extension F of K containing all the roots of f(X) divides d!, hence F ⊂ K^{t} = K. In other words, we can write y = a_{k} · ∏_{i=1}^{d}(x − α_{i}) for some α_{1}, ..., α_{d} ∈ K.

We distinguish two cases: first, suppose that there exists a ∈ K with |x − a| = ρ(x). Replacing x by x − a we may achieve that |x| ≤ |x − a| for every a ∈ K. Then the constant sequence (a_{n} := 0 | n ∈ ℕ) fulfills the condition of lemma 6.1.9. Thus, if y = f(x) as above is in E^{+}, we must have |f(X)|_{(0,ρ(x))} ≤ 1; in other terms:

(6.3.13)  |b_{i}| · ρ(x)^{i} ≤ 1 for every i ≤ d.

Now, if ρ(x) ∈ Γ_{K}, we can find c ∈ K such that x_{0} := x · c still generates E and |x_{0}| = 1, whence |b_{i}/c^{i}| ≤ 1 for every i ≤ 1; however, y = b_{0} + (b_{1}/c) · x_{0} + (b_{2}/c^{2})x_{0}^{2} + ... + (b_{d}/c^{d})x^{d}, thus y ∈ K^{+}[x_{0}], so in this case, E^{+} itself is one of the E_{i}^{+}.

In case ρ(x) ∉ Γ_{K}, since anyway Γ_{K} is of rank one and not discrete, we can find a sequence of elements c_{1}, c_{2}, ... ∈ K such that, letting x_{i} := x · c_{i}, we have

|x_{j} − a| ≥ |x_{j}| for every a ∈ K, j ∈ ℕ; |x_{j}| < 1 and |x_{j}| → 1.

Claim 6.3.14. If x ∉ Γ_{K}, then |x^{l}| ∉ Γ_{K} for every 0 < l < p. Proof of the claim: Indeed, suppose that |x^{l}| ∈ Γ_{K} for some 0 < l < p; since Γ_{K} is l-divisible, we can multiply x by some a ∈ K to have |x^{l}| = 1, therefore |x| = 1, a contradiction.

From (6.3.13) and claim 6.3.14 we deduce that actually |b_{i}| · ρ(x)^{i} < 1 whenever i > 0. It follows that, for j sufficiently large, we will have 1 > |x_{j}^{i}| > |b_{i}|·ρ(x)^{i} for every i > 0. Writing y = b_{0} + (b_{1}/c_{j})x_{j} + (b_{2}/c_{j}^{2})x_{j}^{2} + ... + (b_{d}/c_{j}^{d})x_{j}^{d} we deduce y ∈ K^{+}[x_{j}], therefore the sequence of K^{+}-subalgebra K^{+}[c_{i} · x] will do in this case.

Finally we have to consider the case where the infimum ρ(x) is not attained for any x ∈ E. In this case, since the valuation is not discrete and of rank 1, we can find, for every x ∈ E, a sequence of elements a_{0}, a_{1}, a_{2}, ... ∈ K such that

(6.3.15)  γ_{j} := |x − a_{j}| → ρ(x).

In particular, for j sufficiently large we will have |x| > |x − a_{j}|, therefore |x| = |a_{j}|. This shows:

(6.3.16)  Γ_{E} = Γ_{K}.

Now, pick x ∈ E \ K and any sequence of elements a_{i} ∈ K such that (6.3.15)holds; it is clear that (a_{i} | i ∈ ℕ) fulfills the condition of lemma 6.1.9. Consequently

(6.3.17)  |y| = |f(X)|_{(a_{j},γ_{j})} for every sufficiently large j.

Let f(X) = b_{0,j} + b_{1,j}(X − a_{j}) + ... + b_{d,j}(X − a_{j})^{d}. (6.3.17)says that |b_{i,j}| · γ_{j}^{i} ≤ 1 whenever j is sufficiently large. However, from (6.3.16) we know that γ_{j} ∈ Γ_{K}. Pick c_{j} ∈ K such that |c_{j}| = γ_{j}^{−1} and set x_{j} := c_{j}(x − a_{j}). It follows that |b_{i,j}/c_{j}^{i}| ≤ 1 and y = b_{0,j} + (b_{1,j}/c_{1,j})x_{j} + ... + (b_{d,j}/c_{j}^{d})x_{j}^{d}. Hence y ∈ K^{+}[x_{j}]. It is then easy to verify that the family of all such K^{+}- subalgebras is filtered by inclusion, and thus conclude the proof of case (b).

At last, we turn to case (d). Notice that, by remark 6.2.7(ii), this case can occur only if E is inseparable over K, and then l = p. Let x ∈ E \ K; let a ∈ 𝔪 be a uniformizer; for given n ∈ ℕ, suppose that b_{n} ∈ K has been found such that |x − b_{n}| ≤ |a^{n}|. Since κ(E) = κ, we can find an element c ∈ K^{+} such that c ≡ (x − b_{n})/a^{n} (mod 𝔪). Set b_{n+1} := b_{n} + c · a_{n}; then |x − b_{n+1}| ≤ |a^{n+1}|. This shows that ρ(x) = 0, and the resulting sequence (b_{n} | n ∈ ℕ) converges to x in the 𝔪-adic topology. Let y ∈ E; we can write y = f(x) for a polynomial f(X) ∈ K[X] of degree d < p. Let F be the minimal field extension of K that contains all the roots of f(X). Notice that [F : K] divides d!, hence F is separable over K, and [E ·F : F] = p. Let f(X) = c·∏_{i=0}^{d}(X −α_{i}) be the factorization of f(X) in F [X]. By lemma 6.1.9 we deduce that, for every sufficiently large n ∈ ℕ we have: |y| = |f(X)|_{(b_{n},|x−b_{n}|)}, where | · |_{(b_{n},|x−b_{n}|)} is the Gauss valuation on F (X). One then argues as in the proof of case (b), to show that y ∈ E_{n}^{+} := K^{+}[c_{n}(x − b_{n})], with c_{n} ∈ K such that |c_{n}(x − b_{n})| = 1. Again, it is easy to verify that E_{i}^{+} ⊂ E_{i+1}^{+} for every i ∈ ℕ, so the proof is complete. □

Corollary 6.3.18. Let E be a finite field extension of K of prime degree l.

(i) If E satisfies condition (a) of proposition 6.3.11, and the valuation of K is not discrete (but still of rank one), then Ω_{E^{+}/K^{+}} = 0, 𝕃_{E^{+}/K^{+}} ≃ 0 and 𝒟_{E^{+}/K^{+}} = E^{+a}.

(ii) If E satisfies condition (c) of proposition 6.3.11, then we have : F_{0}(Ω_{E^{+}/K^{+}}) = 𝒟_{E^{+}/K^{+}} and H_{i}(𝕃_{E^{+}/K^{+}}) = 0 for i > 0.

Proof. (i): Since condition (a) holds, proposition 6.3.11 and its proof show that there exists a ∈ K such that E^{+} is the increasing union of all K^{+}-subalgebras of the form E_{b}^{+} := K^{+}[b · a^{1/l}], where b ∈ K^{+} ranges over all elements such that |b^{l} · a| < 1. Consequently, Ω_{E^{+}/K^{+}} = colim_{b} Ω_{E_{b}^{+}/K^{+}}, and 𝕃_{E^{+}/K^{+}} = colim_{b} 𝕃_{E_{b}^{+}/K^{+}}. Then, again from proposition 6.3.11 it follows

that H_{j}(𝕃_{E^{+}/K^{+}}) = 0 for every j > 0. Hence, in order to show the first two assertions, it suffices to show that the filtered system of the Ω_{E_{b}^{+}/K^{+}} is essentially zero. However, the E_{b}^{+}- module Ω_{E_{b}^{+}/K^{+}} is generated by ω_{b} := d(b · a^{1/l}), and clearly l · (b^{l} · a)^{(l−1)/l} · ω_{b} = 0. Since (l, p) = 1, it follows that (b^{l} · a)^{(l−1)/l} · ω_{b} = 0. On the other hand, for |b| < |c| we can write: ω_{b} = b·c^{−1}·ω_{c}. Therefore, the image of ω_{b} in Ω_{E_{c}^{+}/K^{+}} vanishes, whenever |b·c^{−1}| < |c^{l}·a|^{(l−1)/l}, i.e., whenever |b · a^{1/l}| < |c^{l} · a| < 1. Since the valuation of K is of rank one and not discrete, such a c can always be found. To show the last stated equality, let us recall the following general fact (for whose proof we refer to [60, Ch.VII, §1]). Claim 6.3.19. Suppose that E = K[w] for some w ∈ E, and let f(X) ∈ K[X] be its minimal polynomial; the elements 1, w, w^{2}, ..., w^{l−1} form a basis of the K-vector space E. Let e_{1}^{∗}, ..., e_{n}^{∗} be the corresponding dual basis under the trace pairing; then the bases S := {e_{1}^{∗}, ..., e_{n}^{∗}} and S′ := {w^{l−1}/f′(w), w^{l−2}/f′(w), ..., 1/f′(w)} span the same E^{+}-submodule of E.

Let us take w = b · a^{1/l} for some b ∈ V such that |b^{l} · a| < 1. It follows from claim 6.3.19 that (𝒟_{E^{+}/K^{+}})^{−1} ⊂ f′(w)^{−1} · E^{+a}, whence f′(w) ∈ 𝒟_{E^{+}/K^{+}∗}. However, f′(w) = l · w^{l−1}, and from the definition of w we see that |f′(w)| can be made arbitrarily close to 1, by choosing |b| closer and closer to |a|^{1/l}.

(ii): the claim about the cotangent complex is just a restatement of proposition 6.3.11(iii),(iv). By proposition 6.3.11(ii) we can write V_{E} = K^{+}[w] for some w ∈ E^{+}. Let f(x) ∈ K^{+}[X] be the minimal polynomial of w. Claim 6.3.19 implies that 𝒟_{E^{+}/K^{+}} = (f′(w)); a standard calculation yields Ω_{E^{+}/K^{+}} ≃ E^{+}/(f′(w)), so the assertion holds. □

Theorem 6.3.20. Let (E, | · |_{E}) be a finite separable valued field extension of (K, | · |) and suppose that K^{+} has rank one. Then F_{0}(Ω_{E^{+a}/K^{+a}}) = 𝒟_{E^{+}/K^{+}} and H_{i}(𝕃_{E^{+}/K^{+}}) = 0 for i > 0.

Proof. We begin with a few reductions: Claim 6.3.21. We can assume that E is a Galois extension of K. Proof of the claim: Indeed, let (L, | · |_{L}) be a Galois valued field extension of K extending (E, | · |_{E}). We obtain by transitivity ([46, II.2.1.2]) a distinguished triangle

(6.3.22)  σ^{−1}𝕃_{L^{+}/E^{+}} → 𝕃_{E^{+}/K^{+}} ⊗_{E^{+}} L^{+} → 𝕃_{L^{+}/K^{+}} → 𝕃_{L^{+}/E^{+}}.

Suppose that the theorem is already known for the Galois extensions K ⊂ L and E ⊂ L. Then (6.3.22)implies that H_{i}(𝕃_{E^{+}/K^{+}}) = 0 for i > 0 and moreover provides a short exact sequence

0 → Ω_{E^{/}K^{+}} ⊗_{E^{+}} L^{+} → Ω_{L^{+}/K^{+}} → Ω_{L^{+}/E^{+}} → 0.

However, on one hand, by lemma 6.3.10(ii) the different is multiplicative in towers of extensions, and the other hand, the Fitting ideal F_{0} is multiplicative for short exact sequences, by virtue of remark (6.3.5)(ii), so the claim follows.

Claim 6.3.23. We can assume that K^{+} is strictly henselian. Proof of the claim: Indeed, let K^{+sh} be the strict henselisation of K^{+} and K^{sh} := Frac(K^{+sh}). It is known that K^{+sh} is an ind-étale extension of K^{+}, therefore E^{+} ⊗_{K^{+}} K^{+sh} is a reduced normal semilocal integral and flat K^{+sh}-algebra, whence a product of reduced normal local integral and flat K^{+sh}-algebras W_{1}, ..., W_{k}. Each such W_{i} is necessarily the integral closure of K^{+sh} in E_{i} := Frac(W_{i}). It follows that 𝕃_{E^{+}/K^{+}} ⊗_{K^{+}} K^{+sh} ≃ 𝕃_{E^{+}⊗_{K^{+}}K^{+sh}/K^{+sh}} ≃ ⊕_{i=1}^{k}𝕃_{W_{i}⊗_{K^{+}}K^{+sh}}. Furthermore: 𝒟_{E^{+}/K^{+}} ⊗_{K^{+a}} (K^{+sh})^{a} ≃ ⊕_{i=1}^{k}𝒟_{E_{i}^{+}/K^{+sh}} and similarly for the modules of differentials. We remark as well that the formation of Fitting ideals commutes with arbitrary base changes. In conclusion, it is clear that the assertions of the theorem hold for the extension K ⊂ E if and only if they hold for each extension K^{sh} ⊂ E_{i}.

Claim 6.3.24. Suppose K = K^{sh}. We can assume that Gal(E/K) is a p-group. Proof of the claim: Indeed, let P be the kernel of (6.2.11). By proposition 6.2.12, P is p-group; let L be the fixed field of P. Then L ⊂ K^{t} and, by virtue of corollary 6.2.14, we see that L admits a chain of subextensions K := L_{0} ⊂ L_{1} ⊂ ... ⊂ L_{k} := L such that each L_{i} ⊂ L_{i+1} satisfies either condition (a) or (c) of proposition 6.3.11. Then, by corollary 6.3.18 it follows that the assertions of the theorem are already known for the extensions L_{i} ⊂ L_{i+1}. From here, using transitivity of the cotangent complex and multiplicativity of the different in towers of extensions, and of the Fitting ideals for short exact sequences, one shows that the assertions hold also for the extension K ⊂ L (cp. the proof of claim 6.3.21). Now, if the assertions are known to hold as well for the extension L ⊂ E, again the same argument proves them for K ⊂ E.

Claim 6.3.25. The theorem holds if the valuation of K is discrete. Proof of the claim: By claim 6.3.24, we can suppose that Gal(E/K) is a p-group. Hence, we can find a sequence of subextensions E_{0} := K ⊂ E_{1} ⊂ E_{2} ⊂ ... ⊂ E_{n} := E with [E_{i+1} : E_{i}] = p, for every i = 0, ..., n − 1. Arguing like in the proof of claim 6.3.24 we see that it suffices to prove the claim for each of the extensions E_{i} ⊂ E_{i+1}. In this case we are left to dealing with an extension K ⊂ E of degree p, which is taken care of by corollary 6.3.18(ii).

Claim 6.3.26. Suppose K = K^{sh}, that Gal(E : K) is a p-group and that the valuation of K is not discrete. Let L be a finite Galois extension of K such that ([L : K], p) = 1. Then the natural map E^{+} ⊗_{K^{+}} L^{+} → (E · L)^{+} is an isomorphism. Proof of the claim: By corollary 6.2.14 we know that L admits a tower of subextensions of the form K := L_{0} ⊂ L_{1} ⊂ ... ⊂ L_{k} := L, such that, for each i ≤ k we have L_{i+1} = L_{i}[a^{1/l}] for some a ∈ L_{i} and some prime l ≠ p. By induction on i, we can then reduce to the case where L = K[a^{1/l}] for some a ∈ K and a prime l ≠ p. Under the above assumptions, we must have E∩L = K, hence a ∉ E. Then E·L = E[a^{1/l}] and by proposition 6.3.11 and its proof, (E·L)^{+} is the filtered union of all its subalgebras of the form E^{+}[b · a^{1/l}], where b ∈ E ranges over all the elements such that |b^{m} · a| < 1. However, since the valuation of K is not discrete and has rank one, Γ_{K} is dense in Γ_{E}, and consequently the subfamily consisting of the E^{+}[b · a^{1/l}] with b ∈ K is cofinal. Finally, for b ∈ K we have E^{+}[b · a^{1/l}] ≃ E^{+} ⊗_{K^{+}} K^{+}[b · a^{1/l}]. By taking colimits, it follows that (E · L)^{+} ≃ E^{+} ⊗_{K^{+}} L^{+}.

Claim 6.3.27. We can assume that K is equal to K^{t}. Proof of the claim: By claim 6.3.23 we can and do assume that K = K^{sh}, in which case K^{t} is the filtered union of all the finite Galois extension L of K such that ([L : K], p) = 1. Then K^{t+} = ⋃_{L} L^{+} and (E ·K^{t})^{+} = ⋃_{L}(E ·L)^{+}, where L ranges over all such extensions. By claim 6.3.24 we can also assume that Gal(E/K) is a p-group, in which case, by claim 6.3.26, we have E^{+}⊗_{K^{+}}L^{+} →^{∼} (E·L)^{+} for every L as above. Taking colimit, we get E^{+}⊗_{K^{+}}K^{t+} →^{∼} (E·K^{t})^{+}. Since K^{t+} is faithfully flat over K^{+}, this shows that, in order to prove the theorem, we can replace K by K^{t}; however, by (6.2.18)we have (K^{t})^{t} = K^{t}, whence the claim.

After this preparation, we are ready to finish the proof of the theorem. We are reduced to considering a Galois extension E of K = K^{t} such that Gal(E/K) is a p-group; moreover, we can assume that the valuation of K is not discrete. Then, arguing as in the proof of claim 6.3.25, we can further reduce to dealing with an extension K ⊂ E of degree p; furthermore, the condition K = K^{t} still holds, by virtue of (6.2.18). In this situation, condition (b) of proposition 6.3.11 is fulfilled, hence H_{j}(𝕃_{E^{+}/K^{+}}) = 0 for j > 0, by proposition 6.3.11(iii). It remains to show the identity F_{0}(Ω_{E^{a+}/K^{+a}}) = 𝒟_{E^{+}/K^{+}}. By proposition 6.3.11(i), there exists x ∈ E such that E^{+} is the filtered union of a family of finite K^{+}-subalgebras E_{i}^{+} := K^{+}[a_{i}x + b_{i}] (i ∈ ℕ) of E^{+}. Let f(X) ∈ K^{+}[X] be the minimal polynomial of x. By construction of E_{i}^{+}, it is clear that they form a Cauchy net in ℐ_{K^{+a}}(E^{+a}) converging to E^{+a}. It then follows from lemma 2.5.26, that the net {Ω_{E_{i}^{+}/K^{+}} ⊗_{E_{i}^{+}} E^{+} | i ∈ ℕ} converges to Ω_{E^{+}/K^{+}} in ℳ (E^{+}). In particular, F_{0}(Ω_{E^{+}/K^{+}}) = lim_{i→∞}F_{0}(Ω_{E_{i}^{+}/K^{+}} ⊗_{E_{i}^{+}} E^{+}). The minimal polynomial of a_{i}x_{i} + b_{i} is f_{i}(X) :=

f(a_{i}^{−1}X − b_{i}), therefore: Ω_{E_{i}^{+}/K^{+}} = E_{i}^{+}/(f′_{i}(a_{i}x_{i} + b_{i})) = E_{i}^{+}/(a_{i}^{−1}f′(x)). Consequently, F_{0}(Ω_{E^{+}/K^{+}}) = lim_{i→∞}(a_{i}^{−1}f′(x)). On the other hand, claim 6.3.19 yields: 𝒟_{E_{i}^{+}/K^{+}} = (a_{i}^{−1}f′(x)) for every i ∈ ℕ. Then the claim follows from lemma 4.1.30. □

The final theorem of this section completes and extends theorem 6.3.20 to include valuations of arbitrary rank. Theorem 6.3.28. Let (K, | · |) be any valued field and (E, | · |_{E}) any algebraic valued field extension of (K, | · |). We have :

(i) H_{i}(𝕃_{E^{+}/K^{+}}) = 0 for i > 1 and H_{1}(𝕃_{E^{+}/K^{+}}) is a torsion-free E^{+}-module.
(ii) If moreover, E is a separable extension of K, then H_{i}(𝕃_{E^{+}/K^{+}}) = 0 for i > 0.

Proof. Let us show first how to deduce (ii) from (i). Indeed, suppose that E is separable over K. Then 𝕃_{E/K} ≃ 0. However, by (i), the natural map H_{1}(𝕃_{E^{+}/K^{+}}) → H_{1}(𝕃_{E^{+}/K^{+}}) ⊗_{K^{+}} K ≃ H_{1}(𝕃_{E/K}) is injective, so the assertion follows.

In order to prove (i), we reduce easily to the case of a finite algebraic extension. Let us write K as the filtered union of its subfields L_{α} that are finitely generated over the prime field. For each such L_{α}, let K_{α} := (L_{α})^{a} ∩ K and E_{α} := (L_{α})^{a} ∩ E. Then E_{α} is a finite extension of K_{α} and K is the filtered union of the K_{α}. It follows easily that we can replace the extension K ⊂ E by the extension K_{α} ⊂ E_{α}, thereby reducing to the case where the transcendence degree of K over its prime field is finite. In this situation, the rank r of K is finite (cp. (6.1.24)). We argue by induction on r. Suppose first that r = 1. We can split into a tower of extensions K ⊂ K^{s} ∩ E ⊂ E; then, by using transitivity (cp. the proof of claim 6.3.21), we reduce easily to prove the assertion for the subextensions K ⊂ K^{s} and K^{s} ∩ E ⊂ E. However, the first case is already covered by theorem 6.3.20, so we can assume that E is purely inseparable over K. In this case, we can further split E into a tower of subextensions of degree equal to p; thus we reduce to the case where [E : K] = p. We apply transitivity to the tower K ⊂ K^{t} ⊂ E · K^{t} = E^{t}: by proposition 6.3.11(iv) we know that H_{i}(𝕃_{E^{t+}/K^{t+}}) vanishes for i > 1 and is torsion-free for i = 1; by theorem 6.3.20, we have H_{i}(𝕃_{K^{t+}/K^{+}}) = 0 for i > 0, therefore H_{i}(𝕃_{E^{t+}/K^{+}}) vanishes for i > 1 and is torsion-free for i = 1. Next we apply transitivity to the tower K ⊂ E ⊂ E^{t} : by theorem 6.3.20 we have H_{i}(𝕃_{E^{t+}/E^{+}}) = 0 for i > 0, and the claim follows easily.

Next suppose that r > 1, and that the theorem is already known for ranks < r. Arguing as in the proof of claim 6.3.23, we can even reduce to the case where K^{+} is henselian, and then E^{+} is the integral closure of K^{+} in E. Let 𝔭_{r} := (0) ⊂ 𝔭_{r−1} ⊂ ... ⊂ 𝔭_{0} be the chain of prime ideals of K^{+}, and for every i ≤ r let 𝔮_{i} be the unique prime ideal of E^{+} lying over 𝔭_{i}. The valuation ring E_{𝔮_{1}}^{+} has rank r − 1, thus, by inductive assumption, the desired assertions are known for the extension K_{𝔭_{1}}^{+} ⊂ E_{𝔮_{1}}^{+}. It suffices therefore to show that H_{i}(𝕃_{E^{+}/K^{+}}) ⊂ H_{i}(𝕃_{E_{𝔮_{1}}^{+}/K_{𝔭_{1}}^{+}}) for every i ≥ 0. Pick a ∈ 𝔭_{0} \ 𝔭_{1}. Then K_{𝔭_{1}}^{+} = K^{+}[a^{−1}] and E_{𝔮_{1}}^{+} = E^{+}[a^{−1}] and 𝕃_{E_{𝔮_{1}}^{+}/K_{𝔭_{1}}^{+}} = 𝕃_{E^{+}/K^{+}} ⊗_{K^{+}} K^{+}[a^{−1}]. Hence, we are reduced to show that multiplication by a is injective on the homology of 𝕃_{E^{+}/K^{+}}. Let R := K^{+}/aK^{+} and R_{E} := E^{+} ⊗_{K^{+}} R. We have a short exact sequence 0 → K^{+} →^{a} K^{+} → R → 0, therefore, after tensoring by 𝕃_{E^{+}/K^{+}}, a distinguished triangle:

𝕃_{E^{+}/K^{+}} →^{a} 𝕃_{E^{+}/K^{+}} → 𝕃_{E^{+}/K^{+}} ⊗^{L}_{K^{+}} R → σ𝕃_{E^{+}/K^{+}}.

On the other hand, according to remark 6.1.12(ii), E^{+} is flat over K^{+}, therefore 𝕃_{E^{+}/K^{+}} ⊗^{L}_{K^{+}} R ≃ 𝕃_{R_{E}/R} (by [46, II.2.2.1]). Consequently, it suffices to show that H_{i}(𝕃_{R_{E}/R}) = 0 for i ≥ 2. However, R = (K^{+}/𝔭_{1}) ⊗_{K^{+}} R, and R_{E} = (E^{+}/𝔮_{1}) ⊗_{K^{+}} R; moreover, E^{+}/𝔮_{1} is the integral closure of the valuation ring K^{+}/𝔭_{1} in the finite field extension Frac(E^{+}/𝔮_{1}) of Frac(K^{+}/𝔭_{1}). Therefore we can replace K^{+} by K^{+}/𝔭_{1} and E^{+} by K^{+}/𝔮_{1}. This turns us back to the case where r = 1. Then the vanishing of H_{i}(𝕃_{E^{+}/K^{+}}) for i ≥ 2 yields the vanishing of H_{i}(𝕃_{R_{E}/R}) for i > 2. Moreover, since H_{1}(𝕃_{E^{+}/K^{+}}) is torsion-free, multiplication by a on H_{1}(𝕃_{E^{+}/K^{+}}) is injective, therefore H_{2}(𝕃_{R_{E}/R}) vanishes as well. □

6.4. Logarithmic differentials. In this section K^{+} is a valuation ring of arbitrary rank. We keep the notation of (6.2.1). We start by reviewing some facts on logarithmic structures, for which the general reference is [48].

6.4.1. Let Mnd_{X} (reps. ℤ-Mod_{X}) be the category of sheaves of commutative monoids (resp. of abelian groups) on a topological space X. The forgetful functor ℤ-Mod_{X} → Mnd_{X} admits a left adjoint functor \underline{M} ↦ \underline{M}^{gp}. If \underline{M} is a sheaf of monoids, \underline{M}^{gp} is the sheaf associated to the presheaf defined by : U ↦ \underline{M}(U)^{gp} for every open subset U ⊂ X.

The functor Γ : Mnd_{X} → Mnd that associates to every sheaf of monoids its global sections, admits a left adjoint Mnd → Mnd_{X} : M ↦ M_{X}. For a monoid M, M_{X} is the sheaf associated to the constant presheaf with value M.

6.4.2. Recall that a pre-log structure on a scheme X is a morphism of sheaves of commutative monoids : α : \underline{M} → 𝒪_{X}, where the monoid structure of 𝒪_{X} is induced by multiplication of local sections. We denote by pre-log_{X} the category of pre-log structures on X.

To a monoid M and a morphism of monoids ϕ : M → Γ(X, 𝒪_{X}), one can associate a pre-log structure ϕ_{X} : M_{X} → 𝒪_{X} by composing the induced morphism of constant sheaves M_{X} → Γ(X, 𝒪_{X})_{X} with the counit of the adjunction Γ(X, 𝒪_{X})_{X} → 𝒪_{X}.

6.4.3. To a morphism ϕ : Y → X of schemes, one can associate a pair of adjoint functors ϕ^{∗} : pre-log_{X} → pre-log_{Y} and ϕ_{∗} : pre-log_{Y} → pre-log_{X}. Let (\underline{M}, α : \underline{M} → 𝒪_{X}) (resp. (\underline{N}, β : \underline{N} → 𝒪_{Y})) be a pre-log structure on X (resp. on Y) and ϕ^{♭} : 𝒪_{X} → ϕ_{∗}𝒪_{Y} ϕ^{♯} : ϕ^{−1}𝒪_{X} → 𝒪_{Y} the natural morphisms (unit and counit of the adjunction (ϕ^{−1}, ϕ_{∗}) on

sheaves of ℤ-modules); then ϕ^{−1}\underline{M} ⟶^{ϕ^{−1}α} ϕ^{−1}𝒪_{X} ⟶^{ϕ^{♯}} 𝒪_{Y} defines ϕ^{∗}(\underline{M}, α : \underline{M} → 𝒪_{X}) and ϕ_{∗}(\underline{N}, β : \underline{N} → 𝒪_{Y}) is the morphism of sheaves of monoids γ : ϕ_{∗}\underline{N} ×_{ϕ_{∗}𝒪_{Y}} 𝒪_{X} → 𝒪_{X} which makes commute the cartesian diagram

\xymatrix{ϕ_{∗}\underline{N} ×_{ϕ_{∗}𝒪_{Y}} 𝒪_{X} \ar[r]^{γ} \ar[d] & 𝒪_{X} \ar[d]^{ϕ^{♭}} \\ ϕ_{∗}\underline{N} \ar[r]^{ϕ_{∗}β} & ϕ_{∗}𝒪_{Y} .}

6.4.4. A pre-log structure α is said to be a log structure if α^{−1}(𝒪_{X}^{×}) ≃ 𝒪_{X}^{×}. We denote by log_{X} the category of log structures on X. The forgetful functor log_{X} → pre-log_{X} : \underline{M} ↦ \underline{M}^{pre-log} admits a left adjoint

(6.4.5)  pre-log_{X} → log_{X} : \underline{M} ↦ \underline{M}^{log}

and the resulting the diagram:

(6.4.6)  \xymatrix{α^{−1}(𝒪_{X}^{×}) \ar[r] \ar[d] & \underline{M} \ar[d] \\ 𝒪_{X}^{×} \ar[r] & _{\underline{M}_{log}}}

is cocartesian in the category of pre-log structures. From this, one can easily verify that the unit of the adjunction : \underline{M} ↦ (\underline{M}^{pre-log})^{log} is an isomorphism for every log structure \underline{M}.

6.4.7. The category log_{X} admits arbitrary colimits; indeed, since the unit of the adjunction (6.4.5)is an isomorphism, it suffices to construct such colimits in the category of pre-log structures, and then apply the functor (−) ↦ (−)^{log} which preserves colimits, since it is a left adjoint. In particular, log_{X} admits arbitrary direct sums, and for any family (\underline{M}_{i} | i ∈ I) of pre-log structures we have (⊕_{i∈I} \underline{M}_{i})^{log} ≃ ⊕_{i∈I} \underline{M}_{i}^{log}.

6.4.8. For any morphism of schemes Y → X we remark that, if (\underline{M}, α) is a log structure on Y, then the pre-log structure ϕ_{∗}(\underline{M}, α) is actually a log structure (this can be checked on the stalks). We deduce a pair of adjoint functors (ϕ^{∗}, ϕ_{∗}) for log structures, as in (6.4.3). These are formed by composing the corresponding functors for pre-log structures with the functor (6.4.5).

6.4.9. We say that a log structure \underline{M} is regular if \underline{M} = (M_{X})^{log} for some free monoid M, and the associated morphism of monoids ϕ : M → Γ(X, 𝒪_{X}) maps M into the set of non-zero-divisors of Γ(X, 𝒪_{X}).

6.4.10. For an 𝒪_{X}-module ℱ, denote by Hom_{𝒪_{X}} (ℱ , ∗) the category of all homomorphisms of 𝒪_{X}-modules ℱ → 𝒜 (for any 𝒪_{X}-module 𝒜). A morphism from ℱ → 𝒜 to ℱ → ℬ is a morphism 𝒜 → ℬ of 𝒪_{X}-modules which induces the identity on ℱ. This category admits arbitrary colimits.

6.4.11. Given a pre-log structure α : \underline{M} → 𝒪_{X}, one defines the sheaf of logarithmic differentials Ω_{X/ℤ}(log \underline{M}) as the quotient of the 𝒪_{X}-module Ω_{X/ℤ} ⊕ (𝒪_{X} ⊗_{ℤ_{X}} \underline{M}^{gp}) by the 𝒪_{X}- submodule generated by the local sections of the form (dα(m), −α(m) ⊗ m), for every local section m of \underline{M}. (The meaning of this is, that one adds to Ω_{X/ℤ} the logarithmic differentials α(m)^{−1}dα(m)). For every local section m of \underline{M}, we denote by d log(m) the image of 1 ⊗ m in Ω_{X/ℤ}(log \underline{M}). The assignment \underline{M} ↦ (Ω_{X/ℤ} → Ω_{X/ℤ}(log \underline{M})) defines a (covariant) functor :

Ω : pre-log_{X} → Hom_{𝒪_{X}} (Ω_{X/ℤ}, ∗).

Lemma 6.4.12. Let X be a scheme.

(i) The functor Ω commutes with all colimits.

(ii) The functor Ω factors through the functor (6.4.5).

(iii) Let j : U → X be a formally étale morphism of schemes and \underline{M} a log structure on X.

Then the natural morphism: j^{∗}Ω_{X/ℤ}(log \underline{M}) → Ω_{U/ℤ}(log j^{∗}\underline{M}) is an isomorphism. (iv) If \underline{M} is a regular log structure, then Ω(\underline{M}) is a monomorphism of 𝒪_{X}-modules.

Proof. (i): It is clear that Ω commutes with filtered colimits. Thus, to show that it commutes with all colimits, it suffices to show that it commutes with finite direct sums and with coequalizers. We consider first direct sums. We have to show that, for any two pre-log structures \underline{M}_{1} and \underline{M}_{2}, the natural morphism

Ω_{X/ℤ}(log \underline{M}_{1}) ⨿_{Ω_{X/ℤ}} Ω_{X/ℤ}(log \underline{M}_{2}) → Ω_{X/ℤ}(log \underline{M}_{1} ⊕ \underline{M}_{2})

is an isomorphism. Notice that the functor (−) ↦ (−)^{gp} of (6.4.1) commutes with colimits, since it is a left adjoint. It follows that the diagram

\xymatrix{Ω_{X/ℤ} \ar[r] \ar[d] & Ω_{X/ℤ} ⊕ (𝒪_{X} ⊗_{ℤ_{X}} \underline{M}_{1}^{gp}) \ar[d] \\ Ω_{X/ℤ} ⊕ (𝒪_{X} ⊗_{ℤ_{X}} \underline{M}_{2}^{gp}) \ar[r] & Ω_{X/ℤ} ⊕ (𝒪_{X} ⊗_{ℤ_{X}} (\underline{M}_{1} ⊕ \underline{M}_{2})^{gp})}

is cocartesian. Thus, we are reduced to show that the kernel of the map

Ω_{X/ℤ} ⊕ (𝒪_{X} ⊗_{ℤ_{X}} \underline{M}^{gp}) → Ω_{X/ℤ}(log \underline{M})

is generated by the images of the kernels of the corresponding maps relative to \underline{M}_{1} and \underline{M}_{2}. However, any section of \underline{M}_{1} ⊕ \underline{M}_{2} can be written locally in the form x · y for two local sections x of \underline{M}_{1} and y of \underline{M}_{2}. Then we have :

\begin{matrix}(dα(x · y), −α(x · y) ⊗ (x · y)) = & (α(x) · dα(y) + α(y) · dα(x), −α(x) · α(y) ⊗ (x · y)) \\ = & α(x) · (dα(y), −α(y) ⊗ y) + α(y) · (dα(x), −α(x) ⊗ x) \end{matrix}

so the claim is clear. Next, suppose that ϕ, ψ : \underline{M} → \underline{N} are two morphisms of pre-log structures. Let : α : \underline{Q} → 𝒪_{X} be the coequalizer of ϕ and ψ. Clearly, \underline{Q} is the coequalizer of ϕ and ψ in the category of sheaves of monoids. The functor \underline{M} ↦ \underline{M}^{gp} preserves colimits, so we are reduced to consider the cokernel of β := ϕ^{gp} − ψ^{gp}. Moreover, clearly we have Coker(β) ⊗_{ℤ_{X}} 𝒪_{X} ≃ Coker(β ⊗_{ℤ_{X}} 𝒪_{X}); the claim follows easily.

(ii): Let us apply the functor Ω to the cocartesian diagram (6.4.6). In view of (i), the resulting diagram of 𝒪_{X}-modules is cocartesian. However, it is easy to check that Ω_{X/ℤ}(log α^{−1}(𝒪_{X}^{×})) ≃ Ω_{X/ℤ}(log 𝒪_{X}^{×}) ≃ Ω_{X/ℤ}. The assertion follows directly.

(iii): one uses [30, Ch.IV, Cor. 17.2.4]; the details will be left to the reader.

(iv): By (ii), the functor Ω descends to a functor

(6.4.13)  log_{X} → Hom_{𝒪_{X}} (Ω_{X/ℤ}, ∗).

Since the unit of the adjunction (6.4.5) is an isomorphism, it follows easily that (6.4.13) commutes with all colimits of log structures. Hence, to verify that Ω(\underline{M}) is a monomorphism when \underline{M} is regular, we are immediately reduced to the case when \underline{M} is the regular log structure associated to a morphism of monoids ϕ : ℕ → Γ(X, 𝒪_{X}). Let f := ϕ(1). It is easy to check that in this case, the diagram

\xymatrix{𝒪_{X} \ar[r]^{f} \ar[d]_{df} & 𝒪_{X} \ar[d]^{d log f} \\ Ω_{X/ℤ} \ar[r]^{β} & Ω_{X/ℤ}(log \underline{M})}

is cocartesian. By assumption, f is a non-zero-divisor, thus multiplication by f is a monomorphism of 𝒪_{X}-modules, so the assertion follows. □

6.4.14. This general formalism will be applied here to the following situation. We consider the submonoid M := K^{+} \ {0} of K^{+}. The imbedding M ⊂ K^{+} induces a log structure on Spec K^{+}, which we call the total log structure on K^{+}. More generally, we consider the natural projection π : M → Γ^{+} (see (6.1.25)); then for every submonoid N ⊂ Γ^{+}, we have a log structure \underline{N} corresponding to the imbedding π^{−1}(N) ⊂ K^{+}. To ease notation, we will denote by Ω_{K^{+}/ℤ}(log N) the corresponding K^{+}-module of logarithmic differentials. Proposition 6.4.15. In the situation of (6.4.14), let Δ ⊂ Γ be any subgroup, N a prime ideal of Δ^{+} (cp. (6.1.27)) and suppose that the convex rank of Σ := Δ/(Δ^{+} \ N)^{gp} equals one. Then we have a short exact sequence

0 → Ω_{K^{+}/ℤ}(log Δ^{+} \ N) →^{j} Ω_{K^{+}/ℤ}(log Δ^{+}) →^{ρ} Σ ⊗_{ℤ} (K^{+}/π^{−1}(N) · K^{+}) → 0.

Proof. Let us first remark that the assumptions and the notation make sense : indeed, since N is a prime ideal of Δ^{+}, it follows that M := Δ^{+} \ N is a convex submonoid of Δ^{+}, hence M = (M^{gp})^{+} and M^{gp} is a convex subgroup of Δ (cp. (6.1.26)), therefore Σ is an ordered group (cp. example (6.1.21)(v)), and hence it makes sense to say that its convex rank equals one.

Let us show that j is injective. We can write Δ^{+} as the colimit of the filtered family of its finitely generated submonoids F_{α}. For each such F_{α}, theorem 6.1.31 gives us a free finitely generated submonoid L_{α} ⊂ Δ^{+} such that F_{α} ⊂ L_{α}. Clearly Δ^{+} is the colimit of the L_{α}, and M is the colimit of the M_{α} := M ∩L_{α}. Thus \underline{Δ}^{+} is the colimit of the \underline{L}_{α} and \underline{M} is the colimit of the \underline{M}_{α}. Let S_{α} be a basis of L_{α}. Since M is convex in Δ, we see that M_{α} is free with basis S_{α} ∩M and L_{α} = M_{α} ⊕N_{α}, where N_{α} is the free submonoid spanned by S_{α} \M. For each e ∈ S_{α} \M, pick arbitrarily an element x_{e} ∈ K^{+} such that |x_{e}| = e. The map e ↦ x_{e} can be extended to a map of monoids N_{α} → K^{+}, and then to a pre-log structure ν_{α} : (N_{α})_{SpecV} → 𝒪_{SpecK^{+}}. Clearly we have an isomorphism of pre-log structures: \underline{L}_{α} = \underline{M}_{α} ⊕ ν_{α}. Since the formation of logarithmic differentials commutes with colimits of monoids, we are reduced to showing that the analogous map

j_{α} : Ω_{K^{+}/ℤ}(log M_{α}) → Ω_{K^{+}/ℤ}(log L_{α})

is injective. By lemma 6.4.12(i), we have Ker(j_{α}) ≃ Ker(Ω_{K^{+}/ℤ} → Ω_{K^{+}/ℤ}(log ν_{α})). By lemma 6.4.12(iv), the latter map is injective, whence the assertion.

Next we proceed to show how to construct ρ. Define a map

ρ̃: X := Ω_{K^{+}/ℤ} ⊕ (π^{−1}Δ ⊗_{ℤ} K^{+}) → Σ ⊗_{ℤ} (K^{+}/π^{−1}(N) · K^{+})

by the rule : (ω, a ⊗ b) ↦ \overline{π(a)} ⊗ \overline{b}, for any ω ∈ Ω_{K^{+}/ℤ}, a ∈ π^{−1}Δ, b ∈ K^{+}. Claim 6.4.16. Kerρ̃ contains the kernel of the surjection X → Ω_{K^{+}/ℤ}(log Δ^{+}). Proof of the claim: It suffices to show that a ⊗ |a| ∈ Kerρ̃ whenever π(a) ∉ (Δ^{+} \ N)^{gp}. However, π(a) ∉ (Δ^{+} \ N)^{gp} ⇔ π(a) ∉ Δ^{+} \ N ⇔ π(a) ∈ N ⇔ a ∈ π^{−1}(N), so the claim follows.

By claim 6.4.16 we deduce that ρ̃ descends to the map ρ as desired. It is now obvious that ρ is surjective and that its kernel contains the image of j. To conclude the proof, it suffices to show that the cokernel of j is annihilated by π^{−1}N. We are thus reduced to showing that π^{−1}(N) annihilates the classes in Coker(j) of the elements d log(e), for every e ∈ π^{−1}N. Let a ∈ π^{−1}(N). Since the convex rank of Σ equals one, and π(e) ∈ N, there exists k ≥ 0 and b ∈ K^{+} such that e = a^{k} · b and |b| < |a|. In particular, |b| ∈ Δ^{+}, and we can write: d log(e) = d log(a^{k} · b) = b · k · d log(a) + a^{k} · d log(b), and it is clear that a annihilates each of the terms of this expression. □

Corollary 6.4.17. In the situation of (6.4.14), we have :

(i) The natural map : β_{K^{+}} : Ω_{K^{+}/ℤ} → Ω_{K^{+}/ℤ}(log Γ^{+}) is injective.

(ii) Suppose moreover that K^{+} has finite rank. Let 𝔭_{r} := 0 ⊂ 𝔭_{r−1} ⊂ ... ⊂ 𝔭_{0} := 𝔪_{K} be the chain of all the prime ideals of K^{+}. Denote by Δ_{r} := Γ_{K} ⊃ Δ_{r−1} ⊃ ... ⊃ Δ_{0} := 0 the corresponding ascending chain of convex subgroups of Γ_{K} (see (6.1.22)). Then Coker β_{K^{+}} admits a finite filtration Fil^{•}(Coker β_{K^{+}}) indexed by the totally ordered set Spec(K^{+}), such that :

gr^{𝔭_{i}}(Coker β_{K^{+}}) ≃ (Δ_{i+1}/Δ_{i}) ⊗_{ℤ} (K^{+}/𝔭_{i}) for every 𝔭_{i} ∈ Spec K^{+}.

Proof. (i): Since the formation of differentials and logarithmic differentials commutes with colimits of ℤ-algebras and log structures, we can reduce to the case where K is a field of finite type over its prime field. In this case the convex rank of Γ is finite, so the assertion follows from proposition 6.4.15 and an easy induction.

(ii): is a straightforward consequence of proposition 6.4.15. □

6.5. Transcendental extensions. In this section we extend the results of section 6.3 to the case of arbitrary extensions of valued fields.

6.5.1. We fix the following notation throughout this section. For a valued field extension (K, | · |) ⊂ (E, | · |_{E}), we let

ρ_{E^{+}/K^{+}} : E^{+} ⊗_{K^{+}} Ω_{K^{+}/ℤ}(log Γ^{+}) → Ω_{E^{+}/ℤ}(log Γ_{E}^{+})

be the natural morphism. One of the main results of this section states that Coker(ρ_{E^{+}/K^{+}}) is injective with torsion-free cokernel when K is algebraically closed (theorem 6.5.20) or when K has characteristic zero (lemma 6.5.12). Lurking behind the results of this sections there should be some notion of ”logarithmic cotangent complex”, which however is not currently available.

6.5.2. Let 𝔊 := (G, j, N, ≤) be a datum as in (6.1.5). We wish to study the total log structure of the valued field (K(𝔊), | · |_{𝔊}). We consider the morphism of monoids

(6.5.3)  G → K[𝔊] : g ↦ [g].

Let K[𝔊]^{+} be the subring of the elements x ∈ K[𝔊] such that |x|_{𝔊} ≤ 1. Let π : G → Γ_{𝔊} be the projection; for every submonoid M ⊂ Γ_{𝔊}^{+}, the preimage π^{−1}M is a submonoid of G, and the restriction of (6.5.3) induces a morphism of monoids π^{−1}(M) → K[𝔊]^{+}, whence a pre-log structure π^{−1}M_{X} on X := Spec K[𝔊]^{+} (see (6.4.2)). To ease notation, we set \underline{M} := (π^{−1}M_{X})^{log} and we will write Ω_{X/ℤ}(log M) for the associated sheaf of log differentials. Lemma 6.5.4. Resume the notation of (6.1.5). Then the natural diagram

\xymatrix{K^{×} ⊗_{ℤ} K[𝔊]^{+} \ar[r]^{α} \ar[d]_{β} & G ⊗_{ℤ} K[𝔊]^{+} \ar[d]^{η} \\ Ω_{K^{+}/ℤ}(log Γ^{+}) ⊗_{K^{+}} K[𝔊]^{+} \ar[r] & Ω_{K[𝔊]^{+}/ℤ}(log Γ_{𝔊}^{+})}

is cocartesian.

Proof. Let P be the push out of α and β. We already have a map ϕ : P → Ω_{K[𝔊]^{+}/ℤ}(log Γ_{𝔊}^{+}), and by inspecting the definition of K[𝔊]^{+} one verifies easily that ϕ is surjective; thus we need only find a left inverse for ϕ. Let us remark also that β, and consequently η, is surjective, hence it suffices to exhibit:

(a) a derivation δ : K[𝔊]^{+} → P such that η(δa) = da for every a ∈ K[𝔊]^{+};

(b) a ℤ-linear map ψ : (π^{−1}Γ_{𝔊}^{+})^{gp} → G such that η∘ψ(γ) = d log(g) for every g ∈ π^{−1}Γ_{𝔊}^{+}. Of course we can take for ψ the natural identification (π^{−1}Γ_{𝔊}^{+})^{gp} →^{∼} G. To define δ, choose arbitrarily a set of representatives (g_{γ} | γ ∈ G/K^{×}) for the classes of G/K^{×}. Then every a ∈ K[𝔊] can be written in a unique way as a K-linear combination a = ∑_{γ∈G/K^{×}} a_{γ} · [g_{γ}]; we define δ′ : K[𝔊]^{+} → G ⊗_{ℤ} K[𝔊]^{+} by the rule: a ↦ ∑_{γ∈G/K^{×}} g_{γ} ⊗ a_{γ}. It is easy to check that the image of δ′(a) in P does not depend on the choices of representatives, and this defines our sought derivation δ. □

6.5.5. Let K(𝔊)^{+} be the valuation ring of the valuation | · |_{𝔊}. It is easy to see that K(𝔊)^{+} = K[𝔊]_{𝔭}^{+}, where 𝔭 is the ideal of elements x ∈ K[𝔊] such that |x|_{𝔊} < 1. It then follows from lemma 6.4.12(i),(iii) that the diagram of lemma 6.5.4 remains cocartesian when we replace everywhere K[𝔊]^{+} by K(𝔊)^{+}. Proposition 6.5.6. Suppose that K is algebraically closed, let (E, | · |_{E}) be a purely transcendental valued field extension of (K, | · |) with tr.deg(E : K) = 1. Then:

(i) Ω_{E^{+}/K^{+}} is a torsion-free E^{+}-module and H_{i}(𝕃_{E^{+}/K^{+}}) = 0 for every i > 0.

(ii) The natural map of E^{+}-modules: Ω_{K^{+}/ℤ}(log Γ^{+}) ⊗_{K^{+}} E^{+} → Ω_{E^{+}/ℤ}(log Γ_{E}^{+}) is injec-

tive with torsion-free cokernel. (iii) Suppose that Γ_{E} = Γ. Then the natural diagram

\xymatrix{Ω_{K^{+}/ℤ} ⊗_{K^{+}} E^{+} \ar[r] \ar[d] & Ω_{E^{+}/ℤ} \ar[d] \\ Ω_{K^{+}/ℤ}(log Γ^{+}) ⊗_{K^{+}} E^{+} \ar[r]^{ρ_{E^{+}/K^{+}}} & Ω_{E^{+}/ℤ}(log Γ_{E}^{+})}

is cocartesian.

Proof. Let 𝔪_{E} be the maximal ideal of E^{+} and X ∈ E such that E = K(X). Following (6.1.10), we distinguish two cases, according to whether there exists or there does not exist an element a ∈ K which minimizes the function K → Γ_{E} : b ↦ |X − b|_{E}. Suppose first that such an element does not exist. We pick a net (a_{i} | i ∈ I) satisfying the conditions of lemma 6.1.9, relative to the element x := X. For a given b ∈ K, choose i ∈ I such that |X − a_{i}|_{E} < |X − b|_{E}; then we have |a_{i} − b| = |X − b|_{E} and it follows easily that Γ_{E} = Γ_{K} in this case. Then, for every i ∈ I we can find b_{i} ∈ K such that |X − a_{i}|_{E} = |b_{i}|. Let f(X)/g(X) ∈ E^{+} be the quotient of two elements f(X), g(X) ∈ K[X]. By lemma 6.1.9, we have |f(X)/g(X)|_{(a_{i},|b_{i}|)} ≤ 1 and γ := |g(X)|_{E} = |g(X)|_{(a_{i},|b_{i}|)} for every sufficiently large i ∈ I. Pick a ∈ K such that |a| = γ. Arguing as in the proof of case (b) of proposition 6.3.11(i), we deduce that a^{−1} · g(X), a^{−1} · f(X) ∈ A_{i} := K^{+}[(X − a_{i})/b_{i}], and if we let 𝔭_{i} := A_{i} ∩ 𝔪_{E^{+}}, then f(X)/g(X) ∈ E_{i}^{+} := A_{i,𝔭_{i}}. It is also easy to see that the family of the K^{+}-algebras E_{i}^{+} is filtered by inclusion. Clearly Ω_{E_{i}^{+}/K^{+}} is a free E_{i}^{+}-module of rank one, and H_{i}(𝕃_{E_{i}^{+}/K^{+}}) = 0 for every i > 0, so (i) follows easily in this case. Notice that, since Γ = Γ_{E}, the log structure \underline{Γ}_{E}^{+} on Spec E^{+} (notation of (6.4.14)) is the log structure associated to the morphism of monoids (K^{+}) \ {0} → E^{+}. It follows easily that, for every i ∈ I, we have a cocartesian diagram

(6.5.7)  \xymatrix{Ω_{K^{+}/ℤ} ⊗_{K^{+}} A_{i} \ar[r]^{α_{i}} \ar[d] & Ω_{A_{i}/ℤ} \ar[d] \\ Ω_{K^{+}/ℤ}(log Γ^{+}) ⊗_{K^{+}} A_{i} \ar[r] & Ω_{A_{i}/ℤ}(log Γ^{+})}

where moreover, α_{i} is split injective; the diagram of (iii) is obtained from (6.5.7), by localizing at 𝔭 and taking colimits over the family I; since both operations preserve colimits, we get (ii) and (iii) in this case. Finally, suppose that there exists an element a ∈ K such that |X − a| is minimal; we can replace X by X−a, and thus assume that a = 0. By (6.1.10)it follows that |·|_{E} is a Gauss valuation; then this case can be realized as the valuation | · |_{𝔊} associated to the datum 𝔊 := (K^{×}⊕ℤ, j, N, ≤), where j is the obvious imbedding, and N is either ℤ or {0}, depending on whether |X|_{E} ∈ Γ or otherwise. In either case, (6.5.5) tells us that the map of (ii) is split injective, with cokernel isomorphic to E^{+}, so (ii) holds. Suppose first that |X|_{E} ∈ Γ. Then we can find b ∈ K such that |X/b|_{E} = 1, and one verifies easily that E^{+} is the localization of A := K^{+}[X/b] at the prime ideal 𝔪_{K} ·E^{+}. Clearly (6.5.7)remains cocartesian when we replace A_{i} by A and α_{i} by the corresponding map α; the latter is still split injective, so (iii) follows easily. (i) is likewise obvious in this case. In case |X|_{E} ∉ Γ, we distinguish three cases. First, suppose that |X|_{E} < |b| for every b ∈ K^{×}. Then K^{+}[X/b] ⊂ E^{+} for every a ∈ K^{×}, and indeed it is easy to check that E^{+} is the filtered union of its K^{+}-subalgebras of the form K^{+}[X/b]_{𝔭_{b}}, where 𝔭_{b} is the prime ideal generated by 𝔪_{K} and X/b. Again (i) follows. The second case, when |X|_{E} > |b| for every b ∈ K^{×}, is reduced to the former, by replacing X with X^{−1}. It remains only to consider the case where there exist a_{0}, b_{0} ∈ K such that |a_{0}| < |X|_{E} < |b_{0}|; then we can find a net (a_{i}, b_{i} | i ∈ I) consisting of pairs of elements of K^{×}, such that |a_{i}| < |X|_{E} < |b_{i}| for every i ∈ I, and moreover, for every a, b ∈ K^{×} such that |a| < |X| < |b|, there exists i_{0} ∈ I with |a| < |a_{i}| and |b_{i}| < |b| whenever i ≥ i_{0}. In such a situation, one verifies easily that E^{+} is the filtered union of its K^{+}-subalgebras of the form E_{i}^{+} := K^{+}[a_{i}/X, X/b_{i}]_{𝔭_{i}}, where 𝔭_{i} is the prime ideal generated by 𝔪_{K} and the elements a_{i}/X, X/b_{i}. Each ring E_{i}^{+} is a complete intersection K^{+}-algebra, isomorphic to K^{+}[X, Y]/(X · Y − a_{i}/b_{i}). It follows that 𝕃_{E_{i}^{+}/K^{+}} is acyclic in degrees > 0, and Ω_{E_{i}^{+}/K^{+}} ≃ X · E_{i}^{+} ⊕ Y ⊕ E_{i}^{+}/(XdY + Y dX). We leave to the reader the verification that this E_{i}^{+}-module is torsion-free. □

Theorem 6.5.8. Let (K, | · |) ⊂ (E, | · |_{E}) be any extension of valued fields. Then (i) H_{i}(𝕃_{E^{+}/K^{+}}) = 0 for every i > 1 and H_{1}(𝕃_{E^{+}/K^{+}}) is a torsion-free E^{+}-module.

(ii) If K is perfect, then H_{i}(𝕃_{E^{+}/K^{+}}) = 0 for every i > 0.

Proof. Let us show first how to deduce (ii) from (i). Indeed, we reduce easily to the case where E is finitely generated over K. Then, if K is perfect, we can find a subextension F ⊂ E which is purely transcendental over K, and such that E is separable over F; by transitivity, we deduce that 𝕃_{E/K} ≃ E ⊗_{F} 𝕃_{F/K}; moreover H_{i}(𝕃_{F/K}) = 0 for i > 0; by (i) we know that H_{1}(𝕃_{E^{+}/K^{+}}) imbeds into H_{1}(𝕃_{E^{+}/K^{+}}) ⊗_{E^{+}} E ≃ H_{1}(𝕃_{E/K}), so the assertion follows.

To show (i), let | · |_{E^{a}} be a valuation on the algebraic closure E^{a} of E, which extends | · |_{E}; recall that E^{a+} is a faithfully flat K^{+}-module by remark 6.1.12(ii). We apply transitivity to the tower K^{+} ⊂ E^{+} ⊂ E^{a+} to see that the theorem holds for the extension (K, | · |) ⊂ (E, | · |_{E}) if and only if it holds for (K, | · |) ⊂ (E^{a}, | · |_{E^{a}}) and for (E, | · |_{E}) ⊂ (E^{a}, | · |_{E^{a}}). For the latter extension the assertion is already known by theorem 6.3.28(i), so we are reduced to prove the theorem for the case (K, | · |) ⊂ (E^{a}, | · |_{E^{a}}). Similarly, we apply transitivity to the tower K^{+} ⊂ K^{a+} ⊂ E^{a+} to reduce to the case where both K and E are algebraically closed. Then we can write E as the filtered union of the algebraic closures E_{i}^{a} of its finitely generated subfields E_{i}, thereby reducing to prove the theorem for the extensions K ⊂ E_{i}^{a}; hence we can assume that tr.d(E : K) is finite. Again, by transitivity, we further reduce to the case where the transcendence degree of E over K equals one. In this case, we can pick an element X ∈ E transcendent over K, and write E = K(X)^{a}. Using once more transitivity, we reduce to show the assertion for the purely transcendental extension K ⊂ K(X), in which case proposition 6.5.6(i) applies, and concludes the proof. □

Lemma 6.5.9. Let R → S be a ring homomorphism.

(i) Suppose that 𝔽_{p} ⊂ R, denote by Φ_{R} : R → R the Frobenius endomorphism of R, and define similarly Φ_{S}. Let R_{(Φ)} := Φ_{R}^{∗}R and S_{(Φ)} := Φ_{S}^{∗}S (cp. (3.5.7)). Suppose moreover that the natural morphism :

R_{(Φ)} ⊗^{L}_{R} S → S_{(Φ)}

is an isomorphism in D(R-Mod). Then 𝕃_{S/R} ≃ 0 in D(s.S-Mod). (ii) Suppose that S is a flat R-algebra and let p be a prime integer, b ∈ R a non-zero-divisor such that p · R ⊂ b^{p} · R. Suppose moreover that the Frobenius endomorphisms of R′ := R/b^{p} · R and S′ := S/b^{p} · S are surjective. Then the natural morphism :

𝕃_{S/R} → 𝕃_{S[b^{−1}]/R[b^{−1}]}

is an isomorphism in D(s.S-Mod).

Proof. (i): Let P^{•} := P_{R}^{•}(S) be the standard simplicial resolution of S by free R-algebras. Then 𝕃_{S/R} ≃ Ω_{P^{•}/R} ⊗_{P^{•}} S. Let Φ_{P^{•}} : P^{•} → P_{(Φ)}^{•} be the termwise Frobenius endomorphism of the simplicial algebra P^{•}. As usual, we can write Φ_{P^{•}} = (Φ_{R} ⊗_{R} 1_{P^{•}}) ∘ Φ_{P^{•}/R}, where the relative Frobenius Φ_{P^{•}/R} : R_{(Φ)} ⊗_{R} P^{•} → P_{(Φ)}^{•} is a morphism of simplicial R_{(Φ)}-algebras. Concretely, if P^{k} = R[X_{i} | i ∈ I] is a free algebra on generators (X_{i} | i ∈ I), then

(6.5.10)  Φ_{P^{k}/R}(X_{i}) = X_{i}^{p} for every i ∈ I.

Under the assumption of the lemma, Φ_{P^{•}/R} is a quasi-isomorphism of simplicial R_{(Φ)}-algebras. It then follows from [46, Ch.II, Prop.1.2.6.2] that Φ_{P^{•}/R} induces an isomorphism

(6.5.11)  R_{(Φ)} ⊗_{R} 𝕃_{S/R} →^{∼} 𝕃_{S_{(Φ)}/R_{(Φ)}} ≃ 𝕃_{S/R}.

However, (6.5.10)shows that (6.5.11)is represented by a map of simplicial complexes which is termwise the zero map, so the claim follows.

(ii): Under the stated assumptions, the Frobenius map induces an isomorphism of R-algebras: R/b · R →^{∼} R′_{(Φ)} (resp. of S-algebras: S/b · S →^{∼} S′_{(Φ)}). Thus the map

S′ ⊗_{R′} (R′)_{(Φ)} → S′_{(Φ)} : x ⊗ y ↦ Φ_{S′}(x) · y

is an isomorphism. Since moreover S′ is a flat R′-algebra, we see that the assumption of (i) is satisfied, whence 𝕃_{S′/R′} ≃ 0. If we now tensor the short exact sequence 0 → R → R → R′ → 0 by 𝕃_{S/R}, we obtain a distinguished triangle

𝕃_{S/R} ⟶^{b^{p}} 𝕃_{S/R} → 𝕃_{S/R} ⊗^{L}_{R} R′ → σ𝕃_{S/R}.

However, 𝕃_{S/R} ⊗^{L}_{R} R′ ≃ 𝕃_{S′/R′} by [46, II.2.2.1], so we have shown that b^{p} acts as an isomorphism on 𝕃_{S/R}. In other words, 𝕃_{S/R} ≃ 𝕃_{S/R} ⊗_{S} S[b^{−1}] ≃ 𝕃_{S[b^{−1}]/R[b^{−1}]}, as claimed. □

Lemma 6.5.12. Let (K, | · |) ⊂ (E, | · |_{E}) be an extension of valued fields and suppose that ℚ ⊂ κ(K). Then the map ρ_{E^{+}/K^{+}} of (6.5.1)is injective with torsion-free cokernel.

Proof. To begin with, let (F, | · |_{F}) be any valued field extension of (E, | · |_{E}). We remark that:

(6.5.13)  ρ_{F^{+}/E^{+}} ∘ (ρ_{E^{+}/K^{+}} ⊗_{E^{+}} 1_{F^{+}}) = ρ_{F^{+}/K^{+}}.

Claim 6.5.14. Suppose moreover that E is an algebraic extension of K. Then ρ_{E^{+}/K^{+}} is an isomorphism. Proof of the claim: Applying (6.5.13), with F := E^{a}, we reduce easily to prove the claim in case E is algebraically closed. Let K^{sh} be the field of fractions of the strict henselization of K^{+} (which we see as imbedded in E^{+}). Let j : Spec K^{sh+} → Spec K^{+} be the morphism induces by the imbedding K ⊂ K^{sh}; In view of lemma 6.2.5, the log structure \underline{Γ}_{K^{sh}}^{+} on Spec K^{sh+} (notation of (6.4.14)) equals j^{∗}\underline{Γ}. Since moreover K^{sh+} is local ind-étale over K^{+}, we deduce from 6.4.12(iii) that ρ_{K^{sh+}/K^{+}} is an isomorphism. Then arguing as in the foregoing, we see that it suffices to prove the claim for the case when K = K^{sh}. Since everything in sight commutes with filtered unions of field extensions, we can even reduce to the case where E is a finite (Galois) extension of K. Then, by corollary 6.2.14, this case can be realized as the extension associated to some datum 𝔊 := (G, j, N, ≤), where moreover G/K^{×} is a finite torsion group. Since by assumption ℚ ⊂ K[𝔊]^{+}, the claim follows by lemma 6.5.4 and (6.5.5).

Now, if K ⊂ E is an arbitrary extension, we can apply (6.5.13) with F := E^{a} and claim 6.5.14 to the extension E ⊂ E^{a} to reduce to the case where E is algebraically closed. Then we can apply again claim 6.5.14 to the extension K ⊂ K^{a} and (6.5.13) with E := K^{a} and F := E, to reduce to the case where also K is algebraically closed. Then, by the usual argument we reduce to the case of an extension of finite transcendence degree, and even to the case of transcendence degree equal to one. We factor the latter as a tower of extensions K ⊂ K(X) ⊂ E, where X is transcendental over K, hence E algebraic over K(X). So finally we are reduced to the case E = K(X), in which case we conclude by proposition 6.5.6(ii). □

Theorem 6.5.15. Let (K, |·|) ⊂ (E, |·|_{E}) be an extension of valued fields, with K algebraically closed. Then Ω_{E^{+}/K^{+}} is a torsion-free E^{+}-module.

Proof. Pick a valuation | · |_{E^{a}} of the field E^{a} extending | · |_{E}. We have an exact sequence: H_{1}(𝕃_{E^{a+}/E^{+}}) → E^{a+} ⊗_{E^{+}} Ω_{E^{+}/K^{+}} → Ω_{E^{a+}/K^{+}}, where the leftmost term is torsion-free by theorem 6.3.28, so it suffices to show that Ω_{E^{a+}/K^{+}} is torsion-free, and we can therefore assume that E is algebraically closed. In this case, if now char(K) > 0, it follows that the Frobenius endomorphism of E^{+} is surjective; then, for any a ∈ E^{+} we can write da = d(a^{1/p})^{p} = p · a^{(p−1)/p} · da^{1/p} = 0, so actually Ω_{E^{+}/K^{+}} = 0. In case char(K) = 0 and char(κ(K)) = p, let us pick an element b ∈ K^{+} such that |b^{p}| ≥ |p|. Since K and E are algebraically closed, the Frobenius endomorphisms on K^{+}/b^{p}K^{+} and E^{+}/b^{p}E^{+} are surjective, so 𝕃_{E^{+}/K^{+}} ≃ 𝕃_{E^{+}[b^{−1}]/K^{+}[b^{−1}]} by lemma 6.5.9(ii). Now, K^{+}[b^{−1}] is the valuation ring of a valuation | · |′ on K, which extends to a valuation | · |′_{E} on E^{+} whose valuation ring is E^{+}[b^{−1}]. Furthermore, the residue fields of these valuations are fields of characteristic zero. Hence, we have reduced the proof of the theorem to the case where κ(K) ⊃ ℚ. We can further reduce to the case where tr.d(E : K) is finite and K is the algebraic closure of an extension of finite type of its prime field. By lemma 6.5.12 we have a commutative diagram with exact rows:

\xymatrix{ & E^{+} ⊗_{K^{+}} Ω_{K^{+}/ℤ} \ar[r] \ar[d]^{1_{E^{+}}⊗β_{K^{+}}} & Ω_{E^{+}/ℤ} \ar[r] \ar[d]^{β_{E^{+}}} & Ω_{E^{+}/K^{+}} \ar[r] \ar[d]^{γ} & 0 \\ 0 \ar[r] & E^{+} ⊗_{K^{+}} Ω_{K^{+}/ℤ}(log Γ^{+}) \ar[r] & Ω_{E^{+}/ℤ}(log Γ_{E}^{+}) \ar[r] & Coker(ρ_{E^{+}/K^{+}}) \ar[r] & 0}

where β_{K^{+}} and β_{E^{+}} are the maps of corollary 6.4.17(i). By virtue of lemma 6.5.12, it suffices to show that γ is injective. Since β_{E^{+}} is injective by corollary 6.4.17(i), the snake lemma reduces us to prove : Claim 6.5.16. The induced map E^{+} ⊗_{K^{+}} Coker β_{K^{+}} → Coker β_{E^{+}} is injective. Proof of the claim: Under our current assumptions, K^{+} and E^{+} are valuation rings of finite Krull dimension, by (6.1.24). Let 𝔭_{r} := 0 ⊂ 𝔭_{r−1} ⊂ ... ⊂ 𝔭_{0} := 𝔪_{E} be the chain of all the prime ideals of E^{+}. Denote by Δ_{r} := Γ_{E} ⊃ Δ_{r−1} ⊃ ... ⊃ Δ_{0} := 0 the corresponding ascending chain of convex subgroups of Γ_{E} (see (6.1.22)). Let Fil^{•}(Coker β_{E^{+}}) (resp. Fil^{•}(Coker β_{K^{+}})) be the finite filtration indexed by the totally ordered set Spec E^{+} (resp. Spec K^{+}), provided by corollary 6.4.17(ii). Since it is preferable to work with a single indexing set, we use the surjection Spec E^{+} → Spec K^{+}, to replace by Spec E^{+} the indexing of the filtration on Coker β_{K^{+}}; of course in this way some of the graded subquotients become trivial, but we do not mind. With this notation we can write down the identities:

E^{+} ⊗_{K^{+}} gr^{𝔭_{i}}(Coker β_{K^{+}}) ≃ E^{+} ⊗_{K^{+}} ((Δ_{i+1} ∩ Γ)/(Δ_{i} ∩ Γ)) ⊗_{ℤ} (K^{+}/𝔭_{i} ∩ K^{+})

for every 𝔭_{i} ∈ Spec(E^{+}). Furthermore, our map ϕ : E^{+}⊗_{K^{+}} Coker β_{K^{+}} → Coker β_{E^{+}} respects these filtrations. If now 𝔭_{i} ∈ Spec(E^{+}) is the radical of the extension of a prime ideal of K^{+}, then clearly the map gr^{𝔭_{i}}(ϕ) : E^{+} ⊗_{K^{+}} gr^{𝔭_{i}}(Coker β_{K^{+}}) → gr^{𝔭_{i}}(Coker β_{E^{+}}) is induced by the imbeddings (Δ_{i+1} ∩ Γ)/(Δ_{i} ∩ Γ) ⊂ Δ_{i+1}/Δ_{i} and K^{+}/𝔭_{i} ∩ K^{+} ⊂ E^{+}/𝔭_{i}, and it is therefore injective. On the other hand, if 𝔭_{i} is not the radical of an ideal extended from K^{+}, we have gr^{𝔭_{i}}(Coker β_{K^{+}}) = 0, so gr^{𝔭_{i}}(ϕ) is trivially injective in this case as well. Since the map gr^{•}(ϕ) is injective, the same holds for ϕ, which concludes the proof of the claim and of the theorem.

□

Theorem 6.5.17. Let | · |_{K^{s}} be a valuation on the separable closure K^{s} of K, extending the valuation of K. Then the map ρ := ρ_{K^{s+}/K^{+}} is injective.

Proof. Suppose first that Γ is divisible. By the usual reductions, we can assume that K is finitely generated over its prime field, hence that the convex rank of Γ_{K^{s}} is finite. We consider the commutative diagram :

\xymatrix{Ω_{K^{+}/ℤ} ⊗_{K^{+}} K^{s+} \ar[r]^{α} \ar[d]_{β_{K^{+}}⊗1_{K^{s+}}} & Ω_{K^{s+}/ℤ} \ar[d]^{β_{K^{s+}}} \\ Ω_{K^{+}/ℤ}(log Γ^{+}) ⊗_{K^{+}} K^{s+} \ar[r]^{ρ} & Ω_{K^{s+}/ℤ}(log Γ_{K^{s}}^{+})}

where β_{K^{+}} and β_{K^{s+}} are the maps of corollary 6.4.17(i). By theorem 6.3.28(ii), α is injective, and the same holds for β_{K^{s+}}, by corollary 6.4.17(i). It follows that Im(β_{K^{+}}⊗_{K^{+}}1_{K^{s+}})∩Ker ρ = {0}, in other words, the induced map

(6.5.18)  Ker ρ → K^{s+} ⊗_{K^{+}} Coker β_{K^{+}}

is injective. By corollary 6.4.17(ii), there is a filtration Fil^{•}(Coker β_{K^{+}}) on Coker β_{K^{+}}, indexed by Spec K^{+}, such that gr^{𝔭_{i}}(Coker β_{K^{+}}) ≃ (Δ_{i+1}/Δ_{i}) ⊗_{ℤ} (K^{+}/𝔭_{i}), where Δ_{i}, Δ_{i+1} are two convex subgroups of Γ_{K}. However, since we assume that Γ is divisible, the same holds for Δ_{i+1}/Δ_{i}; we deduce that gr^{𝔭_{i}}(Coker β_{K^{+}}) vanishes whenever Frac(K^{+}/𝔭_{i}) is a field of positive characteristic. What this means is that the filtration Fil^{•}(Coker β_{K^{+}}) is actually indexed by Spec K^{+} ⊗_{ℤ} ℚ ⊂ Spec K^{+}, and the natural map :

(6.5.19)  Cokerβ_{K^{+}} → ℚ ⊗_{ℤ} Coker β_{K^{+}}

is an isomorphism. The same holds also for Coker β_{K^{s+}}. If K_{ℚ}^{+} := K^{+} ⊗_{ℤ} ℚ = {0}, then Coker β_{K^{+}} = Coker β_{K^{s+}} = {0}, hence Ker ρ = 0, which is what we had to show.

In case K_{ℚ}^{+} ≠ {0}, then K_{ℚ}^{+} is a valuation ring of K with residue field of characteristic zero. However, by lemma 6.4.12(iii) it follows easily that ℚ ⊗_{ℤ} Coker β_{K^{+}} ≃ Coker β_{K_{ℚ}^{+}}, and likewise ρ_{K^{s+}/K^{+}} ⊗_{ℤ} 1_{ℚ} = ρ_{K_{ℚ}^{s+}/K_{ℚ}^{+}}, where K_{ℚ}^{s+} := K^{s+} ⊗_{ℤ} ℚ is a valuation ring of K^{s} whose valuation extends that of K_{ℚ}^{+}. Since (6.5.19)is an isomorphism, (6.5.18)factors through Kerρ_{K_{ℚ}^{s+}/K_{ℚ}^{+}}; however, the latter vanishes by lemma 6.5.12. Since (6.5.18) is injective, we derive Ker ρ = 0, so the theorem holds in this case.

In case Γ is not necessarily divisible, let us choose a datum 𝔊 := (G, j, N, ≤) as in (6.1.5), such that G := (K^{s})^{×} ⊕ F, where F is a torsion-free abelian group (whose composition law we write in multiplicative notation) and N is the graph of a surjective group homomorphism ϕ : F → (K^{s})^{×}. Notice that in this case Γ_{𝔊} ≃ Γ_{K^{s}} ≃ Γ ⊗_{ℤ} ℚ, and the restriction to F of the projection G → Γ_{𝔊} is the map x ↦ |ϕ(x^{−1})|_{K^{s}}. Let now H := K^{×} ⊕ F and define a new datum ℌ := (H, j, H ∩ N, ≤); since ϕ is surjective, clearly we still have Γ_{ℌ} ≃ Γ ⊗_{ℤ} ℚ. Notice as well that K^{s}(𝔊) is separable over K(ℌ). Set ρ_{ℌ} := ρ_{K(ℌ)^{+}/K^{+}}, ρ_{𝔊} := ρ_{K^{s}(𝔊)^{+}/K^{s+}} and ρ_{𝔊/ℌ} := ρ_{K^{s}(𝔊)^{+}/K(ℌ)^{+}}. We consider the diagram :

\xymatrix{Ω_{K^{+}/ℤ}(log Γ^{+}) ⊗_{K^{+}} K^{s}(𝔊)^{+} \ar[d]_{ρ_{ℌ}⊗1_{K^{s}(𝔊)^{+}}} \ar[r]^{ρ⊗1_{K^{s}(𝔊)^{+}}} & Ω_{K^{s+}/ℤ}(log Γ_{K^{s}}^{+}) ⊗_{K^{s+}} K^{s}(𝔊)^{+} \ar[d]^{ρ_{𝔊}} \\ Ω_{K(ℌ)^{+}/ℤ}(log Γ_{ℌ}^{+}) ⊗_{K(ℌ)^{+}} K^{s}(𝔊)^{+} \ar[r]^{ρ_{𝔊/ℌ}} & Ω_{K^{s}(𝔊)^{+}/ℤ}(log Γ_{𝔊}^{+}).}

Since F is torsion-free, it follows easily from (6.5.5)and lemma 6.5.4 that ρ_{ℌ} and ρ_{𝔊} are injective with torsion-free cokernels. Hence, in order to prove that ρ is injective, it suffices to show that ρ_{K^{s}(𝔊)^{+}/K(ℌ)^{+}} is. Finally, let E be the separable closure of K(ℌ) and choose a valuation on E which extends the valuation of K^{s}(𝔊); we notice that Kerρ_{K^{s}(𝔊)^{+}/K(ℌ)^{+}} ⊂ Kerρ_{E^{+}/K(ℌ)^{+}}. Therefore, we can replace K by K(ℌ) and reduce to the case where Γ is divisible, which has already been dealt with. □

Theorem 6.5.20. Let (K, | · |) ⊂ (E, | ⊂ |_{E}) be an extension of valued fields, with K algebraically closed. Then ρ_{E^{+}/K^{+}} is injective with torsion-free cokernel.

Proof. By the usual arguments, we can suppose that E is finitely generated over K. Let | · |_{E^{s}} be an extension of the valuation | · |_{E} to the separable closure E^{s} of E. We have Kerρ_{E^{+}/K^{+}} ⊂ Kerρ_{E^{s+}/K^{+}}, and even Cokerρ_{E^{+}/K^{+}} ⊂ Cokerρ_{E^{s+}/K^{+}}, by theorem 6.5.17. Thus we can replace E by E^{s} and suppose that E is separably closed, hence Γ_{E} divisible, by example 6.1.21(ii). By corollary 6.4.17(i) we have a commutative diagram with exact rows :

\xymatrix{0 \ar[r] & E^{+} ⊗_{K^{+}} Ω_{K^{+}/ℤ} \ar[d]_{α} \ar[r] & E^{+} ⊗_{K^{+}} Ω_{K^{+}/ℤ}(log Γ^{+}) \ar[r] \ar[d]^{ρ_{E^{+}/K^{+}}} & E^{+} ⊗_{K^{+}} Coker β_{K^{+}} \ar[r] \ar[d]^{γ} & 0 \\ 0 \ar[r] & Ω_{E^{+}/ℤ} \ar[r] & Ω_{E^{+}/ℤ}(log Γ_{E}^{+}) \ar[r] & Coker β_{E^{+}} \ar[r] & 0.}

By theorem 6.5.8(ii), the map α is injective. The same holds for γ, in view of claim 6.5.16. It follows already that ρ_{E^{+}/K^{+}} is injective. Moreover, by theorem 6.5.15, Coker α is a torsionfree E^{+}-module. Since both Γ and Γ_{E} are divisible, it follows easily from corollary 6.4.17(ii) that Cokerβ_{K} and Cokerβ_{E} are ℚ-vector spaces (cp. the proof of theorem 6.5.17), hence the same holds for Coker γ. Consequently, Cokerρ_{E^{+}/K^{+}} is a torsion-free ℤ-module, and thus we are reduced to show that ℚ ⊗_{ℤ} Cokerρ_{E^{+}/K^{+}} is a torsion-free E^{+}-module. However, ℚ ⊗_{ℤ} Cokerρ_{E^{+}/K^{+}} ≃ Cokerρ_{E_{ℚ}^{+}/K_{ℚ}^{+}}, where E_{ℚ}^{+} := E^{+} ⊗_{ℤ} ℚ and K_{ℚ}^{+} := K_{ℚ}^{+} are valuation rings with residue fields of characteristic zero (or else they vanish, in which case we are done). But the assertion to prove is already known in this case, by lemma 6.5.12. □

Corollary 6.5.21. Let (K, | · |) be a valued field, and k a perfect field such that k ⊂ K^{+}. Then Ω_{K^{+}/k} and Cokerρ_{K^{+}/k} are torsion-free K^{+}-modules.

Proof. We have k^{a} ⊂ K^{sh+}; let E := k^{a} · K ⊂ K^{sh} and denote by j : Spec E^{+} → Spec K^{+} the morphism induced by the imbedding K ⊂ E. By lemma 6.2.5 the natural map : j^{∗}\underline{Γ}^{+} → \underline{Γ}_{E}^{+} is an isomorphism of log structures; moreover Ω_{k^{a}/k} = 0, since k is perfect. Hence Ω_{K^{+}/k} ⊂ Ω_{E^{+}/k^{a}}, and furthermore, by lemma 6.4.12(iii) we have Cokerρ_{K^{+}/k} ⊂ Cokerρ_{E^{+}/k^{a}}. Then the assertion follows from theorems 6.5.15 and 6.5.20. □

Remark 6.5.22. Notice that corollary 6.5.21 is a straightforward consequence of a standard (as yet unproven) conjecture on the existence of resolution of singularities over perfect fields.

6.6. Deeply ramified extensions. We keep the notation of section 6.3. We borrow the notion of deeply ramified extension of valuation rings from the paper [21], even though our definition applies more generally to valuations of arbitrary rank. Definition 6.6.1. Let (K, | · |) be a valued field, | · |_{K^{s}} a valuation on K^{s} which extends | · |. We say that (K, | · |) is deeply ramified if Ω_{K^{s+}/K^{+}} = 0. Notice that the definition does not depend on the choice of the extension | · |_{K^{s}}. Proposition 6.6.2. Let (K, | · |) be a valued field whose valuation has rank one, | · |_{K^{s}} an extension of | · | to K^{s}. Then the following conditions are equivalent :

(i) (K, | · |) is deeply ramified; (ii) The morphism of almost algebras (K^{+})^{a} → (K^{s+})^{a} is weakly étale; (iii) (Ω_{K^{s+}/K^{+}})^{a} = 0.

Moreover, the above equivalent conditions imply that the valuation of K is not discrete.

Proof. We leave to the reader the verification that (iii) (and, a fortiori, (i)) can hold only in case the valuation of K is not discrete.

Let K^{+sh} be a strict henselization of K^{+} contained in K^{s+}, and K^{sh} its fraction field. It is easy to check that (K, | · |) is deeply ramified if and only if (K^{sh}, | · |_{K^{sh}}) is. Moreover, in view of lemma 3.1.2(iv), condition (ii) holds for (K, | · |) if and only if it holds for (K^{sh}, | · |_{K^{sh}}), and similarly for condition (iii). Hence we can assume that K^{+} is strictly henselian. It is also clear that (i)⇒(iii)⇐(ii). To show that (iii)⇒(ii), let E ⊂ K^{s} be a finite separable extension of K and set E^{+} := K^{s+} ∩ E; it follows from theorem 6.3.20 that the natural map Ω_{E^{+}/K^{+}} ⊗_{E^{+}} K^{s+} → Ω_{K^{s+}/K^{+}} is injective; thus, if (iii) holds, we deduce that (Ω_{E^{+}/K^{+}})^{a} = 0 for every finite separable extension E of K. Again by theorem 6.3.20 we derive that 𝒟_{E^{+}/K^{+}} = E^{+a} for every such E. Finally, lemmata 6.3.10(i) and 4.1.29 show that E^{+a} is étale over K^{+a}, whence (ii). Suppose next that the residue characteristic of K^{+} is zero; then every finite extension E of K factors as a tower of Kummer extensions of prime degree, therefore Ω_{E^{+}/K^{+}} = 0 by corollary 6.3.18(i), which implies that (i)⇔(iii) in this case. Finally, suppose that the residue characteristic is p > 0. Let us choose a ∈ K^{+} such that |a| ≥ |p|. It follows easily from example 6.1.21(iii) that every element x ∈ K^{s+} can be written in the form x = y^{p} + a · z for some y, z ∈ K^{s+}. Hence dx = p · dy^{p−1} + a · dz, which means that Ω_{K^{s+}/K^{+}} = a · Ω_{K^{s+}/K^{+}}. Therefore, even in this case we deduce (iii)⇒(i). □

6.6.3. Let us say that a K^{+a}-module M is K^{+a}-divisible if, for every x ∈ K^{+} \ {0} we have M = x · M. Lemma 6.6.4. Let (K, | · |) be a valued field such that ℚ ⊂ K, and let p := char(κ) > 0. Let (K^{s}, | · |_{K^{s}}) be an extension of the valuation | · | to a separable closure of K. Denote by T the K^{s+}-torsion submodule of Ω_{K^{s+}/ℤ}. Then T ≃ K^{s}/K^{s+}.

Proof. Let (ℚ^{a}, | · |_{ℚ^{a}}) be the restriction of | · |_{K^{s}} to the algebraic closure of ℚ in K^{s}. From theorem 6.5.20 it follows easily that T ≃ Ω_{ℚ^{a+}/ℤ}⊗_{ℚ^{a+}}K^{s+}, hence we can suppose that K = ℚ. From theorem 6.3.20 we deduce that the natural map ℚ^{a+} ⊗_{E^{+}} Ω_{E^{+}/ℤ} → Ω_{ℚ^{a+}/ℤ} is injective for every subextension E ⊂ ℚ^{a}. For every n ∈ ℕ and every subextension E ⊂ ℚ^{a}, let E_{n} := E(ζ_{p^{n}}), where ζ_{p^{n}} is any primitive p^{n}-th root of 1 and set E_{∞} := ⋃_{n>0} E_{n}. Claim 6.6.5. For every finite subextension E ⊂ ℚ^{a}, there exists n ∈ ℕ such that the image of E_{n}^{+} ⊗_{E^{+}} Ω_{E^{+}/ℤ} in Ω_{E_{n}^{+}/ℤ} is included in the image of E_{n}^{+} ⊗_{ℚ_{n}^{+}} Ω_{ℚ_{n}^{+}/ℤ}. Proof of the claim: For every n ∈ ℕ, E_{n}^{+} is a discrete valuation ring and κ(E_{n}) is a finite separable extension of κ(ℚ) = 𝔽_{p}; from the exact sequence 𝔪_{E_{n}}/𝔪_{E_{n}}^{2} → Ω_{E_{n}^{+}/ℤ} → Ω_{κ(E_{n})/𝔽_{p}} = 0 we deduce that Ω_{E_{n}^{+}/ℤ} is a (torsion) cyclic E_{n}^{+}-module. By comparing the annihilators of the modules under consideration, one obtains easily the claim.

A standard calculation shows that Ω_{ℚ_{∞}^{+}/ℤ} ≃ ℚ_{∞}/ℚ_{∞}^{+}. This, together with claim 6.6.5 implies the lemma. □

Proposition 6.6.6. Keep the notation and assumptions of proposition 6.6.2 and suppose moreover that the characteristic p of the residue field κ of K^{+} is positive and that the valuation on K is not discrete. Let (K^{∧}, | · |^{∧}) be the completion of (K, | · |) for the valuation topology. Then the following conditions are equivalent:

(i) (K, | · |) is deeply ramified; (ii) The Frobenius endomorphism of K^{∧+}/pK^{∧+} is surjective; (iii) For some b ∈ K^{+} \ {0} such that 1 > |b| ≥ |p|, the Frobenius endomorphism on

(K^{+}/bK^{+})^{a} is an epimorphism; (iv) Ω_{K^{+}/ℤ}(log Γ^{+}) is a K^{+}-divisible K^{+}-module;

(v) Ω_{K^{+}/ℤ} is a K^{+}-divisible K^{+}-module; (vi) (Ω_{K^{+}/ℤ})^{a} is a K^{+a}-divisible K^{+a}-module; (vii) Coker ρ_{K^{s+}/K^{+}} = 0 (notation of (6.5.1)); (viii) Coker(ρ_{K^{s+}/K^{+}})^{a} = 0.

Proof. Suppose that (i) holds; then by proposition 6.6.2 it follows that the morphism (K^{+})^{a} → (K^{s+})^{a} is weakly étale, so the same holds for the morphism (K^{+}/bK^{+})^{a} → (K^{s+}/bK^{s+})^{a}, for every b ∈ K^{+} with |b| ≥ |p|. In view of example 6.1.21(iii), one sees that the Frobenius endomorphism on K^{s+}/bK^{s+} is an epimorphism. Using theorem 3.5.13(ii) we deduce that the Frobenius endomorphism on (K^{+}/bK^{+})^{a} is an epimorphism as well. This shows that (i)⇒(iii). To show that (iii)⇒(ii), let us choose ε ∈ 𝔪 \ {0} such that |ε^{p}| > |b|; by hypothesis, for every x ∈ K^{+} there exists y ∈ K^{+} such that ε^{p} · x − y^{p} ∈ bK^{+}. It follows easily that the Frobenius endomorphism is surjective on K^{+}/(b · ε^{−p})K^{+}. Replacing b by b · ε^{−p} we can assume that the Frobenius endomorphism is surjective on K^{+}/bK^{+}. Let b_{1} ∈ K^{+} such that 1 > |b_{1}^{p}| ≥ |b|; we let Fil_{1}^{•}(K^{+}/pK^{+}) (resp. Fil_{2}^{•}(K^{+}/pK^{+})) be the b_{1}-adic (resp. b_{1}^{p}-adic) filtration on K^{+}/pK^{+}. The group topology on K^{+}/pK^{+} defined by the filtrations Fil_{i}^{•}(K^{+}/pK^{+}) (i = 1, 2) is the same as the one induced by the valuation topology of K^{+}; moreover, one verifies easily that the Frobenius endomorphism defines a morphism of filtered abelian groups Fil_{1}^{•}(K^{+}/pK^{+}) → Fil_{2}^{•}(K^{+}/pK^{+}) and that the associated morphism of graded abelian groups is surjective. It then follows from [16, Ch.III, §2, n.8, Cor.2] that (ii) holds. Next suppose that (ii) holds; choose b ∈ K^{+} such that 1 > |b| > |b^{p}| ≥ |p|; by hypothesis, the Frobenius endomorphism on K^{+}/b^{p}K^{+} is surjective; the same holds for the Frobenius map on K^{s+}/b^{p}K^{s+}, in view of example 6.1.21(iii). Hence, the assumptions of lemma 6.5.9(ii) are fulfilled, and we deduce that 𝕃_{K^{s+}/K^{+}} ≃ 𝕃_{K^{s+}[b^{−1}]/K^{+}[b^{−1}]}. Now, if char(K) = p, this implies already that Ω_{K^{s+}/K^{+}} ≃ Ω_{K^{s}/K} = 0, which is (i). In case char(K) = 0, we only deduce that Ω_{K^{s+}/K^{+}} ≃ Ω_{K^{s+}[1/p]/K^{+}[1/p]}; however, R := K^{+}[1/p] is a valuation ring of residue characteristic 0. We are therefore reduced to showing that R is deeply ramified. Arguing as in the proof of proposition 6.6.2 we can even assume that R is strictly henselian, in which case the assertion follows from corollary 6.3.18(i).

Furthermore, (ii) implies easily that Ω_{(K^{+}/b^{p}K^{+})/ℤ} = 0 (since dx^{p} = p · dx^{p−1} = 0). Let I := b^{p}K^{+}; it follows that the natural map I/I^{2} → (K^{+}/b^{p}K^{+}) ⊗_{ℤ} Ω_{K^{+}/ℤ} is surjective, i.e. Ω_{K^{+}/ℤ} = b^{p} · Ω_{K^{+}/ℤ} + K^{+} · db^{p} ⊂ b^{p} · Ω_{K^{+}/ℤ}, which implies (v). Next, by corollary 6.4.17(ii), we have Ω_{K^{+}/ℤ}(log Γ^{+})/Ω_{K^{+}/ℤ} ≃ κ ⊗_{ℤ} Γ, and this last term vanishes since (ii) implies that Γ = Γ^{p}. This shows that (ii)⇒(iv) as well. Clearly (v)⇒(vi). Suppose that (vi) holds. We will need the following : Claim 6.6.7. Ω_{K^{s+}/ℤ} is a K^{s+}-divisible module and C := Coker(ρ_{K^{s+}/K^{+}}) is a K^{+}-torsion module (notation of (6.5.1)). Furthermore, C^{a} ≃ (Ω_{K^{s+}/K^{+}})^{a}. Proof of the claim: In view of example 6.1.21(iii), (K^{s}, |·|_{K^{s}}) satisfies condition (iii), hence the first assertion follows from the implications (iii)⇔(ii)⇒(v), which have already been shown. Furthermore, it is clear that Ω_{K^{s+}/K^{+}} is a torsion K^{+}-module and therefore the second assertion follows easily from corollary 6.4.17. The latter corollary also implies that (Ω_{K^{+}/ℤ})^{a} ≃ Ω_{K^{+}/ℤ}(log Γ^{+})^{a}, and similarly for Ω_{K^{s+}/ℤ}, whence the third assertion.

Now, suppose first that 𝔽_{p} ⊂ K^{+}; in this case Ω_{K^{s+}/ℤ} is a torsion-free K^{s+}-module according to corollary 6.5.21. Let b ∈ K^{+} be any element; by theorem 6.5.17 and snake lemma we deduce that the b-torsion submodule C[b]^{a} := Ker(C^{a} → C^{a} : x ↦ b · x) is isomorphic to the cokernel of the scalar multiplication by b on the module K^{s+} ⊗_{K^{+}} Ω_{K^{+}/ℤ}(log Γ^{+}); the latter vanishes by assumption, and by claim 6.6.7 we have C^{a} = ⋃_{b∈K^{+}} C[b]^{a}, whence C^{a} = 0, which is equivalent to (i) by claim 6.6.7 and proposition 6.6.2.

Finally, in case K^{+} is of mixed characteristic, denote by T (resp. T′) the K^{+}-torsion submodule of Ω_{K^{s+}/ℤ} (resp. of K^{s+} ⊗_{K^{+}} Ω_{K^{+}/ℤ}) and define T [b] (resp. T′[b]) as its b-torsion submodule, for any b ∈ K^{+}. The foregoing argument shows that T^{a} is isomorphic to the K^{+a}- torsion submodule of (Ω_{K^{s+}/ℤ}(log Γ_{K^{s}}^{+}))^{a}, and similarly for (T′)^{a}; moreover, by snake lemma we obtain a short exact sequence 0 → T′[b]^{a} → T [b]^{a} → C[b]^{a} → 0 for every b ∈ K^{+}, whence a short exact sequence 0 → (T′)^{a} → T^{a} → C^{a} → 0. Under (vi), (T′)^{a} is a divisible module; however, it is clear from lemma 6.6.4 that the only divisible (K^{s+})^{a}-submodules of T^{a} are 0 and T^{a}. Consequently, in light of claim 6.6.7 and proposition 6.6.2, in order to prove that (vi)⇒(i), it suffices to show that (T′)^{a} ≠ 0. In turns, this is implied by the following : Claim 6.6.8. The image in Ω_{K^{+}/ℤ}(log Γ) of d log(p) ∈ Ω_{ℚ^{+}/ℤ}(log Γ_{ℚ}^{+}), has annihilator pK^{+}.

Proof of the claim: (Of course, ℚ^{+} := \overline{ℚ}^{+} ∩ ℚ). By theorem 6.5.17 it suffices to consider the image of d log(p) in Ω_{K^{s+}/ℤ}(log Γ_{K^{s}}^{+}). Then, by theorem 6.5.20, we reduce to consider the case K^{s} = \overline{ℚ}. Then, once more by theorem 6.5.17, it suffices to look at the annihilator of d log(p) in Ω_{ℚ^{+}/ℤ}(log Γ_{ℚ}^{+}) itself, and the claim follows.

Since (vi) is implied by both (iv) and (v), we deduce at once that all the conditions (i)-(vi) are equivalent. Furthermore, it is clear from claim 6.6.7 and proposition 6.6.2 that both (vii) and (viii) are equivalent to (i), so the proposition follows. □

Remark 6.6.9. By inspecting the proof of proposition 6.6.6, we see that the argument for (ii)⇒(i) still goes through for valued fields (K, | · |) of arbitrary rank and characteristic p > 0. Lemma 6.6.10. Let (K, | · |) be a valued field and b ∈ K an element with 0 < |b| < 1. Denote by 𝔮(b) the radical of the ideal bK^{+} and set 𝔭(b) := ⋂_{r>0} b^{r} · K^{+}. Then 𝔭(b) and 𝔮(b) are consecutive prime ideals, i.e. there are no prime ideals strictly contained between 𝔭(b) and 𝔮(b). Equivalently, the ring W (b) := (K^{+}/𝔭(b))_{𝔮(b)} is a valuation ring of rank one and the image of b is topologically nilpotent in the valuation topology of W (b).

Proof. It is easy to verify that 𝔭(b) and 𝔮(b) are prime ideals, and using (6.1.22) one deduces that W (b) is a valuation ring of rank one, which means that 𝔭(b) and 𝔮(b) are consecutive, □ Theorem 6.6.11. Let (K, | · |) be a valued field, (K^{∧}, | · |^{∧}) its completion. The following conditions are equivalent :

(i) (K, | · |) is deeply ramified.

(ii) For every valued extension (E, | · |_{E}) of (K, | · |), for every b ∈ K^{+} \ {0} and for every

i > 0 we have H_{i}(𝕃_{(E^{+}/bE^{+})/(K^{+}/bK^{+})}) = 0. (iii) For every pair of consecutive prime ideals 𝔭 ⊂ 𝔮 ⊂ K^{+}, the valuation ring (K^{+}/𝔭)_{𝔮}

is deeply ramified. (iv) For every pair of convex subgroups H_{1} ⊂ H_{2} ⊂ Γ, the quotient H_{2}/H_{1} is not iso-

morphic to ℤ, and moreover, if p := char(κ) > 0, the Frobenius endomorphism on K^{∧+}/pK^{∧+} is surjective.

Proof. To show that (ii)⇒(i), we take E := K^{s} and we choose a valuation on K^{s} extending | · |. Then, by arguing as in the proof of lemma 6.5.9(ii), we deduce from (ii) that the scalar multiplication by b on Ω_{K^{s+}/K^{+}} is injective. Since the latter is a torsion K^{s+}-module, we deduce (i). To show (i)⇒(ii), we reduce first to the case where E = E^{a}; indeed, let | · |_{E^{a}} be a valuation on E^{a} extending |·|_{E} and suppose that the sought vanishing is known for the extension (K, | · |) ⊂ (E^{a}, | · |_{E^{a}}); by transitivity, it then suffices to show : Claim 6.6.12. H_{i}(𝕃_{(E^{a+}/bE^{a+})/(E^{+}/bE^{+})}) = 0 for every i > 1.

Proof of the claim: By [46, II.2.2.1] we have 𝕃_{(E^{a+}/bE^{a+})/(E^{+}/bE^{+})} ≃ 𝕃_{E^{a+}/E^{+}} ⊗^{L}_{E^{a+}} E^{a+}/bE^{a+}, whence a spectral sequence

E_{pq}^{2} := Tor_{p}^{E^{a+}}(H_{q}(𝕃_{E^{a+}/E^{+}}), E^{a+}/bE^{a+}) ⇒ H_{p+q}(𝕃_{(E^{a+}/bE^{a+})/(E^{+}/bE^{+})}).

Since E^{a+}/bE^{a+} is an E^{a+}-module of Tor-dimension ≤ 1, we see that E_{pq}^{2} = 0 for every p > 1; furthermore, by theorem 6.3.28(i), it follows that E_{pq}^{2} = 0 whenever p, q > 0, so the claim follows.

Thus, we can suppose that E is algebraically closed. A spectral sequence analogous to the foregoing computes H_{i}(𝕃_{(E^{+}/bE^{+})/(K^{a+}/bK^{a+})}), and using theorems 6.5.8(ii) and 6.5.15, we find that the latter vanishes for i > 0. Consequently, by applying transitivity to the tower of extensions K^{+}/bK^{+} ⊂ K^{a+}/bK^{a+} ⊂ E^{+}/bE^{+}, we reduce to show the assertion for the case E = K^{a}. However, by example 6.1.21(iii) we have K^{a+}/bK^{a+} ≃ K^{s+}/bK^{s+} for every b ∈ K^{+} \ {0}, so we can further reduce to the case E = K^{s}. In this case, one concludes the proof by another spectral sequence argument, this time using assumption (i) and theorem 6.3.28(ii) to show that the relevant terms E_{pq}^{2} vanish.

To show that (iii)⇒(iv), we consider two subgroups H_{1} ⊂ H_{2} as in (iv); if c.rk(H_{2}/H_{1}) > 1, then clearly H_{2}/H_{1} cannot be isomorphic to ℤ, so we can assume that H_{1} and H_{2} are consecutive, so that the corresponding prime ideals are too (see (6.1.22)). In this case, (iii) and proposition 6.6.2 show that H_{2}/H_{1} is not isomorphic to ℤ, which is the first assertion of (iv). To prove the second assertion, it will suffice to show the following : Claim 6.6.13. Suppose that p := char(κ) > 0 and that (iii) holds. Then, for every b ∈ K^{+} \{0} with 1 > |b| ≥ |p|, the Frobenius endomorphism on K^{+}/bK^{+} is surjective. Proof of the claim: For such a b as above, define 𝔭(b), 𝔮(b) and W (b) as in lemma 6.6.10; then W (b) is a valuation ring of rank one, so it is deeply ramified by assumption (iii). Then, by proposition 6.6.6 it follows that the Frobenius endomorphism is surjective on W (b)/b · W (b) ≃ K_{𝔮(b)}^{+}/bK_{𝔮(b)}^{+}. We remark that bK_{𝔮(b)}^{+} ⊂ K^{+}; there follows a natural imbedding: K^{+}/bK_{𝔮(b)}^{+} ⊂ W (b)/b · W (b), commuting with the Frobenius maps. It is then easy to deduce that the Frobenius endomorphism is surjective on K^{+}/bK_{𝔮(b)}^{+}. Moreover, by proposition 6.6.2, the valuation of W (b) is not discrete, hence its value group is isomorphic to a dense subgroup of (ℝ, ≥) (see example 6.1.21(vi)); therefore, by (6.1.22) and example 6.1.21(v), we deduce that there exists an element c ∈ K^{+} such that |b| > |c^{3p}| and |b| < |c^{2p}|. These inequalities have been chosen so that c^{p}K_{𝔮(b)}^{+} ⊂ K^{+} and bK_{𝔮(b)}^{+} ⊂ c^{2p}K_{𝔮(b)}^{+}, whence bK_{𝔮(b)}^{+} ⊂ c^{p}K^{+}, and finally we conclude that the Frobenius endomorphism induces a surjection: K^{+}/cK^{+} → K^{+}/c^{p}K^{+}. We let Fil_{1}^{•}(K^{+}/bK^{+}) (resp. Fil_{2}^{•}(K^{+}/bK^{+})) be the c-adic (resp. c^{p}-adic) filtration on K^{+}/bK^{+}. The foregoing implies that the Frobenius endomorphism induces a morphism of filtered modules Fil_{1}^{•}(K^{+}/bK^{+}) → Fil_{2}^{•}(K^{+}/bK^{+}) which is surjective on the associated graded modules; by [16, Ch.III, §2, n.8, Cor.2] the claim follows.

Next, assume (iv) and let W := (K^{+}/𝔭)_{𝔮}, for two consecutive prime ideals 𝔭 ⊂ 𝔮 ⊂ K^{+}. By assumption the Frobenius map is surjective on K^{+}/bK^{+}, whenever b ∈ K^{+} \{0} and |b| ≥ |p|; we deduce easily that the Frobenius endomorphism is surjective on W/bW, which implies (iii), in view of proposition 6.6.6.

(i)⇒(iii): indeed, let 𝔭 ⊂ 𝔮 be as in (iii); we need to show that (K^{+}/𝔭)_{𝔮} is deeply ramified. After replacing K^{+} by K_{𝔮}^{+} we can assume that 𝔮 is the maximal ideal of K^{+}. The ring k^{+} := K^{+}/𝔭 is a valuation ring; let k := Frac(k^{+}), | · |_{k} the valuation on k corresponding to k^{+}, and (k′, | · |_{k′}) a finite separable valued extension of (k, | · |_{k}). It suffices to show that Ω_{k^{′+}/k^{+}} = 0. We have k′ ≃ k[X]/(f(X)) for some irreducible monic polynomial f(X) ∈ k[X]; let f̃(X) ∈ K_{𝔭}^{+}[X] be a lifting of f(X) to a monic polynomial. Then E^{+} := K_{𝔭}^{+}[X]/(f̃(X)) is the integral closure of K^{+} in the finite separable extension E := Frac(E^{+}) of K, and E^{+}/𝔭E^{+} ≃ k′, so that E^{+} is a valuation ring, by lemma 6.1.13. Furthermore, the preimage of k^{′+} in E^{+} is a valuation ring R of E with R ∩ K = K^{+} and R/𝔭R ≃ k^{′+}. From (i) and theorem 6.3.28(ii) we deduce that Ω_{R/K^{+}} = 0, whence Ω_{k^{′+}/k^{+}} = 0 as required.

Finally we show that (iv) implies (i). We distinguish several cases. The case when p := char(K) > 0 has already been dealt with, in view of remark 6.6.9. Next suppose that char(κ) = 0; we will adapt the argument given for the rank one case to prove corollary 6.3.18(i). As usual, we reduce to the case where K is strictly henselian; it suffices to show that Ω_{E^{+}/K^{+}} = 0 for every finite extension (E, | · |_{E}) of K. Then E factors as a tower of subextensions E_{0} := K ⊂ E_{1} ⊂ E_{2} ⊂ ... ⊂ E_{n} := E such that E_{i+1} = E_{i}[b_{i}^{1/l_{i}}] for every i = 0, ..., n − 1, where l_{i} := [E_{i+1} : E_{i}] is a prime number and b_{i} ∈ E_{i} such that |b_{i}| ∉ l_{i} · Γ_{E_{i}}. It is easy to see that assumption (iv) is inherited by every finite algebraic extension of K, hence we can reduce to the case E = K[b^{1/l}], with b := b_{1}, l := l_{1}. One verifies as in the proof of proposition 6.3.11(i) that E^{+} consists of the elements of the form ∑_{i=0}^{l−1} x_{i} · b^{i/l} such that x_{i} ∈ K and |x_{i} · b^{i/l}|_{E} ≤ 1 for every i = 0, ..., l − 1 and we have to show that d(x_{i} · b^{i/l}) = 0 for every i ≤ l − 1. We may assume that i > 0, and up to replacing b by b^{i} · x_{i}^{l}, we can obtain that b ∈ E^{+}; we have then to verify that db^{1/l} = 0. Define 𝔭(b), 𝔮(b) as in lemma 6.6.10, so that 𝔭(b) and 𝔮(b) are consecutive prime ideals, therefore (E^{+}/𝔭(b))_{𝔮(b)} is a deeply ramified rank one valuation ring; in particular, its value group is not discrete. Then, using (6.1.22)and example 6.1.21(v), we deduce that there exists an element c ∈ K^{+} such that |b| > |c^{l+1}| and |b| < |c^{l}|. We can write b = x · c^{l} for some x ∈ K^{+}, whence db^{1/l} = c · dx^{1/l}. However, |c| ≤ |b|^{(l−1)/l(l+1)} ≤ |a^{1/l} · b^{−1}|^{l−1} = |x|^{(l−1)/l}. Since x^{(l−1)/l} · dx^{1/l} = 0, the claim follows. Finally, suppose that p := char(κ) > 0 and char(K) = 0. Arguing as in the previous case, we produce an element b ∈ K^{+} such that |b^{p}| > |p| and |b^{p+1}| < |p|. The Frobenius map is surjective on K^{+}/b^{p}K^{+} by assumption, and on K^{s+}/b^{p}K^{s+} by example 6.1.21(iii), hence Ω_{K^{s+}/K^{+}} ≃ Ω_{K^{s+}[1/p]/K^{+}[1/p]}, by lemma 6.5.9(ii). Now it suffices to remark that K^{+}[1/p] is a valuation ring with residue field of characteristic zero, so we are reduced to the previous case, and the proof is concluded. □

Remark 6.6.14. By inspection of the proof, it is easy to check that condition (ii) of theorem 6.6.11 is equivalent to the following. There exists a subset S ⊂ K^{+} \ {0} such that the convex subgroup generated by |S| := {|s| | s ∈ S} equals Γ_{K} and H_{i}(𝕃_{E^{+}/K^{+}} ⊗_{K^{+}} K^{+}/s · K^{+}) = 0 for every valued field extension (E, | · |_{E}) of (K, | · |), every s ∈ S and every i > 0.

7. Analytic geometry

In this final chapter we bring into the picture p-adic analytic geometry and formal schemes. The first three sections develop a theory of the analytic cotangent complex : we show how to attach a complex 𝕃_{X/Y}^{an} to any morphism of locally finite type ϕ : X → Y of formal schemes or of R.Huber’s adic spaces. This complex is obtained via derived completion from the usual cotangent complex of the morphism of ringed spaces underlying ϕ. We prove that 𝕃_{X/Y}^{an} controls the analytic deformation theory of the morphism ϕ, in the same way as the usual cotangent complex computes the deformations of the map of ringed spaces underlying ϕ. We hope the reader will agree with us that these sections - though largely independent from the rest of the monograph - are not misplaced, in view of the prominence of the cotangent complex construction throughout our work. Some of what we do here had been already anticipated in an appendix of André’s treatise [2, Suppl.(c)].

The main result of the remaining sections 7.4 and 7.5 is a kind of weak purity statement valid for affinoid varieties over a deeply ramified valued field of rank one (theorem 7.4.17). The occurence of analytic geometry in purity issues (and in p-adic Hodge theory at large) is rather natural; indeed, the literature on the subject is littered with indications of the relevance of analytic varieties, and already in [64], Tate explicitly asked for a p-adic Hodge theory in the framework of rigid varieties. We elect instead to use the language of adic spaces, introduced by R.Huber in [44]. Adic spaces are generalizations of Zariski-Riemann spaces, that had already made a few cameo appearences in earlier works on rigid analytic geometry. Recall that Zariski-Riemann spaces were introduced originally by Zariski in his quest for the resolution of singularities of algebraic varieties. Zariski’s idea was to attach to any (singular, reduced and irreducible) variety X defined over a field k, the ringed space (X̃, 𝒪X̃) defined as the projective limit of the cofiltered system of all blow-up maps X_{α} → X; so a point of X̃ is a compatible system x̃ := (x_{α} ∈ X_{α} | X_{α} → X). It is easy to verify that, for any such x̃, the stalk 𝒪_{X̃,x̃} is a valuation ring dominating the image of x̃ in X. The strategy to construct a regular model for X was broken up in two stages : first one sought to show that for any point x̃ ∈ X̃ one can find some x_{α} under x̃ which is non-singular in the blow-up X_{α}. This first stage goes under the name of local uniformization; translated in algebraic terms, this means that for every valuation ring v of the field k(X) of rational functions on X, there is a model of k(X) on which the center of v is a non-singular point (a model is a reduced irreducible k-scheme Y of finite type such that k(Y) = k(X)). Local uniformization and the quasi-compactness of the Zariski-Riemann space imply that there is a “finite resolving system”, i.e. a finite number of models such that every valuation of k(X) has a non-singular center on one of them (notice that the local uniformization of a point x_{α} “spreads around” to an open neighborhood U_{α} of x_{α} in X_{α}, so we achieve uniformization not just for x̃, but for all the valuations contained in the preimage Ũ of U_{α}, which is open in X̃).

The second step is to try to reduce the number of models in a finite resolving system; restricting to open subvarieties one reduces this to the question of going from a resolving system of cardinality two to a non-singular model. This is what Zariski called the “fundamental theorem” (see [68, §2.5]). Zariski showed local uniformization for all valuation rings containing the field ℚ, but for the fundamental theorem he could only find proofs in dimensions ≤ 3.

Our theorem 7.4.17 is directly inspired by Zariski’s strategy : rather than looking at the singularities of a variety, we try to resolve the singularities of an étale covering Y → X of smooth affinoid adic spaces over a deeply ramified non-archimedean field K (so the map is singular only on the special fibre of a given integral model defined on the valuation ring of K). We assume that X admits generically étale coordinates t_{1}, ..., t_{d} ∈ 𝒪_{X}(X), and the role of Zariski’s X̃ is played by the projective system of all finite coverings of the form X_{n} := X[t_{1}^{1/n}, ..., t_{d}^{1/n}], where n ranges over the positive integers. We have to show that the étale covering Y_{n} := X_{n} ×_{X} Y → X_{n} becomes “less and less” singular (that is, on its special fibre) as n grows; so we need a numerical invariant that quantifies the singularity of the covering : this is the discriminant 𝔡_{Y/X}. The analogue of local uniformization is our proposition 7.4.13, whose proof uses the results of section 6.6. Next, in order to exploit the quasi-compactness of the affinoid adic spaces Y_{n}, we have to show that the estimates furnished by proposition 7.4.13 “spread around” in an open subset; to this aim we prove that the discriminant function is semicontinuous : this is the purpose of section 7.5. Notice that in our case we do not need an analogue of Zariski’s difficult fundamental theorem; this is because the discriminant function always decreases when n grows : what makes the problem of resolution of singularities much harder is that a non-singular point x_{α} on a blow up model X_{α} may be dominated by singular points x_{β} on some further blow up X_{β} → X_{α} (so, for instance one cannot trivially reduce the cardinality of a resolving system by forming joins of the various models of the system). Besides, our spaces are not varieties, but adic spaces, i.e. morally we work only “at the level of the Zariski-Riemann space” and we do not need - as for Zariski’s problem - to descend to (integral) models of our spaces; so we are dealing exclusively with valuation rings (or mild extensions thereof), rather than more complicated local rings.

In essence, this is the complete outline of the method; however, our theorem is - alas - much weaker than Faltings’ and does not yield by itself the kind of Galois cohomology vanishings that are required to deduce comparison theorems for the cohomology of algebraic varieties; we explain more precisely the current status of the question in (7.5.27).

Throughout this chapter we fix a valued field (K, | · |) with valuation of rank one, complete for its valuation topology. As usual, 𝔪 denotes the maximal ideal of K^{+}. We also let a be a topologically nilpotent element in K^{×}.

7.1. Derived completion functor. Let A be a complete K^{+}-algebra of topologically finite presentation. For any A-module M, we denote by M^{∧} the (separated) a-adic completion of M. Proposition 7.1.1. Let A be as in (7.1).

(i) Every finitely generated A-module which is torsion-free as a K^{+}-module, is finitely presented.

(ii) A is a coherent ring.

(iii) Let N be a finitely presented A-module, N′ ⊂ N a submodule. Then there exists an integer c ≥ 0 such that

(7.1.2)  a^{k}N ∩ N′ ⊂ a^{k−c}N′

for every k ≥ c. In particular, the topology on N′ induced by the a-adic topology on N, agrees with the a-adic topology of N′.

(iv) Every finitely generated A-module is a-adically complete and separated.

(v) Every submodule of a free A-module F of finite type is closed for the a-adic topology of F.

(vi) Every A-algebra of topologically finite type is separated.

Proof. (i) is an easy consequence of [13, Lemma 1.2]. To show (ii), one chooses a presentation A := K^{+}⟨T_{1}, ..., T_{n}⟩/I for some finitely generated ideal I, and then reduces to prove the statement for K^{+}⟨T_{1}, ..., T_{n}⟩, in which case it follows from (i). Next, let N, N′ be as in (iii) and define T to be the K^{+}-torsion submodule of N′′ := N/N′; clearly T is an Asubmodule, and the A-module N′′/T is K^{+}-torsion-free, therefore is finitely presented by (i). Since N is finitely generated, this implies that M := Ker(N → N′′/T) is finitely generated. Hence, there exists an integer c ≥ 0 such that a^{c}M ⊂ N′. If now k ≥ c, we have a^{k}N ∩ N′ ⊂ a^{k}N ∩ M = a^{k}M ⊂ a^{k−c}N′, which shows (iii). Next let us show:

Claim 7.1.3. Assertion (iv) holds for every finitely presented A-module. Proof of the claim: Let N be a finitely presented A-module and choose a presentation 0 → K → A^{n} → N → 0. By (ii), K is again finitely presented, and by (iii), the topology on K induced by the a-adic topology on A^{n} coincides with the a-adic topology of K. Hence, after taking a-adic completion, we obtain a short exact sequence : 0 → K^{∧} → A^{n} → N^{∧} → 0 (see [54, Th.8.1]). It follows that the natural map K → K^{∧} is injective, which shows that the map N → N^{∧} is surjective for every finitely presented A-module N. In particular, this holds for K, whence K ≃ K^{∧}, and N ≃ N^{∧}, as claimed.

Finally, let M be a submodule of A^{n}. By (iii), the topology on M induced by A^{n} coincides with the a-adic topology. Consequently, if M is finitely presented, then M is complete for the a-adic topology by claim 7.1.3, hence complete as a subspace of A^{n}, hence closed in A^{n}. For an arbitrary M, define \overline{M} := ⋃_{n>0}(M : a^{n}); then \overline{M} is a submodule of A^{n} and A/\overline{M} is torsionfree as a K^{+}-module, so it is finitely presented by (i), therefore \overline{M} is finitely presented by (ii). It follows that a^{c}\overline{M} ⊂ M for some c ≥ 0, whence a^{k}A^{n} ∩ \overline{M} = a^{k}A^{n} ∩ M for every k ≥ c. By the foregoing, \overline{M} is complete, so a^{k}A^{n} ∩ \overline{M} is also complete, and finally M is complete, hence closed. This settles (v) and (iv) follows as well. (vi) is an immediate consequence of (v). □

Lemma 7.1.4. Let A → B be a map of K^{+}-algebras of topologically finite presentation. Then B is of topologically finite presentation as an A-algebra. More precisely, if ϕ : A⟨T_{1}, ..., T_{n}⟩ → B is any surjective map, Ker ϕ is finitely generated.

Proof. By proposition 7.1.1(vi), B is complete and separated, hence we can find a surjective map ϕ : A⟨T_{1}, ..., T_{n}⟩ → B. It remains to show that Ker ϕ is finitely generated for any such ϕ. We can write A := K^{+}⟨T_{n+1}, ..., T_{m}⟩/I for some finitely generated ideal I, and thus reduce to the case where A = K^{+} and ϕ : K^{+}⟨T_{1}, ..., T_{n}⟩ → B. We will need the following : Claim 7.1.5. Let α : K^{+}⟨Y_{1}, ..., Y_{r+s}⟩ → B be a surjective map and β : K^{+}⟨Y_{1}, ..., Y_{r}⟩ → K^{+}⟨Y_{1}, ..., Y_{r+s}⟩ the natural imbedding. Suppose that γ := α ∘ β is surjective as well. Then Ker α is finitely generated if and only if Ker γ is finitely generated. Proof of the claim: For i = r + 1, ..., r + s, choose f_{i} ∈ K^{+}⟨Y_{1}, ..., Y_{r}⟩ such that γ(f_{i}) = α(Y_{i}). We define a surjective map δ : K^{+}⟨Y_{1}, ..., Y_{r+s}⟩ → K^{+}⟨Y_{1}, ..., Y_{r}⟩ by setting δ(Y_{i}) := Y_{i} for i ≤ r and δ(Y_{i}) := f_{i} for i > r. Clearly γ ∘ δ = α. There follows a short exact sequence 0 → Ker δ → Ker α → Ker γ → 0. However, Ker δ is the closure of the ideal I generated by Y_{i} −f_{i} for i = r +1, ..., r +s. By proposition 7.1.1(v), we deduce that Ker δ = I, and the claim follows easily.

By hypothesis there is at least one surjection ψ : K^{+}⟨Y_{1}, ..., Y_{r}⟩ → B with finitely generated kernel. Let μ : B ˆ⊗_{K^{+}}B → B be the multiplication map and set θ := μ ∘ (ϕˆ⊗_{K^{+}}ψ) : K^{+}⟨T_{1}, ..., T_{n}, X_{1}, ..., X_{k}⟩ → B. Applying twice claim 7.1 we deduce first that Ker θ is finitely generated, and then that Ker ϕ is too, as required. □

Lemma 7.1.6. Let F be a flat A-module. Then:

(i) F^{∧} is a flat A-module.

(ii) For every finitely presented A-module M, the natural map

(7.1.7)  M ⊗_{A} F^{∧} → (M ⊗_{A} F)^{∧}

is an isomorphism.

Proof. To begin with, we claim that the functor N ↦ (N ⊗_{A} F)^{∧} is exact on the abelian category of finitely presented A-modules. Indeed, let \underline{E} := (0 → N′ → N → N′′ → 0) be an exact sequence of finitely presented A-modules; we have to show that (\underline{E} ⊗_{A} F)^{∧} is still exact. Obviously \underline{E} ⊗_{A} F is exact, so the assertion will follow by [54, Th.8.1(ii)], once we know:

Claim 7.1.8. The topology on N′ ⊗_{A} F induced by the imbedding into N ⊗_{A} F agrees with the a-adic topology. Proof of the claim: By proposition 7.1.1(iii), we can find c ≥ 0 such that (7.1.2)holds. Since F is flat, we derive

a^{k}(N ⊗_{A} F) ∩ (N′ ⊗_{A} F) ⊂ a^{k−c}(N′ ⊗_{A} F)

which implies the claim.

(ii): clearly (7.1.7)is an isomorphism in case M is a free module of finite type. For a general M, one chooses a resolution \underline{R} := (A^{n} → A^{m} → M → 0); by the foregoing, the sequence (\underline{R} ⊗_{A} F)^{∧} is still exact, so one concludes by applying the 5-lemma to the map of complexes \underline{R} ⊗_{A} F^{∧} → (\underline{R} ⊗_{A} F)^{∧}.

(i): we have to show that, for every injective map of A-modules f : N′ → N, f ⊗_{A} 1_{F^{∧}} is still injective. By the usual reductions, we can assume that both N and N′ are finitely presented. In view of (ii), this is equivalent to showing that the induced map (N′ ⊗_{A} F)^{∧} → (N ⊗_{A} F)^{∧} is injective, which is already known. □

7.1.9. We will need to consider the left derived functor of the a-adic completion functor, which we denote:

(7.1.10)  D^{−}(A-Mod) → D^{−}(A-Mod) : (K^{•}) ↦ (K^{•})^{∧}.

As usual, it can be defined by completing termwise bounded above complexes of projective A-modules. However, the following lemma shows that it can also be computed by arbitrary flat resolutions. Lemma 7.1.11. Let ϕ : K_{1}^{•} → K_{2}^{•} be a quasi-isomorphism of bounded above complexes of flat A-modules and denote by (K_{i}^{•})^{∧} the termwise a-adic completion of K_{i}^{•} (i = 1, 2). Then the induced morphism

(7.1.12)  (K_{1}^{•})^{∧} → (K_{2}^{•})^{∧}

is a quasi-isomorphism.

Proof. Since K_{1}^{•} and K_{2}^{•} are termwise flat, we deduce quasi-isomorphisms

ϕ_{n} : K_{1,n}^{•} := K_{1}^{•} ⊗_{A} A/a^{n}A → K_{2,n} := K_{2}^{•} ⊗_{A} A/a^{n}A

for every n ∈ ℕ. The map of inverse system of complexes (K_{1,n}^{•})_{n∈ℕ} → (K_{2,n}^{•})_{n∈ℕ} can be viewed as a morphism of complexes of objects of the abelian category (A-Mod)^{ℕ} of inverse systems of A-modules. As such, it induces a morphism (ϕ_{n})_{n∈ℕ} in the derived category D((A-Mod)^{ℕ}), and it is clear that (ϕ_{n})_{n∈ℕ} is a quasi-isomorphism. Let

R lim : D((A-Mod)^{ℕ}) → D(A-Mod)

be the right derived functor of the inverse limit functor lim : (A-Mod)^{ℕ} → A-Mod. We remark that, for every j ∈ ℤ, the inverse systems (K_{i,n}^{j})_{n∈ℕ} (i = 1, 2) are acyclic for the functor lim, since their transition maps are surjective. We derive that R lim(K_{i,n}^{•})_{n∈ℕ} ≃ (K_{i}^{•})^{∧}, and, under this identification, the morphism (7.1.12) is the same as R lim(ϕ_{n})_{n∈ℕ}. Since the latter preserves quasi-isomorphisms, the claim follows. □

7.1.13. We denote by D^{−}(A-Mod)^{∧} the essential image of the functor (7.1.10). Corollary 7.1.14. (i) For any object K^{•} of D^{−}(A-Mod), the natural morphism

(K^{•})^{∧} → ((K^{•})^{∧})^{∧}

is a quasi-isomorphism.

(ii) D^{−}(A-Mod)^{∧} is a full triangulated subcategory of D^{−}(A-Mod).

Proof. Notice first that there are two natural morphisms as in (i), which coincide : namely, for any complex E in D^{−}(A-Mod) one has a natural morphism u_{E} : E → E^{∧}; then one can take either (u_{K})^{∧} or u_{K^{∧}}. Now, (i) is an immediate consequence of lemmata 7.1.11 and 7.1.6. Clearly D^{−}(A-Mod)^{∧} is preserved by shift and by taking cones of arbitrary morphisms; furthermore, it follows from (i) that it is a full subcategory of D^{−}(A-Mod). □

We will need some generalities on pseudo-coherent complexes of R-modules (for an arbitrary ring R), which we borrow from [11, Exp.I]. In our situation, the definitions can be simplified somewhat, since we are only concerned with sheaves over the one-point site that are pseudocoherent relative to the subcategory of free A-modules of finite type.

7.1.15. For given n ∈ ℤ, one says that a complex K^{•} of R-modules is n-pseudo-coherent if there exists a quasi-isomorphism E^{•} → K^{•} where E^{•} is a complex bounded above such that E^{i} is a free R-module of finite type for every i ≥ n. One says that K^{•} is pseudo-coherent if it is n-pseudo-coherent for every n ∈ ℤ.

7.1.16. Let K^{•} be a n-pseudo-coherent (resp. pseudo-coherent) complex of R-modules, and F^{•} → K^{•} a quasi-isomorphism. Then F^{•} is n-pseudo-coherent (resp. pseudo-coherent) ([11, Exp.I, Prop.2.2(b)]). It follows that that the pseudo-coherent complexes form a (full) subcategory D(R-Mod)_{coh} of D(R-Mod).

7.1.17. Furthermore, let X → Y → Z → X[1] be a distinguished triangle in D^{−}(R-Mod). If X and Z are n-pseudo-coherent (resp. pseudo-coherent), then the same holds for Y ([11, Exp.I, Prop.2.5(b)]). Lemma 7.1.18. Let n, p ∈ ℕ, K^{•} a n-pseudo-coherent complexes in D^{≤0}(R-Mod), and ℱ_{p} one of the functors ⊗_{R}^{p}, Sym_{R}^{p}, Λ_{R}^{p}, Γ_{R}^{p} defined in [46, I.4.2.2.6]. Then Lℱ_{p}(K^{•}) is an npseudo-coherent complex.

Proof. It is well known that ℱ_{p} sends free R-modules of finite type to free R-modules of finite type. It follows easily that the assertion of the lemma can be checked by inspecting the definition of the unnormalized chain complex associated to a simplicial complex, and of the simplicial complex associated to a chain complex via the Dold-Kan correspondence. We omit the details.

□

7.1.19. Let K^{•} be a pseudo-coherent complex. By ([11, Exp.I, Prop.2.7]) there exists a quasiisomorphism E^{•} → K^{•} where E^{•} is a bounded above complex of free R-modules of finite type.

7.1.20. Suppose now that R is coherent; then we deduce easily that a complex K^{•} of Rmodules is pseudo-coherent if and only if H^{i}(K^{•}) is a coherent R-module for every i ∈ ℤ and H^{i}(K^{•}) = 0 for every sufficiently large i ∈ ℤ ([11, Exp.I, Cor.3.5]). By proposition 7.1.1(iv) it follows also that D^{−}(A-Mod)_{coh} ⊂ D^{−}(A-Mod)^{∧} for every K^{+}-algebra A as in (7.1).

7.1.21. Let A be as in (7.1)and M an A-module of finite presentation. We denote by M[0] the complex consisting of the module M placed in degree zero. Any finite presentation of M can be extended to a quasi-isomorphism E^{•} → M[0], where E^{•} is a complex of free A-modules of finite type and E^{i} = 0 for i > 0 ([11, Exp.I, Cor.3.5(a)]). Together with proposition 7.1.1(iv), it follows easily that the natural morphism M[0] → M[0]^{∧} is a quasi-isomorphism. Lemma 7.1.22. Let A → B be a map of complete K^{+}-algebras of topologically finite presentation. Then, for every object K^{•} of D^{−}(A-Mod), the natural morphism

(K^{•} ⊗^{L}_{A} B)^{∧} → (K^{•∧} ⊗^{L}_{A} B)^{∧}

is a quasi-isomorphism.

Proof. We can suppose that K^{•} is a complex of free A-modules. Then we are reduced to showing that, for every free A-module F, the natural map (F ⊗_{A} B)^{∧} → (F^{∧} ⊗_{A} B)^{∧} is an isomorphism. We leave this task to the reader. □

Definition 7.1.23. Let ϕ : A → B be a map of complete K^{+}-algebras of topologically finite presentation. The B-module of analytic differentials relative to ϕ is defined as Ω_{B/A}^{an} := Ω_{B/A}^{∧}. The analytic cotangent complex of ϕ is the complex 𝕃_{B/A}^{an} := (𝕃_{B/A})^{∧}. Directly on the definition we derive a natural isomorphism

(7.1.24)  H_{0}(𝕃_{B/A}^{an}) ≃ Ω_{B/A}^{an}

Notice that 𝕃_{B/A}^{an} is defined here via the standard resolution P_{A}(B) → B, and it is therefore well defined as a complex of B-modules, not just as an object in the derived category D^{−}(B-Mod)^{∧}. This will be essential in order to globalize the construction to formal schemes, in (7.2.2), and to adic spaces, in definition 7.2.31.

The following lemma will be useful in section 7.4. Lemma 7.1.25. Let A → B be a continuous map of K^{+}-algebras of topologically finite presentation. The natural map ϕ_{B/A} : Ω_{B/A} → Ω_{B/A}^{an} is surjective with K^{+}-divisible kernel.

Proof. One writes B = B_{0}/I with B_{0} := A⟨T_{1}, ..., T_{n}⟩ and I ⊂ B_{0}. Directly from the construction of Ω_{B_{0}/A}^{an} one checks that ϕ_{B_{0}/A} is onto. Then, a little diagram chasing shows that ϕ_{B/A} is onto as well, and yields a surjective map B ⊗_{B_{0}} Ker ϕ_{B_{0}/A} → Ker ϕ_{B/A}. This allows to reduce to the case where B = B_{0}. In this case, Ker ϕ_{B/A} is generated by the terms of the form δ(f) := df − ∑_{i=1}^{n}(∂f/∂T_{i}) · dT_{i}, where f ranges over all the elements of B_{0}. For given f ∈ B_{0}, we can write f = f_{0} + af_{1}, with f_{0} ∈ A[T_{1}, ..., T_{n}], f_{1} ∈ B_{0}. It follows easily that δ(f) = δ(af_{1}) = a · δ(f_{1}), whence the claim. □

Remark 7.1.26. In view of (7.1.24), lemma 7.1.25 is also implied by the following more general

observation. Let K_{B/A} := Cone(ψ_{B/A} : 𝕃_{B/A} → 𝕃_{B/A}^{an})[1]; one has: K_{B/A} ⊗^{L}_{B} B/aB ≃ 0.

Indeed, directly on the definition of 𝕃_{B/A}^{an} one sees that ψ_{B/A} ⊗^{L}_{B} 1_{B/aB} is an isomorphism. Proposition 7.1.27. Let ϕ : A → B be a map of complete K^{+}-algebras of topologically finite presentation, and suppose that ϕ is formally smooth for the a-adic topology. Then there is a natural quasi-isomorphism

𝕃_{B/A}^{an} ≃ Ω_{B/A}^{an}[0].

Proof. For every n ∈ ℕ, set A_{n} := A/a^{n} · A and B_{n} := B/a^{n} · B. The hypothesis on ϕ implies that ϕ_{n} := ϕ ⊗_{A} 1_{A_{n}} is of finite presentation and formally smooth for the discrete topology, therefore

(7.1.28)  𝕃_{B_{n}/A_{n}} ≃ Ω_{B_{n}/A_{n}}[0] ≃ Ω_{B/A} ⊗_{A} A_{n}[0]

for every n ∈ ℕ. Moreover, ϕ is flat by [13, Lemma 1.6], hence 𝕃_{B_{n}/A_{n}} ≃ 𝕃_{B/A} ⊗_{A} A_{n}. On the other hand, for every i ∈ ℤ there is a short exact sequence (cp. [67, Th.3.5.8])

0 → lim_{n∈ℕ}^{1}H^{i−1}(𝕃_{B/A} ⊗_{A} A_{n}) → H^{i}(𝕃_{B/A}^{an}) → lim_{n∈ℕ}H^{i}(𝕃_{B/A} ⊗_{A} A_{n}) → 0.

In view of (7.1.28), the inverse system (H^{i−1}(𝕃_{B/A} ⊗_{A} A_{n}))_{n∈ℕ} vanishes for i ≠ 1 and has surjective transition maps for i = 1, hence its lim^{1} vanishes for every i ∈ ℤ, and the claim follows easily. □

Proposition 7.1.29. Let ϕ : A → B be a surjective map of complete K^{+}-algebras of topologically finite presentation. Then 𝕃_{B/A} is a pseudo-coherent complex, in particular it lies in D^{−}(B-Mod)^{∧} and 𝕃_{B/A} ≃ 𝕃_{B/A}^{an}.

Proof. First of all, notice that by lemma 7.1.4, B is of finite presentation, hence it is coherent as an A-module. Let P := P_{A}(B) be the standard simplicial resolution of B by free A-algebras. We obtain a morphism of simplicial B-algebras ϕ : B ⊗_{A} P → B by tensoring with B the augmentation P → B (here B is regarded as a constant simplicial algebra). By the foregoing, P is pseudo-coherent, hence P ⊗_{A} B lies in D(B-Mod)_{coh}. Let J := Ker ϕ. The short exact sequence of complexes 0 → J → P ⊗_{A} B → B → 0 is split, therefore J is also pseudocoherent. Recall that we have natural isomorphisms: J^{i}/J^{i+1} →^{∼} Sym_{B}^{i}(𝕃_{B/A}) for every i ∈ ℕ (where J^{0} := B ⊗_{A} P and Sym_{B}^{0}(𝕃_{B/A}) := B) ([46, Ch.III,§3.3]). Furthermore, we have (see loc.cit.) :

(7.1.30)  H_{n}(J^{i}) = 0 for every n, i ∈ ℕ such that i > n.

We prove by induction on n that 𝕃_{B/A} is n-pseudo-coherent for every n ≤ 1. If n = 1 there is nothing to prove. Suppose that the claim is known for the integer n. It then follows by lemma 7.1.18 that J^{i}/J^{i+1} is n-pseudo-coherent for every i > 0. However, it follows from (7.1.30) that J^{i} is n-pseudo-coherent as soon as i > −n. Hence, by (7.1.17)(and an easy induction), we deduce that J^{i} is n-pseudo-coherent for every i ∈ ℕ. Hence J^{2}[1] is (n − 1)-pseudo-coherent; if we now apply (7.1.17) to the distinguished triangle J → 𝕃_{B/A} → J^{2}[1] → J[1], we deduce that 𝕃_{B/A} is (n − 1)-pseudo-coherent. □

Theorem 7.1.31. Let A → B → C be maps of complete K^{+}-algebras of topologically finite presentation. Then:

(i) 𝕃_{B/A}^{an} lies in D^{−}(B-Mod)_{coh}.

(ii) There is a natural distinguished triangle in D^{−}(C-Mod) :

(7.1.32)  C ⊗_{B} 𝕃_{B/A}^{an} → 𝕃_{C/A}^{an} → 𝕃_{C/B}^{an} → C ⊗_{B} 𝕃_{B/A}^{an}[1].

Proof. (i): by lemma 7.1.4 we can find a surjection B_{0} := A⟨T_{1}, ..., T_{n}⟩ → B from a topologically free A-algebra onto B. If we apply transitivity to the sequence of maps A → B_{0} → B and take the (derived) completion of the resulting distinguished triangle, we end up with the triangle:

(B ⊗_{B_{0}} 𝕃_{B_{0}/A})^{∧} → 𝕃_{B/A}^{an} → 𝕃_{B/B_{0}}^{an} → (B ⊗_{B_{0}} 𝕃_{B_{0}/A})^{∧}[1].

We know already from proposition 7.1.29 that 𝕃_{B/B_{0}} is pseudo-coherent, hence it coincides with 𝕃_{B/B_{0}}^{an}. Lemma 7.1.22 yields a quasi-isomorphism: (B ⊗_{B_{0}} 𝕃_{B_{0}/A})^{∧} →^{∼} (B ⊗_{B_{0}} 𝕃_{B_{0}/A}^{an})^{∧}; in view of proposition 7.1.27, 𝕃_{B_{0}/A}^{an} is a free B_{0}-module of finite rank in degree zero, in particular it is pseudo-coherent, so the same holds for (B ⊗_{B_{0}} 𝕃_{B_{0}/A})^{∧}, and taking into account (7.1.17), the claim follows.

(ii): if we apply transitivity to the sequence of maps A → B → C, and then we complete the distinguished triangle thus obtained, we obtain (7.1.32), except that the first term is replaced by (C ⊗_{B} 𝕃_{B/A})^{∧}, which we can also write as (C ⊗_{B} 𝕃_{B/A}^{an})^{∧}, in view of lemma 7.1.22. However, by (i), 𝕃_{B/A}^{an} is pseudo-coherent, so it remains such after tensoring by C; in particular C⊗_{B}𝕃_{B/A}^{an} is already complete, and the claim follows. □

7.2. Cotangent complex for formal schemes and adic spaces. In this section we show how to globalize the definition of the analytic cotangent complex introduced in section 7.1. We consider two kinds of globalization : first we define the cotangent complex of a morphism 𝔣 : 𝔛 → 𝔜 of formal schemes locally of finite presentation over Spf K^{+}; then we will define the cotangent complex for a morphism of adic spaces locally of finite type over Spa(K, K^{+}). Lemma 7.2.1. Let 𝔛 := SpfA be an affine formal scheme finitely presented over Spf K^{+}. For every f ∈ A, let 𝔇(f) := {x ∈ 𝔛 | f ∉ 𝔪_{x}}. The natural map A → Γ(𝔇(f), 𝒪_{𝔛}) is flat.

Proof. Since Γ(𝔇(f), 𝒪_{𝔛}) is the a-adic completion of A_{f}, the lemma follows from lemma 7.1.6(i). □

7.2.2. Let 𝔣 : 𝔛 → 𝔜 be a morphism of formal schemes locally of finite presentation over Spf K^{+}, and suppose that 𝔜 is separated. For every affine open subset U ⊂ 𝔛, the small category F_{U} of all affine open subsets V ⊂ 𝔜 with 𝔣(U) ⊂ V, is cofiltered under inclusion (or else it is empty). For every V ∈ F_{U}, 𝒪_{𝔜}(V) is a K^{+}-algebra of topologically finite presentation, hence the induced morphism 𝒪_{𝔜}(V) → 𝒪_{𝔛}(U) is of the kind considered in definition 7.1.23. We set

L(U/𝔜) := colim_{V ∈F_{U}^{o}} 𝕃_{𝒪_{𝔛}(U)/𝒪_{𝔜}(V)}^{an} .

Definition 7.2.3. The mapping U ↦ L(U/𝔜) defines a complex of presheaves on a cofinal family of affine open subsets of 𝔛. By applying degreewise the construction of [26, Ch.0, §3.2.1], we can extend the latter to a complex of presheaves of 𝒪_{𝔛}-modules on 𝔛. We define the analytic cotangent complex 𝕃_{𝔛/𝔜}^{an} of the morphism 𝔣 : 𝔛 → 𝔜 as the complex of sheaves associated to this complex of presheaves (this means that we form degreewise the associated sheaf, and we consider the resulting complex).

7.2.4. More generally, if 𝔜 is not necessarily separated, we can choose an affinoid covering 𝔜 = ⋃_{i∈I} 𝔘_{i} and the construction above applies to the restrictions 𝔙_{i} := 𝔣^{−1}(𝔘_{i}) → 𝔘_{i}; since the definition of 𝕃_{𝔙_{i}/𝔘_{i}}^{an} is local on 𝔙_{i}, one can then glue them into a single cotangent complex 𝕃_{𝔛/𝔜}^{an}. Lemma 7.2.5. Let 𝕃_{𝔛/𝔜} denote the (usual) cotangent complex of the morphism

(f, f^{♯}) : (𝔛, 𝒪_{𝔛}) → (𝔜, 𝒪_{𝔜})

of ringed spaces; there exists a natural map of complexes

(7.2.6)  𝕃_{𝔛/𝔜} → 𝕃_{𝔛/𝔜}^{an}

inducing an isomorphism

(7.2.7)  𝕃_{𝔛/𝔜} ⊗_{𝒪_{𝔛}} 𝒪_{𝔛}/a^{n}𝒪_{𝔛} →^{∼} 𝕃_{𝔛/𝔜}^{an} ⊗_{𝒪_{𝔛}} 𝒪_{𝔛}/a^{n}𝒪_{𝔛}

for every n ∈ ℕ.

Proof. It suffices to construct (7.2.6) in case 𝔜 is affine. According to [46, Ch.II, (1.2.3.6)] and [46, Ch.II, (1.2.3.4)], the complex 𝕃_{𝔛/𝔜} is naturally isomorphic to the sheafification of the complex of presheaves defined by the rule:

U ↦ colim_{V ∈F_{U}^{o}} 𝕃_{𝒪_{𝔛}(U)/𝒪_{𝔜}(V)}

and then it is clear how to define 7.2.6. From the construction it is obvious that (7.2.7) is an isomorphism. □

It is occasionally important to know that both 𝕃_{𝔛/𝔜}^{an} and the morphism (7.2.6)are well defined in the category of complexes of 𝒪_{𝔛}-modules (not just in its derived category).

7.2.8. Furthermore, denote by Ω_{𝔛/𝔜}^{an} the sheaf of analytic relative differentials for the morphism 𝔣, which is defined as (ℐ /ℐ^{2})_{|𝔛}, where ℐ is the ideal defining the diagonal imbedding 𝔛 → 𝔛 ×_{𝔜} 𝔛. We see easily that there is a natural isomorphism

(7.2.9)  H_{0}(𝕃_{𝔛/𝔜}^{an}) ≃ Ω_{𝔛/𝔜}^{an}.

Proposition 7.2.10. Let 𝔣 : 𝔛 → 𝔜 be a morphism of formal schemes locally of finite type over Spf(K^{+}). Then:

(i) 𝕃_{𝔛/𝔜}^{an} is a pseudo-coherent complex of 𝒪_{𝔛}-modules.

(ii) If 𝔣 is a closed imbedding, then (7.2.6)is a quasi-isomorphism.

(iii) If 𝔣 is formally smooth, then (7.2.9)induces a quasi-isomorphism: 𝕃_{𝔛/𝔜}^{an} →^{∼} Ω_{𝔛/𝔜}^{an}[0].

Proof. (i): according to [11, Exp.I, Prop.2.1(b)], it suffices to show that H_{i}(𝕃_{𝔛/𝔜}^{an}) is a coherent sheaf of 𝒪_{𝔛}-modules for every i ∈ ℕ. To this aim, let U ⊂ 𝔛 be an affine open subset such that the family F_{U} (notation of (7.2.2)) is not empty; pick any V ∈ F_{U}. After replacing 𝔛 by U, we can suppose that U = 𝔛. Set A := Γ(𝔛, 𝒪_{𝔛}) and let L_{i}^{△} be the sheaf of coherent 𝒪_{𝔛}-modules associated to the coherent A-module L_{i} := H_{i}(𝕃_{𝒪_{𝔛}(𝔛)/𝒪_{𝔜}(V)}^{an} ) (cp. [26, Ch.I, §10.10.1], where this concept is discussed in the case of locally noetherian formal schemes). The assertion will be an immediate consequence of theorem 7.1.31(i) and the following :

Claim 7.2.11. There is a natural isomorphism of 𝒪_{𝔛}-modules: L_{i}^{△} →^{∼} H_{i}(𝕃_{𝔛/𝔜}^{an}).

Proof of the claim: By the definition of 𝕃_{𝔛/𝔜}^{an} we deduce a natural morphism of 𝒪_{𝔛}-modules: α : L_{i}^{△} → H_{i}(𝕃_{𝔛/𝔜}^{an}). It therefore suffices to show that α induces an isomorphism on the stalks. To this aim, we remark first that the natural map L_{i} → H_{i}(L(𝔛/𝔜)) is an isomorphism. Indeed, it suffices to consider another open subset V′ ∈ F_{𝔛} with V′ ⊂ V; we have 𝕃_{𝒪_{𝔜}(V′)/𝒪_{𝔜}(V)}^{an} ≃ 0 by proposition 7.1.27, and then it follows by transitivity (theorem 7.1.31(ii)) that the map L_{i} → H_{i}(𝕃_{𝒪_{𝔛}(𝔛)/𝒪_{𝔜}(V′)}^{an} ) is an isomorphism. More generally, this argument shows that, for every affine open subset U′ ⊂ 𝔛, the natural map H_{i}(𝕃_{𝒪_{𝔛}(U′)/𝒪_{𝔜}(V)}^{an} ) → H_{i}(L(U′/𝔜)) is an isomorphism. However, on one hand we have (L_{i}^{△})_{x} ≃ L_{i} ⊗_{A} 𝒪_{𝔛,x}. On the other hand, we have ([26, Ch.0, §3.2.4]) :

(7.2.12)  H_{i}(𝕃_{𝔛/𝔜}^{an})_{x} ≃ colim_{x∈U′} H_{i}(L(U′/𝔜))

where the colimit ranges over the set S of all affine open neighborhoods of x in 𝔛. We can replace S by the cofinal subset of all open neighborhoods of the form 𝔇(f) (for f ∈ A such that f ∉ 𝔪_{x}). Then, lemma 7.2.1, together with another easy application of transitivity allows to identify the right-hand side of (7.2.12)with H_{i}(L(𝔛/𝔜))⊗_{A}𝒪_{𝔛,x}, so (i) follows. (ii) and (iii) are immediate consequences of proposition 7.1.29 and respectively proposition 7.1.27. □

Proposition 7.2.13. Let 𝔛 →^{𝔣} 𝔜 →^{𝔤} ℨ be two morphisms of formal schemes locally of finite presentation over Spf K^{+}. There is a natural distinguished triangle in D^{−}(𝒪_{𝔛}-Mod)

(7.2.14)  L𝔣^{∗}𝕃_{𝔜/ℨ}^{an} → 𝕃_{𝔛/ℨ}^{an} → 𝕃_{𝔛/𝔜}^{an} → L𝔣^{∗}𝕃_{𝔜/ℨ}^{an}[1].

Proof. As explained in [46, Ch.II, §2.1], for every sequence of ring homomorphisms A → B → C, the transitivity triangle is induced by a functorial exact sequence of complexes 𝕃_{C/B/A} of flat C-modules (a ”true triangle” in loc. cit.). Suppose now that A, B, C are complete K^{+}-algebras of topologically finite type; upon a-adic completion, one deduces a true triangle 𝕃_{C/B/A}^{∧}. Then, to every sequence of affine open subsets U ⊂ 𝔛, V ⊂ 𝔜, W ⊂ ℨ such that 𝔣(U) ⊂ V and 𝔤(V) ⊂ W, one can associate the true triangle 𝕃_{𝒪_{𝔛}(U)/𝒪_{𝔜}(V)/𝒪_{ℨ}(W)}^{∧} ; Since the construction is functorial in all arguments, one derives a presheaf of true triangles on a cofinal family of open subsets of 𝔛, which we can then sheafify in the usual manner. The resulting distinguished triangle in D^{−}(𝒪_{𝔛}-Mod) gives rise to (7.2.14). □

In the following we wish to define the cotangent complex of a morphism of adic spaces (studied in [44] and [45]). For simplicity, we will restrict to adic spaces of topologically finite type over Spa(K, K^{+}), which suffice for our applications. For the convenience of the reader we recall a few basic definitions from [45].

7.2.15. An f-adic ring is a topological ring A that admits an open subring A_{0} such that the induced topology on A_{0} is pre-adic and defined by a finitely generated ideal I ⊂ A_{0}. As an example, every K-algebra of topologically finite type is an f-adic ring. A subring A_{0} with the above properties is called a ring of definition for A, and I is an ideal of definition. One denotes by A^{∘} the open subring of power-bounded elements of A.

7.2.16. Let A → B be complete f-adic rings and ϕ : A → B a ring homomorphism. One says that ϕ is of topologically finite type if there exist rings of definition A_{0} ⊂ A and B_{0} ⊂ B such that ϕ(A_{0}) ⊂ B_{0}, the restriction A_{0} → B_{0} factors through a quotient map (i.e. open and surjective) A_{0}⟨T_{1}, ..., T_{n}⟩ → B_{0} and B is finitely generated over A · B_{0}.

7.2.17. An affinoid ring is a pair A = (A^{▷}, A^{+}) consisting of an f-adic ring A^{▷} and a subring A^{+} ⊂ A^{▷} which is open, integrally closed in A^{▷} and contained in the subring A^{∘}. A^{+} is called the subring of integral elements of A.

7.2.18. The completion A^{∧} of an affinoid ring A = (A^{▷}, A^{+}) is the pair ((A^{▷})^{∧}, (A^{+})^{∧}) (it turns out that (A^{+})^{∧} is integrally closed in (A^{▷})^{∧}).

A homomorphism ϕ : (A^{▷}, A^{+}) → (B^{▷}, B^{+}) of affinoid rings is a ring homomorphism ϕ^{▷} : A^{▷} → B^{▷} such that ϕ(A^{+}) ⊂ B^{+}. One says that ϕ is of topologically finite type if ϕ^{▷} is of topologically finite type and there exists an open subring C ⊂ B^{+} such that B^{+} is the integral closure of C, ϕ(A^{+}) ⊂ C and the induced map A^{+} → C is of topologically finite type. (cp. (7.2.16)).

7.2.19. For instance, (K, K^{+}) is an affinoid ring, complete for its valuation topology; notice that in this case we have K^{+} = K^{∘}. A complete f-adic ring of topologically finite type over K is the same as a K-algebra of topologically finite type. Furthermore, suppose that A and B are complete f-adic rings of topologically finite type over K, and let f : A → B be a continuous ring homomorphism. Then there is a unique subring of integral elements B^{+} such that f : (A, A^{∘}) → (B, B^{+}) is a morphism of affinoid rings of topologically finite type; namely one must take B^{+} := B^{∘}. Especially, A^{+} := A^{∘} is the only ring of integral elements of A such that the affinoid ring (A, A^{+}) is of topologically finite type over (K, K^{+}) ([44, Prop.2.4.15]).

7.2.20. Given an arbitrary ring A, a valuation on A is a map | · | : A → Γ ∪ {0} where Γ is an ordered abelian group whose composition law we denote multiplicatively, and the ordering is extended to Γ ∪ {0} as usual. Then | · | is required to satisfy the usual conditions, namely: |x · y| = |x| · |y| and |x + y| ≤ max(|x|, |y|) for every x, y ∈ A, and |0| = 0, |1| = 1.

7.2.21. Now, let A be an f-adic ring, and | · | : A → Γ ∪ {0} a valuation on A. For every γ ∈ Γ, let U_{γ} := {α ∈ Γ | α < γ} ∪ {0}. We endow Γ ∪ {0} with the topology which restricts to the discrete topology on Γ, and which admits (U_{γ} | γ ∈ Γ) as a fundamental system of open neighborhoods of 0. We say that | · | is continuous if it is continuous with respect to the above topology on Γ ∪ {0}. One denotes by Cont(A) the set of all (equivalence classes of) continuous valuations on A. Given a, b ∈ A, let U(a/b) ⊂ Cont(A) be the subset of all valuations | · | such that |a| ≤ |b| ≠ 0. Cont(A) is endowed with the topology which admits the collection (U(a/b) | a, b ∈ A) as a sub-basis. With this topology, Cont(A) is a spectral topological space (see [45, 1.1.13] for the definition of spectral space). In particular, this implies that Cont(A) admits a basis of quasi-compact open subsets. Such a basis is provided by the rational subsets, defined as follows. A subset U ⊂ Cont(A) is called rational if there exist f_{1}, ..., f_{n}, g ∈ A such that the ideal J := f_{1}A + ... + f_{n}A is open in A and U consists of all | · | ∈ Cont(A) such that |f_{i}| ≤ |g| ≠ 0 for every i = 1, ..., n. (Notice that, since we have chosen to restrict to f-adic rings containing K, asking for J to be an open ideal is the same as requiring that J = A). Given f_{1}, ..., f_{n}, g ∈ A with the above property, we denote by R(f_{1}/g, ..., f_{n}/g) the corresponding rational subset.

7.2.22. If A := (A^{▷}, A^{+}), then one defines the subset Spa A := {| · | ∈ Cont(A^{▷}) | |a| ≤ 1 for every a ∈ A^{+}} ⊂ Cont(A^{▷}). Spa A, endowed with the subspace topology, is called the adic spectrum of the affinoid ring A. Spa A is a pro-constructible subset of Cont(A), hence it is a spectral space too. Any continuous map A → B of affinoid rings induces in the obvious way a continuous map on adic spectra: Spa B → Spa A.

7.2.23. For any affinoid ring A, one can endow X := Spa A with a presheaf 𝒪_{X} of topological rings, as follows. First of all, for any f_{1}, ..., f_{n}, g ∈ A^{▷} as in (7.2.21), one defines an affinoid ring A(f_{1}/g, ..., f_{n}/g), such that A(f_{1}/g, ..., f_{n}/g)^{▷} := (A^{▷})_{g} and A(f_{1}/g, ..., f_{n}/g)^{+} is the integral closure of the subring A[f_{1}/g, ..., f_{n}/g] in A(f_{1}/g, ..., f_{n}/g)^{▷}. If B ⊂ A^{▷} is a ring of definition and I ⊂ B an ideal of definition, let B(f_{1}/g, ..., f_{n}/g) be the subring of (A^{▷})_{g} generated by B and f_{1}/g ,..., f_{n}/g; we endow B(f_{1}/g, ..., f_{n}/g) with the pre-adic topology defined by the ideal I · B(f_{1}/g, ..., f_{n}/g); then the f-adic topology on A(f_{1}/g, ..., f_{n}/g)^{▷} is defined to be the unique ring topology for which B(f_{1}/g, ..., f_{n}/g) is a ring of definition. Next, let A⟨f_{1}/g, ..., f_{n}/g⟩ := A(f_{1}/g, ..., f_{n}/g)^{∧} (cp. (7.2.18)). With this preliminaries, one sets:

𝒪_{X}(R(f_{1}/g, ..., f_{n}/g)) := A⟨f_{1}/g, ..., f_{n}/g⟩^{▷}.

In this way, 𝒪_{X} is well defined on every rational subset. One can then extend the definition to an arbitrary open subset of Spa A, following [26, Ch.0, §3.2.1]. It is not difficult to check that, for every open subset U ⊂ Spa A, and every x ∈ U, any valuation | · |_{x} in the equivalence class x extends to the whole of 𝒪_{X}(U), hence to the stalk 𝒪_{X,x}. One denotes by 𝒪_{X}^{+} the sub-presheaf defined by the rule: 𝒪_{X}^{+}(U) := {f ∈ 𝒪_{X}(U) | |f|_{x} ≤ 1 for every x ∈ U}. In the cases of interest, the presheaf 𝒪_{X} is a sheaf (and 𝒪_{X}^{+} is therefore a subsheaf). In such cases, one can show that, for every rational subset R(f_{1}/g, ..., f_{n}/g), the natural map A⟨f_{1}/g, ..., f_{n}/g⟩^{+} → 𝒪_{X}^{+}(R(f_{1}/g, ..., f_{n}/g)) is an isomorphism of topological rings.

This holds notably when A^{▷} is a K-algebra of topologically finite type. One calls the datum (Spa A, 𝒪_{Spa A}, 𝒪_{Spa A}^{+}) an affinoid adic space. General adic spaces are obtained as usual, by gluing affinoids. Adic spaces form a category, whose morphisms f : X → Y are the morphisms of topologically locally ringed spaces (X, 𝒪_{X}) → (Y, 𝒪_{Y}) which induce morphisms of sheaves f^{∗}𝒪_{Y}^{+} → 𝒪_{X}^{+}.

7.2.24. Let f : X → Y be a morphism of adic spaces. One says that f is locally of finite type if for every x ∈ X there exist open affinoid subspaces U ⊂ X, V ⊂ Y such that x ∈ U, f(U) ⊂ V and the induced morphism of affinoid rings (𝒪_{Y} (V), 𝒪_{Y}^{+}(V)) → (𝒪_{X}(U), 𝒪_{X}^{+}(U)) is of topologically finite type.

7.2.25. A morphism f : X → Y between adic spaces (defined over Spa(K, K^{+})) is called smooth (resp. unramified, resp. étale) if f is locally of finite type and if, for any affinoid ring A, any ideal I of A^{▷} with I^{2} = {0} and any morphism Spa A → Y, the mapping Hom_{Y} (Spa A, X) → Hom_{Y} (Spa A/I, X) is surjective (resp. injective, resp. bijective).

7.2.26. In [45, §1.9] it is shown how to associate functorially to every formal scheme 𝔛 (say locally of finite presentation over Spf K^{+}) an adic space d(𝔛), together with a morphism of topologically ringed spaces λ : d(𝔛) → 𝔛, characterized by a certain universal property which we won’t spell out here, but that includes the condition that Im(𝒪_{𝔛} → λ_{∗}𝒪_{d(𝔛)}) ⊂ 𝒪_{d(𝔛)}^{+}. If 𝔛 = SpfA_{0} for a K^{+}-algebra A_{0} of topologically finite type, then d(𝔛) = Spa A, where A is the affinoid ring (A_{0} ⊗_{K^{+}} K, A^{+}), with A^{+} defined as the integral closure of the image of A_{0} in A_{0} ⊗_{K^{+}} K. Moreover, 𝔛 is quasi-compact if and only if d(𝔛) is.

7.2.27. Let 𝔛 be a formal scheme of finite presentation over Spf K^{+}. The collection 𝒞_{𝔛} of all morphisms 𝔣 : 𝔛′ → 𝔛 of formal schemes of finite presentation over Spf K^{+} such that d(𝔣) is an isomorphism, forms a small cofiltered category (with morphisms given as usual by the commutative diagrams). It is shown in [44, §3.9] that there is a natural isomorphism of topologically ringed spaces

(7.2.28)  (d(𝔛), 𝒪_{d(𝔛)}^{+}) →^{∼} lim_{(𝔛′→𝔛)∈𝒞} (𝔛′, 𝒪_{𝔛′}).

(Actually, the argument in loc.cit. is worked out only in the case of noetherian formal schemes, but it is not difficult to adapt it to the present situation).

7.2.29. Let f : A → B be a morphism of affinoid rings of topologically finite type over (K, K^{+}) (especially A^{+} = A^{∘} and B^{+} = B^{∘}, see (7.2.19)). We let 𝒞_{f} be the filtered family consisting of all the pairs (A_{0}, B_{0}) of K^{+}-algebras of topologically finite presentation, such that A_{0} (resp. B_{0}) is an open subalgebra of A^{∘} (resp. of B^{∘}) and f(A_{0}) ⊂ B_{0}. The analytic cotangent complex of the morphism f is the complex of B^{+}-modules

𝕃_{B/A}^{+} := colim_{(A_{0},B_{0})∈𝒞_{f}} 𝕃_{B_{0}/A_{0}}^{an}.

7.2.30. Let f : X → Y be a morphism of adic spaces locally of finite type over Spa(K, K^{+}), and suppose that Y is separated. For every affinoid open subset U ⊂ X, the small category F_{U} of all affinoid open subsets V ⊂ Y with f(U) ⊂ V, is cofiltered under inclusion (or else it is empty). For every V ∈ F_{U}, (𝒪_{Y} (V), 𝒪_{Y} (U)^{+}) is an affinoid (K, K^{+})-algebra of topologically finite type, hence the induced morphism 𝒪_{Y} (V) → 𝒪_{X}(U) is of the kind considered in (7.2.29). We set

L(U/Y) := colim_{V ∈F_{U}^{o}} 𝕃_{𝒪_{X}(U)/𝒪_{Y} (V)}^{+} .

Definition 7.2.31. The mapping U ↦ L(U/Y) defines a complex of presheaves on a cofinal family of affinoid open subsets of X. By applying degreewise the construction of [26, Ch.0, §3.2.1], we can extend the latter to a complex of presheaves of 𝒪_{X}^{+}-modules on X. We define the analytic cotangent complex 𝕃_{X/Y}^{+} of the morphism f : X → Y as the complex of sheaves associated to this complex of presheaves (cp. definition (7.2.3)). We also set 𝕃_{X/Y}^{an} := 𝕃_{X/Y}^{+} ⊗_{K^{+}} K. The definition can be extended to the case of a morphism f : X → Y where Y is not necessarily separated: one argues as in (7.2.4).

7.2.32. Let X^{+} denote the ringed space (X, 𝒪_{X}^{+}) and define likewise Y^{+}. Just as in the proof of lemma 7.2.5, by inspecting the construction we obtain natural maps of complexes

(7.2.33)  𝕃_{X^{+}/Y^{+}} → 𝕃_{X/Y}^{+}  𝕃_{X/Y} → 𝕃_{X/Y}^{an} .

7.2.34. Let A, B be complete f-adic rings and A → B a ring homomorphism of topologically finite presentation. We refer to [45, §1.6] for the construction of a universal A-derivation of B, which is a continuous A-derivation d : B → Ω_{B/A}^{an} from B to a complete topological Bmodule Ω_{B/A}^{an}, universal for A-derivations B → M to complete topological B-modules M. The construction of Ω_{B/A}^{an} can be globalized to a sheaf of relative differentials Ω_{X/Y}^{an} for any morphism of adic spaces X → Y locally of finite presentation. Then one checks easily that:

(7.2.35)  H_{0}(𝕃_{X/Y}^{an} ) ≃ Ω_{X/Y}^{an} .

Proposition 7.2.36. Let X →^{f} Y →^{g} Z be two morphisms of adic spaces locally of finite type over Spa(K, K^{+}). There is a natural distinguished triangle in D^{−}(𝒪_{X}-Mod)

(7.2.37)  Lf^{∗}𝕃_{Y/Z}^{an} → 𝕃_{X/Z}^{an} → 𝕃_{X/Y}^{an} → Lf^{∗}𝕃_{Y/Z}^{an}[1].

Proof. Mutatis mutandis, this is the same as the proof of proposition 7.2.13, so we can leave the details to the reader. □

Remark 7.2.38. In fact, the proof of proposition 7.2.36 shows that (7.2.37) is represented by a functorial true triangle (see [46, Ch.I, §3.2.4]). This will be important in the sequel, when we will need to compute the truncation τ_{[−1}𝕃_{X/Z} of the analytic cotangent in the situation contemplated in proposition 7.2.44. Theorem 7.2.39. Let 𝔣 : 𝔛 → 𝔜 (resp. f : X → Y) be a morphism of formal schemes (resp. of adic spaces) locally of finite presentation over Spf K^{+} (resp. locally of finite type over Spa(K, K^{+})) such that the induced morphism d(𝔣) : d(𝔛) → d(𝔜) is smooth (resp. such that f is smooth). Then:

(i) 𝕃_{𝔛/𝔜}^{an} ⊗_{K^{+}} K ≃ Ω_{𝔛/𝔜}^{an}[0] ⊗_{K^{+}} K in D^{−}((𝒪_{𝔛} ⊗_{K^{+}} K)-Mod).

(ii) 𝕃_{X/Y}^{an} ≃ Ω_{X/Y}^{an} [0] in D^{−}(𝒪_{X}-Mod).

Proof. After the usual reductions, both assertions come down to the following situation. We have a map of K^{+}-algebras of topologically finite presentation ϕ : A_{0} → B_{0}, such that d(Spf ϕ) is a smooth morphism of affinoid adic spaces. We have to show that 𝕃_{B_{0}/A_{0}}^{an} ⊗_{K^{+}} K ≃ Ω_{B_{0}/A_{0}}^{an} ⊗_{K^{+}} K[0] in D(B-Mod). We can write B_{0} = C_{0}/I_{0}, where C_{0} := A_{0}⟨T_{1}, ..., T_{n}⟩ and I_{0} is some finitely generated ideal. Set I := I_{0} ⊗_{K^{+}} K, A := A_{0} ⊗_{K^{+}} K and let 𝔫 be a maximal ideal of C := C_{0} ⊗_{K^{+}} K with I ⊂ 𝔫. Claim 7.2.40. I_{𝔫} is generated by a regular sequence of elements of the local ring C_{𝔫}. Proof of the claim: Let 𝔭 := 𝔫 ∩ A; 𝔭 is a maximal ideal in A, and its residue field K′ is a finite extension of K. Let \overline{𝔫} be the image of 𝔫 in C ⊗_{A} K′ and ℐ ⊂ 𝒪_{d(Spf C_{0})} the sheaf of ideals corresponding to I; the maximal ideal 𝔫 yields a point in d(Spf C_{0}), which we denote by x(𝔫). We have an isomorphism on the 𝔫-adic completions:

(7.2.41)  (ℐ_{x(𝔫)})^{∧} ≃ (I_{𝔫})^{∧}.

Moreover, there are natural maps :

ℐ_{x(𝔫)}/ℐ_{x(𝔫)}^{2} → \overline{𝔫}/\overline{𝔫}^{2} → Ω_{C/A}^{an} ⊗_{C} C/𝔫

and, by [14, Prop.2.5], there exists a set of generators g_{1}, ..., g_{k} for ℐ_{x(𝔫)} such that the images dg_{1}, ..., dg_{k} in Ω_{C/A}^{an} ⊗_{C} C/𝔫 are linearly independent; it follows that the images \overline{g}_{1}, ..., \overline{g}_{k} in \overline{𝔫}/\overline{𝔫}^{2} are also linearly independent. Due to (7.2.41), we can assume that g_{1}, ..., g_{k} ∈ I_{𝔫}, and then it follows that g_{1} ⊗ 1, ..., g_{k} ⊗ 1 are the first k elements of a regular system of parameters for the regular local ring C_{𝔫} ⊗_{A} K′. From lemma 7.1.6 it follows that C_{𝔫} is a flat A_{𝔭}-module; then by [28, Ch.0, Prop.15.1.16] we deduce that g_{1}, ..., g_{n} is a regular sequence of elements of C_{𝔫}, as required.

Set B := B_{0} ⊗_{K^{+}} K; it follows from claim 7.2.40 and [46, Ch.III, Prop.3.2.4] that C_{𝔫} ⊗_{C} 𝕃_{C/B} ≃ 𝕃_{C_{𝔫}/B_{𝔫}} ≃ I_{𝔫}/I_{𝔫}^{2}[1] for every maximal ideal 𝔫, hence 𝕃_{C/B} ≃ I/I^{2}[1]. By proposition 7.1.29, we derive 𝕃_{C_{0}/B_{0}}^{an} ⊗_{K^{+}} K ≃ I/I^{2}[1]. Finally, by theorem 7.1.31(ii) and proposition 7.1.27 we deduce an isomorphism in D^{−}(B-Mod) :

𝕃_{B_{0}/A_{0}}^{an} ⊗_{K^{+}} K ≃ (0 → I/I^{2} → B ⊗_{C_{0}} Ω_{C_{0}/A_{0}}^{an} → 0)

and the latter complex is quasi-isomorphic to Ω_{B/A}^{an}[0] by [14, Prop.2.5]. □

Lemma 7.2.42. Let f : X → Y be a closed imbedding of adic spaces locally of finite type over Spa(K, K^{+}). Then the natural morphism

𝕃_{X/Y} → 𝕃_{X/Y}^{an}

is an isomorphism in D(𝒪_{X}-Mod).

Proof. The question is local on X, so we can reduce to the case where Y = Spa(A, A^{∘}) for some K-algebra of topologically finite type and X = Spa(B, B^{∘}), where B = A/I for some ideal I ⊂ A. Let 𝒞 be the filtered family of all triples of the form (U, A_{0}, B_{0}) where U ⊂ Y is an affinoid open neighborhood of X, A_{0} ⊂ 𝒪_{Y} (U)^{∘}, B_{0} ⊂ B^{∘} are two open K^{+}-subalgebras of topologically finite presentation with Im(A_{0} → B) ⊂ B_{0}.

Claim 7.2.43. The family 𝒞′ ⊂ 𝒞_{π} of all (U, A_{0}, B_{0}) such that Im(A_{0} → B) = B_{0} is cofinal.

Proof of the claim: Let (U, A_{0}, B_{0}) be any triple in 𝒞. We can find finitely many elements \overline{b}_{1}, ..., \overline{b}_{k} ∈ B_{0} such that A_{0}[\overline{b}_{1}, ..., \overline{b}_{k}] is dense in B_{0}. Choose elements b_{1}, ..., b_{k} ∈ A that lift the \overline{b}_{i}; the subset U′ := \{x ∈ U | |b_{i}(x)| ≤ 1; i = 1, ..., k\} is an affinoid open neighborhood of X and the topological closure A′_{0} of A_{0}[b_{1}, ..., b_{k}] is a K^{+}-algebra of topologically finite presentation contained in 𝒪_{X}(U′)^{∘}; moreover Im(A′_{0} → B) = B_{0}, hence (U′, A′_{0}, B_{0}) ∈ 𝒞.

Finally, let (U, A_{0}, B_{0}) ∈ 𝒞′; by proposition 7.1.29 we have 𝕃_{B_{0}/A_{0}}^{an} ≃ 𝕃_{B_{0}/A_{0}}. After taking colimits we deduce:

colim_{X⊂U⊂Y} 𝕃_{B/𝒪_{Y} (U)}^{an} ≃ colim_{X⊂U⊂Y} 𝕃_{B/𝒪_{Y} (U)}

where U runs over the cofiltered family of all open affinoid neighborhoods of X in Y. The lemma follows easily. □

Proposition 7.2.44. Let X →^{f} Y →^{g} Z be morphisms of adic spaces locally of finite type over Spa(K, K^{+}), where f is a closed imbedding with defining ideal ℐ ⊂ 𝒪_{Y}, and g is smooth. Then there is a natural isomorphism in D(𝒪_{X}-Mod)

(7.2.45)  τ_{[−1}𝕃_{X/Z}^{an} →^{∼} (0 → (ℐ /ℐ^{2})_{|X} →^{d} f^{∗}Ω_{Y/Z}^{an} → 0)

where d is the natural map.

Proof. From lemma 7.2.42 and [46, Ch.III, Cor.1.2.8.1] we deduce a natural isomorphism

τ_{[−1}𝕃_{X/Y}^{an} ≃ (ℐ /ℐ^{2})_{|X}[1].

Taking into account proposition 7.2.36, theorem 7.2.39, one can repeat the proof of [46, Ch.III, Cor.1.2.9.1], which yields a distinguished triangle:

(7.2.46)  f^{∗}H_{0}(𝕃_{Y/Z})[0] → τ_{[−1}𝕃_{X/Z} → H_{1}(𝕃_{X/Y})[1] →

such that the connecting morphism (ℐ /ℐ^{2})_{|X} ≃ H_{1}(𝕃_{X/Y}) → f^{∗}H_{0}(𝕃_{Y/Z}) ≃ f^{∗}Ω_{Y/Z}^{an} is naturally identified (up to sign) with the differential map f ↦ df. There follows an isomorphism such as (7.2.45); however, the naturality of (7.2.45) is not explicitly verified in loc.cit. : in general, this kind of manipulations, when carried out in the derived category, do not lead to functorial identifications. The problem is that we can have morphisms of distinguished triangles:

\xymatrix{A^{•} \ar[r] \ar[d]_{f} & B^{•} \ar[r] \ar[d]_{g} & C^{•} \ar[r] \ar[d]^{h} & A^{•}[1] \ar[d]^{f[1]} \\ A^{′•} \ar[r] & B^{′•} \ar[r] & C^{′•} \ar[r] & A^{′•}[1]}

such that both f and g are the zero maps, and yet h is not. The issue can be resolved if one remarks that (7.2.46)is deduced from a transitivity triangle via proposition 7.2.36, and thus it is actually well defined in the derived category of true triangles T(𝒪_{X}-Mod) (cp. remark 7.2.38 and [46, Ch.I, §3.2.4]). It suffices then to apply the following Claim 7.2.47. Let 𝒞 be any abelian category, \underline{T} := (0 → A^{•} → B^{•} → C^{•} → 0) a true triangle of T(𝒞) and n ∈ ℤ an integer such that H^{i+1}(A^{•}) = 0 = H^{i}(C^{•}) for every i ≠ n. Then there is a natural isomorphism in T(𝒞):

\underline{T} ≃ (0 → H^{n+1}(A^{•})[n + 1] → Cone(H^{n}(C^{•})[n] →^{d} H^{n+1}(A^{•})[n + 1]) → H^{n}(C^{•})[n] → 0)

where d is the connecting morphism of the long exact homology sequence associated to \underline{T}. Proof of the claim: Under the stated assumptions, the natural maps of complexes ϕ : τ_{[n}C^{•} → C and ψ : A^{•} → τ_{n+1]}A^{•} are quasi-isomorphisms; we set \underline{T}′ := ψ ∗ \underline{T} ∗ ϕ (notation of (2.5.5). Clearly the pullback and push out maps define a natural isomorphism \underline{T}′ →^{∼} \underline{T} in T(𝒞). Say that \underline{T}′ = (0 → A^{′•} → B^{′•} → C^{′•} → 0); since H^{n+1}(C^{′•}) = 0, it follows that the complex \underline{T}′′ := τ_{[n+1}\underline{T}′ := (0 → τ_{[n+1}A^{′•} → τ_{[n+1}B^{′•} → τ_{[n+1}C^{′•} → 0) is again a true triangle of T(𝒞), naturally isomorphic to \underline{T}′. Likewise, \underline{T}′′ is naturally isomorphic to the true triangle τ_{n+1]}\underline{T}′′, and by inspection, one sees that the latter has the expected shape. □

7.3. Deformations of formal schemes and adic spaces. Lemma 7.3.1. Let (𝔛, 𝒪_{𝔛}) be a formal scheme locally of finite presentation over Spf K^{+}, let ℐ be a coherent 𝒪_{𝔛}-module and (0 → ℐ → 𝒪_{1} → 𝒪_{𝔛} → 0) an extension of sheaves of K^{+}-algebras on 𝔛. Then (𝔛, 𝒪_{1}) is a formal scheme locally of finite presentation over Spf K^{+}.

Proof. We may and do assume that 𝔛 is an affine formal scheme over Spf K^{+}; then the assertion follows from claims 7.3.2 and 7.3.3 below. Claim 7.3.2. 𝒪_{1}(𝔛) is a complete K^{+}-algebra of topologically finite presentation. Proof of the claim: Since 𝔛 is affine, we derive a short exact sequence 0 → ℐ (𝔛) → 𝒪_{1}(𝔛) → 𝒪_{𝔛}(𝔛) → 0. Since ℐ is coherent, the 𝒪_{𝔛}(𝔛)-module ℐ (𝔛) is finitely presented. Let T ⊂ ℐ (𝔛) be the K^{+}-torsion submodule; by proposition 7.1.1(i), it follows that ℐ (𝔛)/T is finitely presented, hence T is finitely generated over 𝒪_{𝔛}(𝔛); likewise, the K^{+}-torsion submodule T′ of 𝒪_{𝔛}(𝔛) is a finitely generated 𝒪_{𝔛}(𝔛)-module, and then the K^{+}-torsion submodule T_{1} of 𝒪_{1}(𝔛) is a finitely generated 𝒪_{1}(𝔛)-module. Let N := Ker(𝒪_{1}(𝔛) → 𝒪_{𝔛}(𝔛)/T′); in the usual way we derive that there exists k_{0} ∈ ℕ such that

a^{k}𝒪_{1}(𝔛) ∩ ℐ (𝔛) = a^{k}N ∩ ℐ (𝔛) ⊂ a^{k−k_{0}}ℐ (𝔛) for every k ≥ k_{0}.

We deduce by [54, Th.8.1(ii)] a short exact sequence of complete K^{+}-algebras 0 → ℐ (𝔛) → 𝒪_{1}(𝔛)^{∧} → 𝒪_{𝔛}(𝔛) → 0, which shows that 𝒪_{1}(𝔛) is complete. Since ℐ (𝔛) and 𝒪_{𝔛}(𝔛) are topologically of finite type, we can then find a continuous surjection A := K^{+}⟨T_{1}, ..., T_{n}⟩ → 𝒪_{1}(𝔛); by lemma 7.1.4 it follows that 𝒪_{1}(𝔛) is a finitely presented A-algebra, hence ℐ (𝔛) is a finitely presented A-module, and then the same holds for 𝒪_{1}(𝔛), so the claim is proved.

Claim 7.3.3. For f ∈ 𝒪_{𝔛}(𝔛), let 𝔇(f) ⊂ 𝔛 be as in lemma 7.2.1. Then the natural map (𝒪_{1}(𝔛)_{f})^{∧} → 𝒪_{1}(𝔇(f)) is an isomorphism. Proof of the claim: The existence of the said map is a consequence of claim 7.3.2; however, we have as well a natural short exact sequence 0 → ℐ (𝔛)_{f}^{∧} → 𝒪_{1}(𝔇(f)) → 𝒪_{𝔛}(𝔛)_{f}^{∧} → 0, so the assertion is immediate. □

The meaning of lemma 7.3.1 is that the square zero deformations of 𝔛 by ℐ in the category of ringed spaces are the same as those in the category of formal Spf K^{+}-schemes locally of finite presentation. Especially, the latter are classified by the appropriate Ext-group of the (usual) cotangent complex of the map of ringed spaces 𝔛 → Spf K^{+}; we aim to show that the same computation can be carried out with the analytic cotangent complex 𝕃_{𝔛/K^{+}}^{an} introduced in definition 7.2.3.

7.3.4. Let T be a topos, I a small category; the category T^{I} of all functors from I to T is a topos in a natural way, and we define a functor c : T → T^{I} by assigning to an object X of T the constant functor c_{X} : I → T of value X (so c_{X}(i) = X for every object i of I, and c_{X}(ϕ) = 1_{X} for every morphism ϕ of I). The functor c admits a right adjoint lim_{I} : T^{I} → T, and the adjoint

pair (c, lim_{I}) defines a morphism of topoi π_{T} : T^{I} → T. If ϕ : S → T is any morphism of topoi, we obtain a commutative diagram of topoi:

\xymatrix{S^{I} \ar[r]^{π_{S}} \ar[d]_{ϕ^{I}} & S \ar[d]^{ϕ} \\ T^{I} \ar[r]^{π_{T}} & T}

whence two spectral sequences:

\begin{matrix}E_{2}^{pq} := & R^{p}ϕ_{∗} lim_{i∈I}^{q}ℱ_{i} ⇒ R^{p+q}(ϕ ∘ π_{S})_{∗}(ℱ_{i} | i ∈ I) \\ F_{2}^{pq} := & lim_{i∈I}^{p} R^{q}ϕ_{∗}ℱ_{i} ⇒ R^{p+q}(ϕ ∘ π_{S})_{∗}(ℱ_{i} | i ∈ I) \end{matrix}

for every abelian sheaf (ℱ_{i} | i ∈ I) on S^{I}. Lemma 7.3.5. Let 𝔛 be an adic formal scheme over Spf K^{+}, ℱ a coherent 𝒪_{𝔛}-module and set ℱ_{n} := ℱ /a^{n}ℱ for every n ∈ ℕ. Then (ℱ_{n} | n ∈ ℕ) defines an abelian sheaf on 𝔛^{ℕ}, and we have lim_{n∈ℕ}^{q}ℱ_{n} = 0 on 𝔛, for every q > 0.

Proof. We apply the spectral sequences of (7.3.4)to the indexing category ℕ and the morphism of topoi ϕ : 𝔘 → Spf K^{+}, where 𝔘 ⊂ 𝔛 is any affine open subset. Hence F_{2}^{pq} = 0 whenever q > 0, therefore the abutment is isomorphic to F_{2}^{p0} = lim_{n∈ℕ}^{p} ℱ_{n}(𝔘); however, the inverse system

(ℱ_{n}(𝔘) | n ∈ ℕ) is surjective, hence F^{p0} = 0 for p > 0, and it follows that E_{∞}^{pq} = 0 whenever p + q > 0 and E_{∞}^{00} ≃ lim_{n∈ℕ}ℱ_{n}(𝔘) ≃ ℱ (𝔘). Since 𝔘 is arbitrary, the claim follows easily. □

7.3.6. With the notation of (7.3.4), let 𝒪_{•} := (𝒪_{i} | i ∈ I) be a ring object of the topos T^{I} (briefly: a T^{I}-ring), or which is the same, a functor 𝒪 from I to the category of T -rings. Let also 𝒪_{T} be a T -ring, and π_{T}^{♯} : 𝒪_{T} → π_{T ∗}𝒪_{•} a morphism of T -rings. Then the pair (π_{T} , π_{T}^{♯}) defines a morphism of ringed topoi π_{T} : (T^{I}, 𝒪_{•}) → (T, 𝒪_{T}). The corresponding functor π_{T ∗} = lim_{I} : 𝒪_{•}-Mod → 𝒪_{T} -Mod admits a left adjoint π_{T}^{∗}, defined by the rule:

ℱ ↦ π_{T}^{−1}ℱ ⊗_{π_{T}^{−1}𝒪_{T}} 𝒪_{•} = (ℱ ⊗_{𝒪_{T}} 𝒪_{i} | i ∈ I).

(Of course, π_{T}^{−1} is the same as the functor c : T → T^{I} of (7.3.4)). The adjoint pair (π_{T ∗}, π_{T}^{∗}) extends to an adjoint pair of derived functors (Rπ_{T ∗}, Lπ_{T}^{∗}); more precisely, for every complex K_{1}^{•} ∈ D^{−}(𝒪_{T} -Mod) and K_{2}^{•} ∈ D^{+}(𝒪_{•}-Mod) there is a natural isomorphism (“trivial duality”)

Hom_{D(𝒪_{•}-Mod)} (π_{T}^{−1}K_{1}^{•} ⊗^{L}_{π_{T}^{−1}𝒪_{T}} 𝒪_{•}, K_{2}^{•}) ≃ Hom_{D(𝒪_{T} -Mod)} (K_{1}^{•}, Rlim_{I}K_{2}^{•})

(see [46, Ch.III, Prop.4.6]). Proposition 7.3.7. Let 𝔛 be formal scheme locally of finite presentation over Spf K^{+} and ℱ any coherent 𝒪_{𝔛}-module. Then the natural morphism 𝕃_{𝔛/K^{+}} → 𝕃_{𝔛/K^{+}}^{an} induces isomorphisms

𝔼xt_{𝒪_{𝔛}}^{i} (𝕃_{𝔛/K^{+}}^{an}, ℱ) →^{∼} 𝔼xt_{𝒪_{𝔛}}^{i} (𝕃_{𝔛/K^{+}}, ℱ) for every i ∈ ℕ.

Proof. We endow the topos 𝔛^{ℕ} with the ring object 𝒪_{•} := (𝒪_{𝔛}/a^{n}𝒪_{𝔛} | n ∈ ℕ); the natural morphism 𝒪_{𝔛} → lim_{ℕ}𝒪_{•} determines a morphism of ringed topoi π : (𝔛^{ℕ}, 𝒪_{•}) → (𝔛, 𝒪_{𝔛}) as in (7.3.6). By lemma 7.3.5, the natural map ℱ → Rlim_{ℕ}π^{∗}ℱ is an isomorphism and by lemma 7.1.7, both 𝕃_{𝔛/K^{+}} and 𝕃_{𝔛/K^{+}}^{an} are complexes of flat 𝒪_{𝔛}-modules, hence the trivial duality isomorphism reads

𝔼xt_{𝒪_{•}}^{i} (π^{∗}𝕃_{𝔛/K^{+}}^{an}, π^{∗}ℱ) ≃ 𝔼xt_{𝒪_{𝔛}}^{i} (𝕃_{𝔛/K^{+}}^{an}, ℱ) for every i ∈ ℕ

and likewise for 𝕃_{𝔛/K^{+}}. To conclude it remains only to remark that π^{∗}𝕃_{𝔛/K^{+}} ≃ π^{∗}𝕃_{𝔛/K^{+}}^{an}. □

Lemma 7.3.1 and proposition 7.3.7 enable one to derive the standard results on nilpotent deformations, along the lines of section 3.2 : the statements are unchanged, except that the topos-theoretic cotangent complex is replaced by the (much more manageable) analytic one. The details will be left to the reader.

7.3.8. Let f : X → Y be a morphism of adic spaces locally of finite type over Spa(K, K^{+}). We would like to show that the infinitesimal deformation theory of f is captured by the analytic cotangent complex 𝕃_{X/Y}^{an} , in analogy with the usual topos-theoretic situation, and with the treatment for formal schemes already presented. This turns out to be indeed the case, however the proofs are rather more delicate than those for formal schemes, due to the existence of square zero deformations of affinoid algebras that are not themselves affinoid. In homological terms, this reflects the fact that the natural map

(7.3.9)  𝔼xt_{𝒪_{X}}^{1} (𝕃_{X/Y}^{an} , ℱ) → 𝔼xt_{𝒪_{X}}^{1} (𝕃_{X/Y} , ℱ)

is not in general an isomorphism for arbitrary coherent 𝒪_{X}-modules ℱ. Nevertheless, we have the following: Proposition 7.3.10. Let f : X → Y be as in (7.3.8). Then the map (7.3.9)is injective for every coherent 𝒪_{X}-module ℱ.

Proof. Using the long exact 𝔼xtsequence, we reduce to showing: Claim 7.3.11. 𝔼xt_{𝒪_{X}}^{0} (Cone (𝕃_{X/Y} → 𝕃_{X/Y}^{an} ), ℱ) = 0. Proof of the claim: Indeed, we have

𝔼xt_{𝒪_{X}}^{0} (Cone (𝕃_{X/Y} → 𝕃_{X/Y}^{an} ), ℱ) ≃ Ext_{𝒪_{X}}^{0} (H_{0}(Cone (𝕃_{X/Y} → 𝕃_{X/Y}^{an} )), ℱ)
≃ Ext_{𝒪_{X}}^{0} (Coker(H_{0}(𝕃_{X/Y}) → H_{0}(𝕃_{X/Y}^{an} )), ℱ)
≃ Ext_{𝒪_{X}}^{0} (Coker(Ω_{X/Y} → Ω_{X/Y}^{an} ), ℱ)

and the latter group vanishes, due to the universal property of Ω_{X/Y}^{an} (see (7.2.34)). □

7.3.12. Let f : X → Y be a morphism of adic spaces locally of finite type over Spa(K, K^{+}). An analytic deformation of X over Y is a datum of the form (j, ℱ , β) consisting of :

(a) a closed imbedding j : X → X′ of Y -adic spaces, such that the ideal ℐ ⊂ 𝒪_{X′} defining j satisfies ℐ^{2} = 0, and

(b) a coherent 𝒪_{X}-module ℱ with an isomorphism of 𝒪_{X}-modules β : j^{∗}ℐ →^{∼} ℱ. One defines in the obvious way a morphism of analytic deformations, and we let Exan_{Y} (X, ℱ) denote the set of isomorphism classes of analytic deformations of X by ℱ over Y. Proposition 7.3.13. Let X be an affinoid adic space of finite type over Spa(K, K^{+}) and (j : X → X′, ℱ , β) an analytic deformation. Then X′ is affinoid of finite type over Spa(K, K^{+}).

Proof. To start out, we notice that j is a homeomorphism on the topological spaces underlying X and X′, hence 𝒪_{X′}(X′) = Γ(X, j^{∗}𝒪_{X′}); we deduce a short exact sequence of continuous maps:

0 → ℱ (X) → 𝒪_{X′}(X′) → 𝒪_{X}(X) → 0.

Claim 7.3.14. Let ϕ : B → A be a surjective map of complete K-algebras of topologically finite type, such that Ker ϕ is a square zero ideal of B. Then B^{∘} = ϕ^{−1}(A^{∘}).

Proof of the claim: Quite generally, let C be any K-algebra of topologically finite type; according to [12, §6.2.3, Prop.1] one has C^{∘} = {f ∈ C | |f(x)| ≤ 1 for every x ∈ Max C}. In our situation, it is clear that Max A = Max B, whence the claim.

Claim 7.3.15. 𝒪_{X′}(X′) is a complete K-algebra of topologically finite type over K.

Proof of the claim: Let \overline{g}_{1}, ..., \overline{g}_{m} be a set of topological generators for 𝒪_{X}(X), and choose arbitrary liftings g_{i}, ..., g_{m} ∈ 𝒪_{X′}(X′). Pick also a finite set of generators g_{m+1}, ..., g_{n} for the 𝒪_{X}(X)-module ℱ (X). We define a map ϕ : K[T_{1}, ..., T_{n}] → 𝒪_{X′}(X′) by the rule T_{i} ↦ g_{i} (i = 1, ..., n). Let X′ = ⋃_{k} U_{k} be a covering of X′ by finitely many of its affinoid domains. We deduce a map ψ : K[T_{1}, ..., T_{n}] → ∏_{k} 𝒪_{X′}(U_{k}). By viewing 0 → j_{∗}ℱ → 𝒪_{X′} → j_{∗}𝒪_{X} → 0 as a short exact sequence of coherent 𝒪_{X′}-modules on X′, we deduce short exact sequences 0 → ℱ (U_{k}) → 𝒪_{X′}(U_{k}) →^{π_{k}} 𝒪_{X}(U_{k}) → 0 for every k. By the open mapping theorem (see [12, §2.8.1]) one deduces easily that the topology of 𝒪_{X}(U_{k}) is the same as the quotient topology deduced from the surjection π_{k}, especially 𝒪_{X}(U_{k}) is a K-algebra of topologically finite type. Since the images of \overline{g}_{1}, ..., \overline{g}_{m} in 𝒪_{X}(U_{k}) are power bounded for every k, it then follows from claim 7.3.14 that the images of g_{1}, ..., g_{m} in 𝒪_{X′}(U_{k}) are power bounded for every k. Hence ψ extends to a map ψ^{∧} : K⟨T_{1}, ..., T_{n}⟩ → ∏_{k} 𝒪_{X′}(U_{k}), and by construction, ψ^{∧} factors through a continuous map ϕ^{∧} : K⟨T_{1}, ..., T_{n}⟩ → 𝒪_{X′}(X′). It is easy to check that ϕ^{∧} is surjective, so the claim follows, again by the open mapping theorem.

Set A := 𝒪_{X′}(X′); by claim 7.3.15 we have the affinoid adic space X′′ := Spa(A, A^{∘}) of finite type over Spa(K, K^{+}). By construction we get morphisms X →^{j} X′ →^{α} X′′ of adic spaces inducing homeomorphisms on the underlying topologies. To conclude it suffices to show:

Claim 7.3.16. The morphism α^{♯} : 𝒪_{X′′} → α_{∗}𝒪_{X′} is an isomorphism of sheaves of topological algebras.

Proof of the claim: Using claim 7.3.22 we see that it suffices to show that α^{♯} is an isomorphism of sheaves of K-algebras. However, for any affinoid open domain in X′′ we have a commutative diagram with exact rows:

[figure: 0; ℱ (U); 𝒪_{X′′}(U); 𝒪_{X}(U); 0; 0; ℱ (U); 𝒪_{X′}(U); 𝒪_{X}(U); 0.]

Since the affinoid open subsets of X′′ form a basis of the topology for both X′′ and X′, the claim follows. □

7.3.17. In the situation of (7.3.12), let (j : X → X′, ℱ , β) be an analytic deformation of X over Y. Clearly j is a homeomorphism on the underlying topological spaces, hence we can view the given deformation as the datum of a map of f^{−1}𝒪_{Y} -algebras 𝒪_{X′} → 𝒪_{X}, whence a transitivity distinguished triangle (proposition 7.2.36)

Lj^{∗}𝕃_{X′/S}^{an} → 𝕃_{X/S}^{an} → 𝕃_{X/X′}^{an} → Lj^{∗}𝕃_{X′/S}^{an}[1]

which in turns yields a distinguished triangle:

RHom_{𝒪_{X}} (𝕃_{X/X′}^{an}, ℱ) → RHom_{𝒪_{X}} (𝕃_{X/S}^{an}, ℱ) → RHom_{𝒪_{X′}} (𝕃^{a}n_{X′/S}, j_{∗}ℱ) → .

Especially, we get a map

(7.3.18)  Ext_{𝒪_{X}}^{1} (𝕃_{X/X′}^{an}, ℱ) → Ext_{𝒪_{X}}^{1} (𝕃_{X/S}^{an}, ℱ).

Let ℐ ⊂ 𝒪_{X′} be the ideal that defines the imbedding j; by proposition 7.2.44 we have a natural isomorphism 𝕃_{X/X′}^{an} →^{∼} j^{∗}ℐ [1], whence an isomorphism

(7.3.19)  Ext_{𝒪_{X}}^{1} (𝕃_{X/X′}^{an}, ℱ) →^{∼} Hom_{𝒪_{X}} (j^{∗}ℐ , ℱ).

Combining (7.3.18)and (7.3.19)we see that the given isomorphism β : j^{∗}ℐ → ℱ determines a unique element e^{an}(X′, β) ∈ Ext_{𝒪_{X}}^{1} (𝕃_{X/S}^{an}, ℱ). One verifies easily that e^{an}(X′, β) depends only on the isomorphism class of the analytic deformation (j, ℱ , β), therefore it defines a map

e^{an} : Exan_{Y} (X, ℱ) → Ext_{𝒪_{X}}^{1} (𝕃_{X/S}^{an}, ℱ).

By inspecting the construction, it is easy to check that e^{an} fits into a commutative diagram

(7.3.20)  \xymatrix{Exan_{Y} (X, ℱ) \ar[r]^{e^{an}} \ar[d] & 𝔼xt_{𝒪_{X}}^{1} (𝕃_{X/S}^{an}, ℱ) \ar[d] \\ Exal_{f^{−1}𝒪_{Y}} (𝒪_{X}, ℱ) \ar[r]^{e} & 𝔼xt_{𝒪_{X}}^{1} (𝕃_{X/Y} , ℱ)}

where e is the isomorphism of [46, Ch.III, Th.1.2.3] and the right vertical arrow is (7.3.9). Theorem 7.3.21. For every coherent 𝒪_{X}-module ℱ, the map e^{an} is a natural bijection.

Proof. In view of (7.3.20), in order to show that e^{an} is injective, it suffices to verify that if (j_{i} : X → X_{i}, β_{i}) for i = 1, 2 are two analytic deformations of X by ℱ over Y, and (j_{1}, β_{1}) →^{∼} (j_{2}, β_{2}) is an isomorphism in the category of extensions of f^{−1}𝒪_{Y} -algebras, then the corresponding map 𝒪_{X_{1}} → 𝒪_{X_{2}} is continuous, i.e. it is an isomorphism of sheaves of topological algebras (since in this case the map restricts to an isomorphism 𝒪_{X_{1}}^{+} →^{∼} 𝒪_{X_{2}}^{+}). This can be checked locally, so we may assume that X is affinoid; in this case both X_{1} and X_{2} are affinoid as well, by proposition 7.3.13. Then the assertion is a straightforward consequence of the following well known: Claim 7.3.22. Every map A → B from a noetherian K-Banach algebra A to K-algebra B of topologically finite type, is continuous. Proof of the claim: This is [12, §6.1.3, Th.1].

The surjectivity is a local issue as well; indeed, any class in the target of e^{an} represents an extension of f^{−1}𝒪_{Y} -algebras 0 → ℱ → ℰ → 𝒪_{X} → 0, and the question amounts to showing that ℰ represents an analytic deformation, which can be checked locally. Thus, suppose that X = Spa(B, B^{∘}), Y = Spa(A, A^{∘}). We can write B = P/I, where P := A⟨T_{1}, ..., T_{n}⟩, and by proposition 7.2.44, the complex τ_{[−1}𝕃_{X/Y}^{an} is naturally isomorphic to the complex of sheaves associated to the complex of B-modules 0 → I/I^{2} → Ω_{P/A}^{an} ⊗_{P} B → 0. Whence, a natural isomorphism

𝔼xt_{𝒪_{X}}^{1} (𝕃_{X/Y}^{an} , ℱ) ≃ Coker(Hom_{𝒪_{X}} (j^{∗}Ω_{Z/Y}^{an} , ℱ) → Hom_{𝒪_{X}} ((I/I^{2})^{∼}, ℱ))
≃ Coker(Hom_{B}(Ω_{P/A}^{an} ⊗_{P} B, ℱ (X)) → Hom_{B}(I/I^{2}, ℱ (X)))

(where we have denoted by j : X → Z := Spa(P, P^{∘}) the induced closed imbedding). However, given α : I/I^{2} → ℱ (X), the corresponding f^{−1}𝒪_{Y} algebra ℰ is the sheaf of algebras associated to the A-algebra E defined as follows. Let β : I/I^{2} → E_{0} := (P/I^{2}) ⊕ ℱ (X) be the map given by the rule: x ↦ (x, α(x)) for every x ∈ I/I^{2}; E_{0} is endowed with an A-algebra structure given by the rule (x, f) · (y, g) = (xy, fy + gx) and one verifies easily that the image of β is an ideal in E_{0}; then set E := E_{0}/Im β. It is clear that E_{0} is an Aalgebra of topologically finite type, and the projection E → B determines a closed imbedding j : X → X′ := Spa(E, E^{∘}) that identifies j^{∗}𝒪_{X′} with ℰ. □

With the foregoing results, one can derive the usual results on existence of deformations and thereof obstructions; again we leave the details to the industrious reader. To conclude this section we want to show that (7.3.9) fails to be surjective already in the simplest situations. To carry out this analysis requires the use of more refined commutative algebra : the following proposition 7.3.23 collects all the information that we shall be needing. Proposition 7.3.23. Let A be a complete K-algebra of topologically finite type, n ∈ ℕ an integer, and set P := A⟨T_{1}, ..., T_{n}⟩. Then:

(i) the natural morphism Spec P → Spec A is regular.

(ii) H_{i}(𝕃_{P/A}) = 0 for every i > 0 and H_{0}(𝕃_{P/A}) is a flat P -module.

Proof. We begin with the following observation: Claim 7.3.24. Let R → S be a faithfully flat morphism of noetherian local rings. If S is regular, then R is regular. Proof of the claim: One verifies with no trouble that the homological dimension of R does not exceed the homological dimension of S.

Claim 7.3.25. Let R → S be a local morphism of local noetherian rings; let 𝔫 ⊂ S be the maximal ideal. Suppose that R is quasi-excellent and that S is formally smooth over R for its 𝔫-adic topology. Then the induced morphism Spec S → Spec R is regular. Proof of the claim: This is [3, Th.].

In view of claim 7.3.25, the proof of (i) is reduced to the following: Claim 7.3.26. Let 𝔫 ⊂ P be any maximal ideal and set 𝔮 := 𝔫 ∩ A. Then:

(i) P_{𝔫} is formally smooth over A_{𝔮} for its 𝔫-adic topology.

(ii) A_{𝔮} is an excellent local ring. Proof of the claim: (i): It is well known that 𝔮 is a maximal ideal and the residue field K′ := A/𝔮 is a finite extension of K. It follows easily from lemma 7.1.6(i) that P_{𝔫} is flat over A, hence by [28, Ch.IV, Th.19.7.1] it suffices to show that P_{𝔫} ⊗_{A} K′ is geometrically regular over K′, which allows to reduce to the case where A = K. Then, in view of claim 7.3.24, it suffices to show that, for every finite field extension K ⊂ K′ there exists a larger finite extension K′ ⊂ K′′ such that the semilocal ring P_{𝔫} ⊗_{K} K′′ is regular. However, the residue field κ := P/𝔫 is a finite extension of K, and if K ⊂ K′ is any finite normal extension with κ ⊂ K′, every maximal ideal of P ⊗_{K} K′ containing 𝔫 ⊗_{K} K′ is of the form 𝔭 := (T_{1} − a_{1}, ..., T_{n} − a_{n}) for certain a_{1}, ..., a_{n} ∈ K′. We are therefore reduced to the case where 𝔫 = (T_{1} − a_{1}, ..., T_{n} − a_{n}) for some a_{i} ∈ K, i = 1, ..., n; in this case the 𝔫-adic completion P_{𝔫}^{∧} of P_{𝔫} is isomorphic to K[[X_{1}, ..., X_{n}]], which is regular. (ii) is a theorem due to Kiehl, whose complete proof is reproduced in [22, Th.1.1.3].

Finally, (ii) follows from (i) and from the following: Claim 7.3.27. Let R → S be a map of noetherian rings. Then the following are equivalent:

(a) H_{i}(𝕃_{S/R}) = 0 for every i > 0 and H_{0}(𝕃_{S/R}) is a flat S-module.

(b) The induced morphism Spec S → Spec R is regular. Proof of the claim: Taking into account [54, Th.28.7], this is seen to be a paraphrase of [2, Suppl.(c), Th.30, p.331]. □

7.3.28. Let us now specialize to the case where P := K⟨T ⟩; it is well known that P is a principal ideal domain; since Ω_{P/K} is a free P -module, we can choose a splitting:

(7.3.29)  Ω_{P/K} ≃ Ω_{P/K}^{an} ⊕ Ω_{P/K}^{na}

where Ω_{P/K}^{na} := Ker(Ω_{P/K} → Ω_{P/K}^{an}) (“na” stays for “not analytic”). Our first aim is to show that P admits nilpotent extensions (by finitely generated P -modules) that are not affinoid algebras. In view of proposition 7.3.23(ii), the square zero extensions of P by a P -module F are classified by Ext_{P}^{1} (Ω_{P/K}, F) (and then it is clear that any non-trivial element of this group cannot represent an analytic deformation, since P admits none). Hence, we come down to computing Ext_{P}^{1} (Ω_{P/K}^{na}, F). Lemma 7.3.30. (i) Ω_{P/K}^{na} is a vector space over the fraction field E of P.

(ii) If either char(K) = 0 or char(K) = p > 0 and [K : K^{p}] = ∞, then Ω_{P/K}^{na} ≠ 0.

Proof. (i): by proposition 7.3.23(ii) we know that Ω_{P/K}^{na} is a flat P -module, especially it is torsion-free. Let f ∈ P be any irreducible element; then κ := P/fP is a finite field extension of K, and we have Hom_{P} (Ω_{P/K}, κ) = Hom_{P} (Ω_{P/K}^{an}, κ) by the universal property of Ω_{P/K}^{an}. In other words, Ω^{na}⊗_{P} κ = 0, so multiplication by f is a bijection on Ω_{P/K}^{na}, and the claim follows.

(ii): suppose first that char(K) = 0; by (i) we have: Ω_{E/K} ≃ E ⊗_{P} Ω_{P/K} ≃ Ω_{P/K}^{na} ⊕ (E ⊗_{P} Ω_{P/K}^{an}). Since Ω_{P/K}^{an} is a free P -module of rank one with generator dT, and since E is a separable extension of K, it suffices to show that tr.deg(E : K(T)) > 0. This can be checked explicitly; for instance, let π ∈ K be a non-zero element such that log(1 + πT) ∈ P; it is well known that this power series is transcendental over K(T). Finally, suppose that char(K) = p > 0 and [K : K^{p}] = ∞. We construct a splitting for the surjection Ω_{E/K} → E ⊗_{P} Ω_{P/K}^{an}, as follows. The tower of field extensions K ⊂ F := E^{p} · K(T) ⊂ E yields an exact sequence

E ⊗_{F} Ω_{F/K} →^{α} Ω_{E/K} →^{β} Ω_{E/F} → 0

and it is easy to check that α factors through a surjection E ⊗_{F} Ω_{F/K} → E ⊗_{P} Ω_{P/K}^{an} and the induced map E ⊗_{P} Ω_{P/K}^{an} → Ω_{E/K} is the sought splitting. Hence, it suffices to exhibit an element f ∈ E such that β(df) ≠ 0. To this aim, we will use the following general remark: Claim 7.3.31. Let E be a field with char(E) = p > 0 and F ⊂ E a subfield with E^{p} ⊂ F. Then: Ker(d : E → Ω_{E/F}) = F. Proof of the claim: Let (x_{i} | i ∈ I) be a p-basis for the extension F ⊂ E, and for every i ∈ I set E_{i} := F [x_{i} | i ∈ I \ {i}]. Clearly E = E_{i}[x_{i}] and the minimal polynomial of x_{i} over E_{i} is m_{i}(X) := X^{p} − x_{i}^{p}. A simple calculation shows that Ker(d : E → Ω_{E/E_{i}}) = E_{i}. Since F = ⋂_{i∈I} E_{i}, it follows that (d : E → Ω_{E/F}) = ⋂_{i∈I} Ker(d : E → Ω_{E/E_{i}}) = F.

In view of claim 7.3.31, we need to exhibit an element f ∈ E such that f ∉ E^{p} · K(T). However, under the standing assumptions, we can find a countable sequence (b_{n} | n ∈ ℕ) of elements b_{n} ∈ K^{∘} that are linearly independent over K^{p}. Set f := ∑_{n∈ℕ} a^{pn}b_{n}T^{n}, so that f ∈ P, and suppose by way of contradiction, that f ∈ E^{p} · K(T); then we could find g ∈ E · K^{1/p}(T^{1/p}) such that g^{p} = f. In turns, this means that g ∈ K′ · E(T^{1/p}) for some finite K-extension K′ ⊂ K^{1/p}. However, any such g can be written uniquely in the form g = ∑_{n∈ℕ} c_{n}T^{n/p} for a sequence (c_{n} | n ∈ ℕ) of elements c_{n} ∈ K′. Then g^{p} = ∑_{n∈ℕ} c_{n}^{p}T^{n}, and consequently c_{n}^{p} = a^{pn}b_{n} for every n ∈ ℕ, in other words, K′ contains all the elements b_{n}^{1/p} for every n ∈ ℕ, which is absurd. □

7.3.32. We shall complete our calculation for F := P. To this aim, recall that E is an injective P -module, therefore

Ext_{P}^{1} (Ω_{P/K}^{na}, P) ≃ Hom_{P} (Ω_{P/K}^{na}, E/P)/Hom_{P} (Ω_{P/K}^{na}, E).

Let A_{E} be the (finite) adele ring of E, i.e. the restricted product of all the completions

A_{E} := \begin{matrix}∏ & ′ \\ 𝔭∈MaxP \end{matrix} E_{𝔭}.

By the strong approximation theorem (cp. [20, Ch.II, §15, Th.]) the natural imbedding E ⊂ A_{E} has everywhere dense image, whence an isomorphism:

E/P ≃ ⊕_{𝔭∈MaxP} E_{𝔭}/P_{𝔭}.

A simple calculation shows that the map E_{𝔭} → Hom_{P} (E, E_{𝔭}/P_{𝔭}) given by the rule: f ↦ (x ↦ xf) is an isomorphism; whence a natural isomorphism:

A_{E} →^{∼} Hom_{P} (E, E/P) a ↦ (x ↦ ax).

Finally, Ext_{P}^{1} (E, P) ≃ A_{E}/E; since Ω_{P/K}^{na} is a direct sum of copies of E, this achieves our aim of producing non-trivial square zero extensions of K-algebras: 0 → P → ℰ → P → 0, whenever the hypotheses of lemma 7.3.30(ii) are fulfilled.

7.3.33. Now we want to carry out the local counterpart of the calculations of (7.3.32). Namely, let X := Spa(P, P^{∘}); we will show that there exist non-trivial deformations of (X, 𝒪_{X}) by the 𝒪_{X}-module 𝒪_{X}, in the category of locally ringed spaces. To this aim, choose any K-rational point p ∈ X, and let 𝒪_{X}(∞) denote the quasi-coherent 𝒪_{X}-module whose local sections on any open subset U ⊂ X are the meromorphic functions on U with poles (of arbitrary finite order) only at p. We deduce an exact sequence

Hom_{𝒪_{X}} (Ω_{X/K}^{na}, 𝒪_{X}(∞)) →^{f} Hom_{𝒪_{X}} (Ω_{X/K}^{na}, 𝒪_{X}(∞)/𝒪_{X}) →^{g} Ext_{𝒪_{X}}^{1} (Ω_{X/K}^{na}, 𝒪_{X})

where Ω_{X/K}^{na} denotes the quasi-coherent sheaf obtained by sheafifying the modules Ω^{na} defined as in (7.3.28). After choosing a global splitting (7.3.29), we can write Ω_{X/K} ≃ Ω_{X/K}^{an} ⊕ Ω_{X/K}^{na}.

Lemma 7.3.34. With the notation of (7.3.33), we have: Hom_{𝒪_{X}} (Ω_{X/K}^{na}, 𝒪_{X}(∞)) = 0.

Proof. Indeed, let ϕ : Ω_{X/K}^{na} → 𝒪_{X}(∞) be any 𝒪_{X}-linear map; we extend ϕ to a map ϕ′ : Ω_{X/K} → 𝒪_{X}(∞) by prescribing that ϕ′ restricts to the zero map on the direct factor Ω_{X/K}^{an}. Then ϕ′ corresponds to a K-linear derivation ∂ : 𝒪_{X} → 𝒪_{X}(∞); the restriction of ∂ to U := X \ {p} is a derivation of 𝒪_{U} with values in the coherent 𝒪_{U}-module 𝒪_{U}; hence the restriction of ϕ′_{|U} to Ω_{U/K}^{na} vanishes identically. It follows that ∂_{|U} vanishes identically. Let j : U → X be the imbedding; since the natural map 𝒪_{X}(∞) → j_{∗}𝒪_{U} is injective, it follows that ∂ must vanish as well, whence ϕ = 0. □

Lemma 7.3.35. With the notation of (7.3.33), let x ∈ X be a K-rational point and suppose that the hypotheses of lemma 7.3.30(ii) hold for K. Then Ω_{X/K,x}^{na} ≠ 0.

Proof. By lemma 7.3.30(ii) we know already that the global sections of Ω_{X/K}^{na} do not vanish. It suffices therefore to show that, for every affinoid subdomain U := Spa(A, A^{∘}) ⊂ X containing x, the natural map Ω_{P/K} ⊗_{P} A → Ω_{A/K} is injective. However, the kernel of this map is a quotient of H_{1}(𝕃_{A/P}), hence it suffices to show the following

Claim 7.3.36. H_{i}(𝕃_{A/P}) = 0 for every i > 0.

Proof of the claim: In light of claim 7.3.27, it suffices to show that the map P → A is regular. Then, in view of claims 7.3.25 and 7.3.26(ii), it suffices to show that, for every maximal ideal 𝔫 ⊂ A, the ring A is formally smooth over P for its 𝔫-adic topology. Let 𝔮 := 𝔫 ∩ P; since A is flat over P, [28, Ch.IV, Th.19.7.1] reduces to showing that the induced morphism P/𝔮 → A/𝔮 is formally smooth, which is trivial, since the latter is an isomorphism. □

7.3.37. It follows easily from proposition 7.3.23(ii) and lemma 7.2.1 that 𝕃_{X/K} ≃ Ω_{X/K}[0], so the extensions 0 → 𝒪_{X} → ℰ → 𝒪_{X} → 0 are classified by Ext_{𝒪_{X}}^{1} (Ω_{X/K}^{na}, 𝒪_{X}). In order to show that the latter is not trivial, it suffices, in view of lemma 7.3.34, to exhibit a nonzero map Ω_{X/K}^{na} → Q := 𝒪_{X}(∞)/𝒪_{X}. However, Q is a skyscaper sheaf sitting at the point p, with stalk equal to F/𝒪_{X,p}, where F := Frac(𝒪_{X,p}). We are therefore reduced to showing the existence of a nonzero map Ω_{X/K,p}^{na} → F. However, we deduce easily from lemma 7.3.30 that Ω_{X/K,p}^{na} is an F -vector space, and by lemma 7.3.35 the latter does not vanish, provided that either char(K) = 0 or K has characteristic p > 0 and [K : K^{p}] = ∞.

7.3.38. The foregoing results notwithstanding, there is at least one situation in which the abstract topos deformation theory of an adic space is the same as its analytic deformation theory. This is explained by the following final proposition. Proposition 7.3.39. Suppose that char(K) = p > 0 and that [K : K^{p}] < ∞. Then, for every morphism X → Y of adic spaces locally of finite type over Spa(K, K^{+}), the natural map 𝕃_{X/Y} → 𝕃_{X/Y}^{an} is an isomorphism in D(𝒪_{X}-Mod).

Proof. It suffices to show the corresponding statement for maps of affinoid rings A → B, hence choose a presentation B ≃ A⟨T_{1}, ..., T_{n}⟩/I. We have to verify that the induced maps H_{i}(𝕃_{B/A}) → H_{i}(𝕃_{B/A}^{an}) are isomorphisms for every i ∈ ℕ. The sequence A → P := A⟨T_{1}, ..., T_{n}⟩ → B yields by transitivity two distinguished triangles : one relative to the usual cotangent complex and one relative to the analytic cotangent complex. Hence, by the five lemma, we are reduced to showing that the maps H_{i}(B ⊗_{P} 𝕃_{P/A}) → H_{i}(B ⊗_{P} 𝕃_{P/A}^{an}) and H_{i}(𝕃_{B/P}) → H_{i}(𝕃_{B/P}^{an} ) are isomorphisms for every i ∈ ℕ. For the latter we appeal to lemma 7.2.42. Also, it follows easily from proposition 7.3.23 and theorem 7.2.39 that both H_{i}(B ⊗_{P} 𝕃_{P/A}) and H_{i}(B ⊗_{P} 𝕃_{P/A}^{an}) vanish for i > 1, hence we are reduced to showing that the map

(7.3.40)  τ_{[−1}𝕃_{B/A} → τ_{[−1}𝕃_{B/A}^{an}

is an isomorphism in D(𝒪_{X}-Mod). By proposition 7.2.44 we have: τ_{[−1}𝕃_{B/A}^{an} ≃ (0 → I/I^{2} → B ⊗_{P} Ω_{P/A}^{an} → 0); on the other hand, proposition 7.3.23 and [46, Ch.III, Cor.1.2.9.1] give: τ_{[−1}𝕃_{B/A} ≃ (0 → I/I^{2} → B ⊗_{P} Ω_{P/A} → 0). Both these isomorphisms are natural, so (7.3.40)is represented by the obvious map between the latter complexes. Thus, we are reduced to showing that the natural map Ω_{P/A} → Ω_{P/A}^{an} is an isomorphism. Let C ⊂ P be the Asubalgebra generated by the image of the Frobenius endomorphism Φ : P → P; a standard calculation shows that Ω_{P/A} ≃ Ω_{P/C}. However, under the standing assumptions, P is finite over its subalgebra C, therefore Ω_{P/A} is a finite P -module, and the claim follows. □

7.4. Analytic geometry over a deeply ramified base. In this section we assume throughout that (K, | · |) is a deeply ramified complete valued field, with valuation of rank one. Recall that a ∈ K^{×} denotes a topologically nilpotent element of K.

If 𝔛 is a formal scheme of finite type over Spf K^{+}, we will sometimes write 𝕃_{𝔛/K^{+}}^{an} instead of 𝕃_{𝔛/Spf K^{+}}^{an}. Similarly we define 𝕃_{X/K}^{+} for an adic space X of finite type over Spa(K, K^{+}), and set Ω_{X/K}^{+} := H_{0}(𝕃_{X/K}^{+}).

Theorem 7.4.1. Let X be a smooth adic space over Spa(K, K^{+}). Then 𝕃_{X/K}^{+} ≃ Ω_{X/K}^{+}[0] in D^{−}(𝒪_{X}^{+}-Mod), and Ω_{X/K}^{+} is a flat sheaf of 𝒪_{X}^{+}-modules.

Proof. Both assertions can be checked on the stalks, therefore let x ∈ X be any point. The stalk 𝒪_{X,x} is a local ring and its residue field κ(x) carries a natural valuation; the preimage in 𝒪_{X,x} of the corresponding valuation ring κ(x)^{+} is the subring 𝒪_{X,x}^{+}. Let I := ⋂_{n∈ℕ} a^{n}𝒪_{X,x}^{+}; it follows from this description that κ(x)^{+} = 𝒪_{X,x}^{+}/I. Especially, we have

(7.4.2)  𝒪_{X,x}^{+}/a𝒪_{X,x}^{+} ≃ κ(x)^{+}/a · κ(x)^{+}.

Claim 7.4.3. Let M be an 𝒪_{X,x}^{+}-module, and suppose that a is regular on M. Then M/IM is a flat κ(x)^{+}-module.

Proof of the claim: By snake lemma we derive Ker(M/IM →^{·a} M/IM) ⊂ Coker(IM →^{·a} IM). However, it is clear that I = aI, so a is regular on M/IM, whence the latter is a torsion-free κ(x)^{+}-module and the claim follows.

Let 𝒰 be the cofiltered system of all affinoid open neighborhoods of x in X; for U ∈ 𝒰 let F_{U} be the filtered system of all K^{+}-subalgebras of 𝒪_{X}^{+}(U) of topologically finite presentation. We derive

(𝕃_{X/K}^{+})_{x} ⊗^{L}_{K^{+}} K^{+}/aK^{+} ≃ colim_{U∈𝒰} colim_{A∈F_{U}} 𝕃_{A/K^{+}}^{an} ⊗^{L}_{K^{+}} K^{+}/aK^{+}
≃ colim_{U∈𝒰} colim_{A∈F_{U}} 𝕃_{A/K^{+}} ⊗^{L}_{K^{+}} K^{+}/aK^{+}
≃ 𝕃_{𝒪_{X,x}^{+}/K^{+}} ⊗^{L}_{K^{+}} K^{+}/aK^{+}
≃ 𝕃_{κ(x)^{+}/K^{+}} ⊗^{L}_{K^{+}} K^{+}/aK^{+}.

Together with theorem 6.6.11, this implies already that scalar multiplication by a is an automorphism of H_{i}(𝕃_{X/K}^{+}), for every i > 0. However, according to theorem 7.2.39(ii), H_{i}(𝕃_{X/K}^{+}) is a K^{+}-torsion sheaf of 𝒪_{X}^{+}-modules, for i > 0, whence the first assertion. It also follows that (Ω_{X/K}^{+})_{x} is a torsion-free, hence flat, K^{+}-module. To prove that (Ω_{X/K}^{+})_{x} is a flat 𝒪_{X,x}^{+}-module, we remark first that (Ω_{X/K}^{an})_{x} ≃ (Ω_{X/K}^{+})_{x} ⊗_{K^{+}} K, and the latter is a flat 𝒪_{X,x}-module, since X is smooth over Spa(K, K^{+}). By lemma 5.2.1 it suffices therefore to show

Claim 7.4.4. (Ω_{X/K}^{+})_{x} ⊗_{K^{+}} K^{+}/aK^{+} is a flat 𝒪_{X,x}^{+} ⊗_{K^{+}} K^{+}/aK^{+}-module.

Proof of the claim: By claim 7.4.3 we know that (Ω_{X/K}^{+})_{x} ⊗_{𝒪_{X,x}^{+}} κ(x)^{+} is a flat κ(x)^{+}-module. In view of (7.4.2), the claim follows after base change to κ(x)^{+}/a · κ(x)^{+}. □

Definition 7.4.5. Let (𝔛_{α} | α ∈ I) be a system of formal schemes of finite presentation over Spf K^{+}, indexed by a small cofiltered category I.

(i) Let 𝔛_{∞} := lim_{α∈I} 𝔛_{α}, where the limit is taken in the category of locally ringed spaces. For every α ∈ I, let π_{α} : 𝔛_{∞} → 𝔛_{α} be the natural morphism of locally ringed spaces. We define Ω_{𝔛_{∞}/K^{+}}^{an} := colim_{α∈I^{o}} π_{α}^{∗}(Ω_{𝔛_{α}/K^{+}}^{an}), which is a sheaf of 𝒪_{𝔛_{∞}}-modules.

More generally, we let 𝕃_{𝔛_{∞}/K^{+}}^{an} := colim_{α∈I^{o}} π_{α}^{∗}(𝕃_{𝔛_{α}/K^{+}}^{an}).

(ii) We say that the cofiltered system (𝔛_{α} | α ∈ I) is deeply ramified if the natural morphism Ω_{𝔛_{∞}/K^{+}}^{an} → Ω_{𝔛_{∞}/K^{+}}^{an} ⊗_{K^{+}} K is an epimorphism.

Lemma 7.4.6. Let (𝔛_{α} | α ∈ I) be a cofiltered system as in definition 7.4.5. For any morphism β → α of I, let 𝔣_{αβ} : 𝔛_{β} → 𝔛_{α} be the corresponding morphism of formal Spf K^{+}-schemes. Moreover, for every α ∈ I, let Ω_{𝔛_{α}/K^{+}}^{tf} be the image of the morphism Ω_{𝔛_{α}/K^{+}}^{an} → Ω_{𝔛_{α}/K^{+}}^{an}⊗_{K^{+}}K (“tf” stays for torsion-free). The following two conditions are equivalent :

(i) The system (𝔛_{α} | α ∈ I) is deeply ramified.

(ii) For every α ∈ I there is a morphism β → α of I, such that the image of the natural morphism 𝔣_{αβ}^{∗}(Ω_{𝔛_{α}/K^{+}}^{tf}) → Ω_{𝔛_{β}/K^{+}}^{tf} is contained in the subsheaf a · Ω_{𝔛_{β}/K^{+}}^{tf}.

Proof. It is clear that (ii)⇒(i). We show that (i)⇒(ii). Under the above assumptions, every 𝔛_{α} is quasi-compact, hence we can cover it by finitely many affine formal schemes 𝔘_{i} := Spf A_{i} (i = 1, ..., n) of finite type over Spf K^{+}. Then, for every i = 1, ..., n, the restriction of Ω_{𝔛_{α}/K^{+}}^{an} to 𝔘_{i} is the coherent sheaf (Ω_{A_{i}/K^{+}}^{an})^{△} (notation of [26, Ch.I, §10.10.1]). Hence Ω_{𝔛_{α}/K^{+}}^{tf} is a coherent sheaf of 𝒪_{𝔛_{α}}-modules. For every morphism β → α, let

U_{αβ} := {x ∈ 𝔛_{β} | Im(𝔣_{αβ}^{∗}(Ω_{𝔛_{α}/K^{+}}^{tf})_{x} → (Ω_{𝔛_{β}/K^{+}}^{tf})_{x}) ⊂ a · (Ω_{𝔛_{β}/K^{+}}^{tf})_{x}}.

U_{αβ} is therefore a constructible open subset of 𝔛_{β}, and we denote its complement by Z_{αβ}. By assumption (i) we know that

lim_{β→α} Z_{αβ} = ⋂_{β→α} π_{α}^{−1}(Z_{αβ}) = ∅.

If we retopologize the reduced schemes Z_{αβ} by their constructible topologies, we get an inverse system of compact spaces, and deduce that some Z_{αβ} is empty by [18, Ch.I, §9, n.6, Prop.8(b)].

□

Example 7.4.7. The prototype of deeply ramified systems is given by the tower of morphisms

(7.4.8)  ... → 𝔹_{K^{+}}^{d}(0, ρ^{1/p^{n}}) →^{ϕ_{n}} 𝔹_{K^{+}}^{d}(0, ρ^{1/p^{n−1}}) →^{ϕ_{n−1}} ... →^{ϕ_{1}} 𝔹_{K^{+}}^{d}(0, ρ)

where, for any r = (r_{1}, ..., r_{d}) ∈ (K^{×})^{d}, we have denoted

𝔹_{K^{+}}^{d}(0, |r|) := Spf K^{+}⟨r_{1}^{−1}T_{1}, ..., r_{d}^{−1}T_{d}⟩

(i.e., the formal d-dimensional polydisc defined by the equations |T_{i}| ≤ |r_{i}|, i = 1, ..., d). The morphisms ϕ_{n} are induced by the ring homomorphisms T_{i} ↦ T_{i}^{p} (i = 1, ..., d). Notice that the tower (7.4.8)is defined whenever ρ_{i} ∈ Γ_{K}^{p^{∞}} := ⋂_{n∈ℕ} Γ_{K}^{p^{n}} for every i = 1, ..., d. We leave to the reader the verification that condition (ii) of lemma 7.4.6 is indeed satisfied. Lemma 7.4.9. Let 𝔛 := (𝔛_{α} | α ∈ I) be a cofiltered system as in definition 7.4.5.

(i) If 𝔜 := (𝔜_{α} | α ∈ I) → (𝔛_{α} | α ∈ I) is a morphism of cofiltered systems such that the induced morphisms of adic spaces d(𝔜_{α}) → d(𝔛_{α}) are unramified for every α ∈ I (cp. (7.2.26)), then 𝔜 is deeply ramified if 𝔛 is.

(ii) Let ℨ := (ℨ_{β} | β ∈ J) be another such cofiltered system, and suppose that 𝔛 and ℨ are isomorphic as pro-objects of the category of formal schemes. Then 𝔛 is deeply ramified if and only if ℨ is.

(iii) If 𝔛 and ℨ := (ℨ_{β} | β ∈ J) are two deeply ramified cofiltered systems, then the fibred product 𝔛 × ℨ := (𝔛_{α} ×_{Spf(K^{+})} ℨ_{β} | (α, β) ∈ I × J) is deeply ramified.

Proof. (i): by [14, Prop.2.2] the natural morphism Ω_{𝔛_{∞}/K^{+}}^{an} ⊗_{K^{+}} K → Ω_{𝔜_{∞}/K^{+}}^{an} ⊗_{K^{+}} K is an epimorphism; the claim follows easily. (ii) and (iii) are easy and shall be left to the reader. □

The counterpart of the above definitions for adic schemes is given in the following: Definition 7.4.10. Let (X_{α} | α ∈ I) be a system of adic spaces of finite type over Spa(K, K^{+}), indexed by a small cofiltered category I.

(i) Let (X_{∞}, 𝒪_{X_{∞}}, 𝒪_{X_{∞}}^{+}) := lim_{α∈I} (X_{α}, 𝒪_{X_{α}}, 𝒪_{X_{α}}^{+}), where the limit is taken in the category of locally ringed spaces. For every α ∈ I, let π_{α} : X_{∞} → X_{α} be the natural morphism of locally ringed spaces. We define Ω_{X_{∞}/K}^{+} := colim_{α∈I^{o}} π_{α}^{∗}(Ω_{X_{α}/K}^{+}), which is a sheaf of

𝒪_{X_{∞}}^{+}-modules. More generally, we let 𝕃_{X_{∞}/K}^{+} := colim_{α∈I^{o}} π_{α}^{∗}(𝕃_{X_{α}/K}^{+}).

(ii) We say that the cofiltered system (X_{α} | α ∈ I) is deeply ramified if the natural morphism Ω_{X_{∞}/K}^{+} → Ω_{X_{∞}/K}^{+} ⊗_{K^{+}} K is an epimorphism.

7.4.11. Let (X_{α} | α ∈ I) be as in definition 7.4.10. For every x ∈ X_{∞}, we let κ(x)^{+} := 𝒪_{X_{∞},x}^{+}/ ⋂_{n∈ℕ} a^{n}𝒪_{X_{∞},x}^{+}. The ring κ(x)^{+} is a filtered colimit of valuation rings, hence it is a valuation ring. Moreover, the image of a in κ(x)^{+} is topologically nilpotent for the valuation topology of κ(x)^{+}.

7.4.12. Given a cofiltered system \underline{𝔛} := (𝔛_{α} | α ∈ I) of formal schemes, one obtains a cofiltered system of adic spaces \underline{X} := (X_{α} := d(𝔛_{α}) | α ∈ I), and using (7.2.27) and lemma 7.4.9(i) one sees easily that \underline{X} is deeply ramified whenever \underline{𝔛} is. Together with example 7.4.7, this yields plenty of examples of deeply ramified systems of adic spaces. Proposition 7.4.13. Let \underline{X} := (X_{α} | α ∈ I) be a deeply ramified cofiltered system of adic spaces. Then, for every point x ∈ X_{∞}, the valuation ring κ(x)^{+} is deeply ramified.

Proof. For every K^{+}-module M, let us denote by T_{n}(M) the submodule of M annihilated by a^{n}. Furthermore, let T (M) := ⋃_{n∈ℕ} T_{n}(M). Since the cofiltered system \underline{X} is deeply ramified, we have: (Ω_{X_{∞}/K}^{+})_{x} = T (Ω_{X_{∞}/K}^{+})_{x}+a·(Ω_{X_{∞}/K}^{+})_{x}. To lighten notation, let 𝒪_{x}^{+} := 𝒪_{X_{∞},x}^{+}. From lemma 7.1.25 one deduces easily that the natural map Ω_{𝒪_{x}^{+}/K^{+}} → (Ω_{X_{∞}/K}^{+})_{x} is surjective with a-divisible kernel. Hence, by snake lemma, the induced map T_{n}(Ω_{𝒪_{x}^{+}/K^{+}} ) → T_{n}(Ω_{X_{∞}/K}^{+})_{x} is surjective for every n, and a fortiori the map T (Ω_{𝒪_{x}^{+}/K^{+}} ) → T (Ω_{X_{∞}/K}^{+})_{x} is onto. It follows easily that Ω_{𝒪_{x}^{+}/K^{+}} = T (Ω_{𝒪_{x}^{+}/K^{+}} ) + a · Ω_{𝒪_{x}^{+}/K^{+}} , and consequently: Ω_{κ(x)^{+}/K^{+}} = T (Ω_{κ(x)^{+}/K^{+}}) + a·Ω_{κ(x)^{+}/K^{+}}. However, it follows easily from theorem 6.6.11 that T (Ω_{κ(x)^{+}/K^{+}}) = 0, so finally Ω_{κ(x)^{+}/K^{+}} = a · Ω_{κ(x)^{+}/K^{+}} and

(7.4.14)  𝕃_{κ(x)^{+}/K^{+}} ⊗^{L}_{K^{+}} K^{+}/aK^{+} ≃ 0.

On the other hand, if (E, | · |_{E}) is any valued field extension of κ(x)^{+}, we have

(7.4.15)  H_{i}(𝕃_{E^{+}/K^{+}} ⊗_{K^{+}} K^{+}/aK^{+}) = 0 for all i > 0

by theorem 6.6.11. From (7.4.14), (7.4.15) and transitivity for the tower K^{+} ⊂ κ(x)^{+} ⊂ E^{+}, we derive H_{i}(𝕃_{E^{+}/κ(x)^{+}} ⊗_{K^{+}} K^{+}/aK^{+}) = 0 for all i > 0. Again by theorem 6.6.11 and remark 6.6.14 we conclude. □

7.4.16. Let \underline{X} be a cofiltered system as in definition 7.4.10. Let 𝒜 be a sheaf of 𝒪_{X_{∞}}^{+}-algebras. We say that 𝒜 is a weakly étale 𝒪_{X_{∞}}^{+}-algebra if, for every x ∈ X_{∞}, the stalk 𝒜_{x}^{a} is a weakly étale 𝒪_{X_{∞},x}^{+a}-algebra. Theorem 7.4.17. Suppose that K is deeply ramified, and let \underline{X} := (X_{α} | α ∈ I) be a deeply ramified cofiltered system. Let also \underline{f} : \underline{Y} := (Y_{α} | α ∈ I) → \underline{X} be a morphism of cofiltered systems, such that the morphisms Y_{α} → X_{α} are finite étale for every α ∈ I. Then f_{∞∗}𝒪_{Y_{∞}}^{+} is a weakly étale 𝒪_{X_{∞}}^{+} algebra.

Proof. To lighten notation, let us write 𝒪^{+} (resp. 𝒜^{+}) instead of 𝒪_{X_{∞}}^{+} (resp. f_{∞∗}𝒪_{Y_{∞}}^{+}). For every α ∈ I consider the cofiltered system \underline{Z}(α) := \underline{X} ×_{X_{α}} Y_{α} indexed by I/α (the category of morphisms β → α), which is defined by setting \underline{Z}(α)_{β→α} := X_{β} ×_{X_{α}} Y_{α}. We have obvious morphisms of cofiltered systems \underline{f}_{/α} : \underline{Z}(α) → \underline{X}; the sheaf 𝒜^{+} is the colimit of the sheaves f_{/α∞∗}𝒪_{Z(α)_{∞}}^{+}. Hence it suffices to prove the assertion for the latter sheaves, and therefore in order to show the theorem we can and do assume that there exists α ∈ I such that, for every β → α, the induced commutative diagram

\xymatrix{Y_{β} \ar[r] \ar[d] & Y_{α} \ar[d] \\ X_{β} \ar[r] & X_{α}}

is cartesian. Let x ∈ X_{∞}.

Claim 7.4.18. 𝒜_{x}^{+} is a flat 𝒪_{x}^{+}-algebra.

Proof of the claim: On one hand, by assumption 𝒜_{x}^{+}[1/a] is a flat 𝒪_{x}^{+}[1/a]-algebra; on the other hand, 𝒜_{x}^{+} ⊗_{𝒪_{x}^{+}} κ(x)^{+} is a flat κ(x)^{+}-module by claim 7.4.3, so that 𝒜_{x}^{+}/a𝒜_{x}^{+} is a flat 𝒪_{x}^{+}/a𝒪_{x}^{+}-module; thus the claim follows from lemma 5.2.1.

Let e ∈ C := 𝒜_{x}^{+} ⊗_{𝒪_{x}^{+}} 𝒜_{x}^{+}[1/a] be the idempotent provided by lemma 3.1.4. In view of claim 7.4.18, we only have to show that ε · e ∈ C^{+} := 𝒜_{x}^{+} ⊗_{𝒪_{x}^{+}} 𝒜_{x}^{+} for every ε ∈ 𝔪. Let \overline{e} be the image of e in C ⊗_{𝒪_{x}} κ(x). Set I := ⋂_{n≥0} a^{n}𝒪_{x}^{+}.

Claim 7.4.19. 𝒜_{x}^{+}/I𝒜_{x}^{+} is the integral closure of of κ(x)^{+} in 𝒜_{x}/I𝒜_{x}.

Proof of the claim: First of all, since I = aI, we have I𝒜_{x} = I𝒜_{x}^{+}, and thus the natural map 𝒜_{x}^{+}/I𝒜_{x}^{+} → 𝒜_{x}/I𝒜_{x} is injective. Next, we remark that 𝒜_{x}^{+} is the integral closure of 𝒪_{x}^{+} in 𝒜_{x}; indeed, this follows from [12, §6.2.2, Lemma 3, Prop.2], after taking colimits. This already shows that 𝒜_{x}^{+}/I𝒜_{x}^{+} is a subalgebra of 𝒜_{x}/I𝒜_{x} integral over κ(x)^{+}. To conclude, suppose that \overline{f} ∈ 𝒜_{x}/I𝒜_{x} satisfies an integral equation: \overline{f}^{n} +\overline{b}_{1} ·\overline{f}^{n−1} +...+\overline{b}_{n} = 0, for certain \overline{b}_{1}, ..., \overline{b}_{n} ∈ 𝒜_{x}^{+}/I𝒜_{x}^{+}; pick arbitrary representatives f ∈ 𝒜_{x}, b_{i} ∈ 𝒜_{x}^{+} of these elements. Then f^{n} + b_{1} · f^{n−1} + ... + b_{n} ∈ I𝒜_{x}. Since I𝒜_{x} = I𝒜_{x}^{+}, we deduce that f is integral over 𝒜_{x}^{+}, so f ∈ 𝒜_{x}^{+} and the claim follows.

Claim 7.4.20. (𝒜_{x}^{+}/I𝒜_{x}^{+})^{a} is an étale κ(x)^{+a}-algebra.

Proof of the claim: In view of propositions 2.4.18(ii) and 2.4.19, it suffices to show that (𝒜_{x}^{+}/I𝒜_{x}^{+})^{a} is weakly étale over κ(x)^{+a}. Let 𝔭 be the height one prime ideal of κ(x)^{+}; then κ(x)_{𝔭}^{+} is a rank one valuation ring and the localization map induces isomorphisms κ(x)^{+a} →^{∼} κ(x)_{𝔭}^{+a}, (𝒜_{x}^{+}/I𝒜_{x}^{+})^{a} →^{∼} (𝒜_{x}^{+}/I𝒜_{x}^{+})_{𝔭}^{a} (recall that the standing basic setup is the standard setup of K^{+}). Taking claim 7.4.19 into account, it suffices then to show that (𝒜_{x}^{+}/I𝒜_{x}^{+})_{𝔮}^{a} is weakly étale over κ(x)_{𝔭}^{+a}, for every prime ideal 𝔮 ⊂ 𝒜_{x}^{+}/I𝒜_{x}^{+} of height one. By proposition 7.4.13, κ(x)^{+} is deeply ramified, hence the same holds for κ(x)_{𝔭}^{+}. Let κ(x)^{s} be a separable closure of κ(x), and κ(x)^{s+} a valuation ring of κ(x)^{s} dominating κ(x)_{𝔭}^{+}; we can assume that (𝒜_{x}^{+}/I𝒜_{x}^{+})_{𝔮} ⊂ κ(x)^{s+}. From claim 7.4.19 we deduce that (𝒜_{x}^{+}/I𝒜_{x}^{+})_{𝔮} is a valuation ring, hence κ(x)^{s+} is a faithfully flat (𝒜_{x}^{+}/I𝒜_{x}^{+})_{𝔮}-algebra; then the claim follows in view of lemma 3.1.2(viii) and proposition 6.6.2.

From claim 7.4.20 we deduce that ε · \overline{e} ∈ C^{+} ⊗_{𝒪_{x}^{+}} κ(x)^{+} for every ε ∈ 𝔪. To conclude, it suffices therefore to remark:

Claim 7.4.21. For every ε ∈ 𝔪, we have ε · e ∈ C^{+} if and only if ε · \overline{e} ∈ C^{+} ⊗_{𝒪_{x}^{+}} κ(x)^{+}.

Proof of the claim: Let e_{ε} ∈ C^{+} be any lifting of ε · \overline{e}; then ε · e − e_{ε} is in the kernel of the projection C → C ⊗_{𝒪_{x}^{+}} κ(x). Let n ∈ ℕ be a large enough integer, so that a^{n} · e ∈ C^{+}; it follows that a^{n} ·(ε·e−e_{ε}) is in the kernel of the projection C^{+} → C^{+} ⊗_{𝒪_{x}^{+}} κ(x)^{+}, consequently a^{n} · (ε · e − e_{ε}) = a^{n} · c for some c ∈ C^{+}. Since a is a non-zero-divisor in C^{+}, it follows that ε · e = c + e_{ε} ∈ C^{+}, as required. □

7.5. Semicontinuity of the discriminant. Definition 7.5.1. Let (V, 𝔪) be a basic setup, A a V^{a}-algebra and P an almost projective Amodule of constant rank r ∈ ℕ. Suppose moreover that P is endowed with a bilinear form b : P ⊗_{A} P → A. We let β : P → P^{∗} be the A-linear morphism defined by the rule: β(x)(y) := b(x ⊗ y) for every x, y ∈ P_{∗}. The discriminant of the pair (P, b) is the ideal

𝔡_{A}(P, b) := Ann_{A} Coker(Λ_{A}^{r}β : Λ_{A}^{r}P → Λ_{A}^{r}P^{∗}).

7.5.2. As a special case, we can consider the pair (B, t_{B/A}) consisting of an A-algebra B which is almost projective of constant rank r over A, and its trace form t_{B/A}. In this situation, we let 𝔡_{B/A} := 𝔡_{A}(B, t_{B/A}), and we call this ideal the discriminant of the A-algebra B. Lemma 7.5.3. Let B be an almost projective A-algebra of constant rank r as an A-module. Then B is étale over A if and only if 𝔡_{B/A} = A.

Proof. By theorem 4.1.14, it is clear that 𝔡_{B/A} = A when B is étale over A. Suppose therefore that 𝔡_{B/A} = A; it follows that Λ_{A}^{r}τ_{B/A} is an epimorphism. However, by proposition 4.3.27, Λ_{A}^{r}B and Λ_{A}^{r}B^{∗} are invertible A-modules. It then follows by lemma 4.1.5(iv) that Λ_{A}^{r}τ_{B/A} is an isomorphism, hence τ_{B/A} is an isomorphism, by virtue of proposition 4.4.28. One concludes again by theorem 4.1.14. □

Lemma 7.5.4. Let (V, 𝔪) be the standard setup associated to a valued field (K, | · |) (cp. (6.1.15), especially, V := K^{+}). Let P′ ⊂ P be two almost projective V^{a}-modules of constant rank equal to r. Let b : P ⊗_{V^{a}} P → V^{a} be a bilinear form, such that b ⊗_{V^{a}} 1_{K^{a}} is a perfect pairing, and denote by b′ the restriction of b to P′ ⊗_{V^{a}} P′. Then we have:

𝔡_{V^{a}}(P′, b′) = F_{0}(P/P′)^{2} · 𝔡_{V^{a}}(P, b).

Proof. Let j : P′ → P be the imbedding, β : P → P^{∗} (resp. β′ : P′ → P^{′∗}) the V^{a}-linear morphism associated to b (resp. to b′). The assumptions implies that Λ_{V^{a}}^{r}j, Λ_{V^{a}}^{r}j^{∗}, Λ_{V^{a}}^{r}β and Λ_{V^{a}}^{r}β′ are all injective, and clearly we have

Λ_{V^{a}}^{r}j^{∗} ∘ Λ_{V^{a}}^{r}β ∘ Λ_{V^{a}}^{r}j = Λ_{V^{a}}^{r}β′.

There follow short exact sequences:

0 → Coker Λ_{V^{a}}^{r}β → Coker Λ_{V^{a}}^{r}(j^{∗} ∘ β) → Coker Λ_{V^{a}}^{r}j^{∗} → 0
0 → Coker Λ_{V^{a}}^{r}j → Coker Λ_{V^{a}}^{r}β′ → Coker Λ_{V^{a}}^{r}(j^{∗} ∘ β) → 0.

Using lemma 6.3.1 and remark 6.3.5(ii), we deduce:

F_{0}(Coker Λ_{V^{a}}^{r}β′) = F_{0}(Coker Λ_{V^{a}}^{r}j) · F_{0}(Coker Λ_{V^{a}}^{r}β) · F_{0}(Coker Λ_{V^{a}}^{r}j^{∗}).

Claim 7.5.5. Let Q be an almost finitely generated projective A-module of constant rank r, which is also uniformly almost finitely generated. Let Q′ ⊂ Q any submodule. Then:

F_{0}(Q/Q′) = Ann_{A}(Coker(Λ_{A}^{r}Q′ → Λ_{A}^{r}Q)).

Proof of the claim: By lemma 2.4.6 and proposition 2.3.24(ii) we can check the identity after a faithfully flat base change, hence we can suppose that Q ≃ A^{r}, in which case the identity holds by definition of the Fitting ideal F_{0}.

Notice that Λ_{V^{a}}^{r}P and Λ_{V^{a}}^{r}P′ are invertible V^{a}-modules by virtue of proposition 4.3.27, consequently claim 7.5.5 applies and yields: F_{0}(Coker Λ_{V^{a}}^{r}β′) = 𝔡_{V^{a}}(P′, b′), F_{0}(Coker Λ_{V^{a}}^{r}β) = 𝔡_{V^{a}}(P, b) and F_{0}(Coker Λ_{V^{a}}^{r}j) = F_{0}(Coker Λ_{V^{a}}^{r}j^{∗}) = F_{0}(P/P′). □

7.5.6. After these generalities, we return to the standard setup (K^{+}, 𝔪) of this chapter, associated to a valued field (K, | · |) of rank one (cp. (6.1.15)). Consider an étale K-algebra L; we denote by W_{L} the integral closure of K^{+} in L. L is the product of finitely many separable field extensions of K, therefore W_{L}^{a} is an almost projective K^{+a}-module of constant rank n, by proposition 6.3.6. Hence, the discriminant of W_{L}^{a} over K^{+a} is defined, and to lighten notation, we will denote it by 𝔡_{L/K}^{+}. Furthermore, since L is étale over K, it is clear that 𝔡_{L/K}^{+} is a fractional ideal of K^{+a} (cp. (6.1.16)). Let | · | : Div(K^{+a}) → Γ_{K}^{∧} be the isomorphism provided by lemma 6.1.19. We obtain an element |𝔡_{L/K}^{+}| ∈ Γ_{K}^{∧}; after choosing (cp. example (6.1.21)(vi)) an order preserving isomorphism

(7.5.7)  ((Γ_{K} ⊗_{ℤ} ℚ)^{∧}, ≤) →^{∼} (ℝ_{>0}, ≤)

on the multiplicative group of positive real numbers, we can then view |𝔡_{L/K}^{+}| ∈ (0, 1].

Lemma 7.5.8. Let K, L be as in (7.5.6)and denote by K^{∧} the completion of K for the valuation topology. Set L^{∧} := K^{∧} ⊗_{K} L. Then K^{∧+} ⊗_{K^{+}} 𝔡_{L/K}^{+} = 𝔡_{L^{∧}/K^{∧}}^{+}.

Proof. Since the base change K → K^{∧} is faithfully flat, everything is clear from the definitions, once we have established that W_{L^{∧}} ≃ K^{∧+} ⊗_{K^{+}} W_{L}. However, both rings can be identified with the a-adic completion (W_{L})^{∧} of W_{L}, so the assertion follows. □

7.5.9. Let X be an adic space locally of finite type over Spa(K, K^{+}). X is a locally spectral space, and every point x ∈ X admits a unique maximal generisation r(x) ∈ X. The valuation ring κ(r(x))^{+} has rank one, and admits a natural imbedding K^{+} ⊂ κ(r(x))^{+}, continuous for the valuation topologies; especially, the image of the topologically nilpotent element a is topologically nilpotent in κ(r(x))^{+}. This imbedding induces therefore a natural isomorphism of completed value groups

(Γ_{K} ⊗_{ℤ} ℚ)^{∧} →^{∼} (Γ_{κ(r(x))} ⊗_{ℤ} ℚ)^{∧}.

In particular, our original choice of isomorphism (7.5.7)fixes univocally a similar isomorphism for every point r(x). We denote by M(X) the set r(X) endowed with the quotient topology induced by the mapping X → r(X) : x ↦ r(x). This topology is coarser than the subspace topology induced by the imbedding into X. The mapping x ↦ r(x) is a retraction of X onto the subset M(X) of its maximal points. If X is a quasi-separated quasi-compact adic space, M(X) is a compact Hausdorff topological space ([45, 8.1.8]).

7.5.10. Let X be as in (7.5.9), and let f : Y → X be a finite étale morphism of adic spaces. For every point x ∈ X, the fibre E(x) := (f_{∗}𝒪_{Y})_{x} ⊗_{𝒪_{X,x}} κ(x) is a finite étale κ(x)-algebra. If now x ∈ M(X), we can consider the discriminant 𝔡_{E(x)/κ(x)}^{+} defined as in (7.5.6) (warning: notice that the definition makes sense when we choose the standard setup associated to the valuation ring κ(x)^{+}; since it may happen that the valuation of K is discrete and that of κ(x) is not discrete, the setups relative to K and to κ(x) may not agree in general). Upon passing to absolute values, we finally obtain a real valued function:

𝔡_{Y/X}^{+} : M(X) → (0, 1] x ↦ |𝔡_{E(x)/κ(x)}^{+}|.

The study of the function 𝔡_{Y/X}^{+} is reduced easily to the case where X (hence Y) are affinoid. In such case, one can state the main result in a more general form, as follows.

Definition 7.5.11. Let A be any (commutative unitary) ring.

(i) We denote by N (A) the set consisting of all multiplicative ultrametric seminorms | · | : A → ℝ. For every x ∈ N (A) and f ∈ A we write usually |f(x)| in place of x(f). N (A) is endowed with the coarsest topology such that, for every f ∈ A, the real-valued map |f| : N (A) → ℝ given by the rule: x ↦ |f(x)|, is continuous.

(ii) For every x ∈ N (A), we let Supp(x) := {f ∈ A | |f(x)| = 0}. Then Supp(x) is a prime ideal and we set κ(x) := Frac(A/Supp(x)). The seminorm x induces a valuation on the residue field κ(x), and as usual we denote by κ(x)^{+} its valuation ring.

(iii) Let A → B be a finite étale morphism. For every x ∈ N (A), we let E(x) := B ⊗_{A} κ(x). Then E(x) is an étale κ(x)-algebra, so we can define 𝔡_{B/A}^{+}(x) := 𝔡_{E(x)/κ(x)}^{+} (cp. the warning in (7.5.10)). By setting x ↦ |𝔡_{B/A}^{+}(x)| we obtain a well-defined function

|𝔡_{B/A}^{+}| : N (A) → (0, 1].

7.5.12. If X = Spa(A, A^{+}), with A a complete K-algebra of topologically finite type, then M(X) is naturally homeomorphic to the subspace M(A) of N (A) consisting of the continuous seminorms that extend the absolute value of K given by (7.5.7). It is shown in [10, §1.2] that M(A) is a compact Hausdorff space, for every Banach K-algebra A.

Proposition 7.5.13. Let A be a ring, B a finite étale A-algebra. Then the function |𝔡_{B/A}^{+}| is lower semi-continuous (i.e. it is continuous for the topology of (0, 1] whose open subsets are of the form (c, 1], c ∈ [0, 1]).

Proof. Let f ∈ A be any element; notice that N (A[1/f]) is naturally homeomorphic to the open subset U(f) := {x ∈ N (A) | |f(x)| ≠ 0}. Hence, after replacing A by some localization, we can assume that B is a free A-module, say of rank n. For every b ∈ B, let χ(b, T) := T^{n} + s_{1}(b) · T^{n−1} + ... + s_{n}(b) be the characteristic polynomial of the A-linear endomorphism B → B given by the rule b′ ↦ b′ · b.

Claim 7.5.14. For every point x ∈ N (A) and every b ∈ B, the following are equivalent:

(i) b ⊗ 1 ∈ W_{E(x)}.
(ii) |s_{i}(b)(x)| ≤ 1 for i = 1, ..., n.

Proof of the claim: Indeed, if (ii) holds, then the image of χ(b, T) in κ(x)[T] is a monic polynomial with coefficients in κ(x)^{+} and b ⊗ 1 is one of its roots (Cayley-Hamilton), hence b ⊗ 1 is integral over κ(x)^{+}, which is (i). Conversely, if (i) holds, let E(x) = ∏_{j=1}^{k} E_{j} be the decomposition of E(x) as product of finite separable extensions of κ(x), and let b_{j} ∈ E_{j} be the image of b. It follows that b_{j} ∈ W_{E_{j}} for every j = 1, ..., k, and moreover the image \overline{χ}(b, T) of χ(b, T) in κ(x)[T] decomposes as a product ∏_{j=1}^{k} χ(b_{j}, T). It suffices therefore to show that the coefficients s_{ij}(b_{j}) satisfy (ii) for every i ≤ n and j ≤ k, so we can assume that E(x) is a field. Let m_{b}(T) ∈ κ(x)[T] be the minimal polynomial of b ⊗ 1; it is well known that \overline{χ}(b, T) divides m_{b}(T)^{n}, hence the roots of \overline{χ}(b, T) are conjugates of b under the action of G := Gal(κ(x)^{a}/κ(x)). Let C be the integral closure of κ(x)^{+} in a finite Galois extension of κ(x) containing E(x); C is an integral κ(x)^{+}-algebra and the Galois conjugates of b ⊗ 1 are all contained in C. Since the latter are the roots of \overline{χ}(b, T), the elements s_{i}(b) ⊗ 1 are symmetric polynomials of the elements σ(b ⊗ 1) (σ ∈ G), so s_{i}(b) ⊗ 1 ∈ C ∩ κ(x) = κ(x)^{+}, which is (ii).

Let t_{B/A} be the trace morphism of the A-algebra B, and let x ∈ N (A). Then the trace morphism t_{x} := t_{E(x)/κ(x)} equals t_{B/A} ⊗_{A} 1_{κ(x)}.

Claim 7.5.15. For every real number ε > 0 we can find a free κ(x)^{+}-submodule W_{ε} of W_{E(x)}, such that W_{ε} ⊗_{κ(x)^{+}} κ(x) = E(x) and

(7.5.16)  |𝔡_{κ(x)^{+}}(W_{ε}, t_{x})| + ε > |𝔡_{B/A}^{+}(x)|.

Proof of the claim: From claim 6.3.8, we derive that, for every positive real number δ < 1 there exists a free finitely generated κ(x)^{+}-submodule W_{δ} ⊂ W_{E(x)} such that |F_{0}(W_{E(x)}^{a}/W_{δ}^{a})| > δ; in view of lemma 7.5.4, the claim follows easily.

Let w_{1}, ..., w_{n} be a basis of W_{ε}; up to replacing A by a localization A[1/g] for some g ∈ A, we can write w_{i} = b_{i} ⊗ 1, for some b_{i} ∈ B (i = 1, ..., n). Consequently:

(7.5.17)  |𝔡_{κ(x)^{+}}(W_{ε}, t_{x})| = | det(t_{B/A}(b_{i} ⊗ b_{j}))(x)|.

By claim 7.5.14 we have |s_{j}(b_{i})(x)| ≤ 1 for i, j = 1, ..., n. Let 1 > δ > 0 be a real number; for every i, j ≤ n, we define an open neighborhood U_{ij} of x in N (A) as follows. Suppose first that |s_{j}(b_{i})(x)| < 1; since the real-valued function y ↦ |s_{j}(b_{i})(y)| is continuous on N (A) (for the standard topology of ℝ), we can find U_{ij} such that |s_{j}(b_{i})(y)| ≤ 1 for all y ∈ U_{ij}.

Suppose next that |s_{j}(b_{i})(x)| = 1; then, up to replacing A by A[1/s_{j}(b_{i})], we can assume that s_{j}(b_{i}) is invertible in A. We pick U_{ij} such that

(7.5.18)  |s_{j}(b_{i})(y)| ≤ 1 + δ for every y ∈ U_{ij}.

We set U := ⋂_{1≤i,j≤n} U_{ij}. Next, we define, for every y ∈ U, an element c_{y} ∈ A, as follows. Choose α, β ≤ n such that |s_{β}(b_{α})(y)| = max_{1≤i,j≤n} |s_{j}(b_{i})(y)|. If |s_{β}(b_{α})(y)| ≤ 1, then set c_{y} := 1; if |s_{β}(b_{α})(y)| > 1, set c_{y} := s_{β}(b_{α})^{−1}. Then |s_{j}(c_{y} · b_{i})(y)| ≤ 1 for every i, j = 1, ..., n and every y ∈ U. Let \overline{W}_{y} be the κ(y)^{+}-submodule of B ⊗_{A} κ(y) spanned by the images of c_{y} · b_{1}, ..., c_{y} · b_{n}. It follows that \overline{W}_{y} ⊂ W_{E(y)} for every y ∈ U. We compute:

\begin{matrix}|𝔡_{Y/X}^{+}(y)| ≥ |𝔡_{κ(y)^{+}}(\overline{W}_{y}, t_{y})| = & | det(t_{B/A}(c_{y} · b_{i} ⊗ c(y) · b_{j}))(y)| \\ = & |c_{y}(y)|^{2n} · | det(t_{B/A}(b_{i} ⊗ b_{j}))(y)|. \end{matrix}

However, the real-valued function y ↦ | det(t_{B/A}(b_{i} ⊗ b_{j}))(y)| is continuous on N (A), therefore, combining (7.5.16) and (7.5.17), we see that, up to shrinking further the open neighborhood U, we can assume that | det(t_{B/A}(b_{i} ⊗ b_{j}))(y)| + ε > |𝔡_{B/A}^{+}(x)| for all y ∈ U, so finally:

|𝔡_{B/A}^{+}(y)| ≥ (1 − δ)^{2n} · (|𝔡_{B/A}^{+}(x)| − ε) for every y ∈ U

which implies the claim. □

Theorem 7.5.19. Let Y → X be as in (7.5.10). Then the map 𝔡_{Y/X}^{+} is lower semi-continuous.

Proof. We can assume that X = Spa(A, A^{+}), where A is a complete K-algebra of topologically finite type, and therefore Y = Spa(B, B^{+}), for a finiteétale A-algebra B. Then there is a natural homeomorphism ω : M(X) →^{∼} M(A), so the theorem follows from proposition 7.5.13 and : Claim 7.5.20. 𝔡_{Y/X}^{+} = 𝔡_{B/A}^{+} ∘ ω.

Proof of the claim: Let x ∈ M(X); x corresponds to a rank one valuation of A, whose value group we identify with (a subgroup of) ℝ_{>0} according to (7.5.9). The resulting multiplicative seminorm is ω(x). We derive easily a natural imbedding ι : κ(ω(x)) ⊂ κ(x), compatible with the identifications of value groups. One knows moreover that ι induces an isomorphism on completions ι^{∧} : κ(ω(x))^{∧} →^{∼} κ(x)^{∧}, so the claim follows from lemma 7.5.8. □

Lemma 7.5.21. Let (K_{α}, |·|_{α} | α ∈ I) be a system of valued field extensions of (K, |·|), indexed by a filtered small category I and such that K_{α}^{+} is a valuation ring of rank one for every α ∈ I. Let moreover L be a finite étale K_{β}-algebra, for some β ∈ I. Set L_{α} := L ⊗_{K_{β}} K_{α} for every morphism β → α in I. Then :

(i) (K_{∞}, | · |_{∞}) := colim_{α∈I} (K_{α}, | · |_{α}) is a valued field extension of (K, | · |), with valuation ring of rank one.

(ii) Set L_{∞} := L ⊗_{K_{β}} K_{∞}. Then, for every sequence of morphisms β → γ → α in I, we

have |𝔡_{L_{α}/K_{α}}^{+}| ≥ |𝔡_{L_{γ}/K_{γ}}^{+}|, and moreover: lim_{(β→α)∈I}|𝔡_{L_{α}/K_{α}}^{+}| = |𝔡_{L_{∞}/K_{∞}}^{+}|.

Proof. (i) is obvious. The proof of the first assertion in (ii) is easy and shall be left to the reader. For the second assertion in (ii) we remark that, due to claim 6.3.8, for every ε < 1 there exists a free K_{∞}^{+}-submodule W_{ε} ⊂ W_{L_{∞}} of finite type, such that |F_{0}(W_{L_{∞}}^{a}/W_{ε}^{a})| > ε. We can find α ∈ I such that W_{ε} = W_{0} ⊗_{K_{α}^{+}} K_{∞}^{+} for some free K_{α}^{+} submodule W_{0} ⊂ L_{α}^{+}. It then follows from lemma 7.5.4 that

|𝔡_{L_{α}/K_{α}}^{+}| ≥ |𝔡_{K_{α}^{+}}(W_{0}, t_{L_{α}/K_{α}})| = |𝔡_{K_{∞}^{+}}(W_{ε}, t_{L_{∞}/K_{∞}})| > ε^{2} · |𝔡_{L_{∞}/K_{∞}}^{+}|

for every morphism α → β in I. □

7.5.22. Suppose now that (K, | · |) is deeply ramified and let X := (X_{α} | α ∈ I) be a deeply ramified cofiltered system of adic spaces of finite type over Spa(K, K^{+}). Suppose furthermore that it is given, for some β ∈ I, a finite étale morphism Y_{β} → X_{β} of adic Spa(K, K^{+})-spaces of finite type. For every morphism α → β of I we set Y_{α} := Y_{β} ×_{X_{β}} X_{α} and denote by f_{α} : Y_{α} → X_{α} the induced morphism of adic spaces. Theorem 7.5.23. In the situation of (7.5.22), for every positive real number ε < 1 there exists a morphism α → β in I such that, for every morphism γ → α we have

|𝔡_{Y_{γ}/X_{γ}}^{+}(x)| > ε for every x ∈ M(X_{γ}).

Proof. Notice that \underline{Y} := (Y_{α} | α → β) is a cofiltered system, hence we can define X_{∞} and Y_{∞} as in (7.4.12), and we obtain a morphism of locally ringed spaces f_{∞} : Y_{∞} → X_{∞}. For every α ∈ I, let π_{α} : X_{∞} → X_{α} be the natural morphism. Moreover, let M(X_{∞}) := lim_{α∈I} M(X_{α}); as

a topological space, it is compact, by Tychonoff’s theorem and the fact that M(X_{α}) is compact (7.5.9)); as a set, it admits an injective (usually non-continuous) map M(X_{∞}) → X_{∞}, so we can identify it as a subset of the latter.

Let x ∈ M(X_{∞}); by proposition 7.4.13, the valuation ring κ(x)^{+} is deeply ramified. Set κ(x) := κ(x)^{+} ⊗_{K^{+}} K; it is clear that the morphism

κ(x) → E(x) := (f_{∞∗}𝒪_{Y_{∞}}^{+})_{x} ⊗_{κ(x)^{+}} κ(x)

is finite and étale. Let W_{x} be the integral closure of κ(x)^{+} in E(x). By proposition 6.6.2 we deduce easily that the induced morphism of K^{+a}-algebras κ(x)^{+a} → W_{x}^{a} is weakly étale, hence étale by proposition 6.3.6. Consequently |𝔡_{E(x)/κ(x)}^{+}| = 1, in light of lemma 7.5.3. For every α → β, let x_{α} := π_{α}(x). Then κ(x) is the colimit of the filtered system (κ(x_{α}) | α → β), and similarly E(x) is the colimit of the finite étale κ(x_{α})-algebras (f_{β∗}𝒪_{Y_{β}})_{x_{β}} ⊗_{𝒪_{X_{β} ,x_{β}}^{+}} κ(x_{α}) (for all α → β). In this situation, lemma 7.5.21 applies and shows that, for every ε < 1 there exists α(ε, x) such that

(7.5.24)  |𝔡_{Y_{α}/X_{α}}^{+}(x_{α})| > ε for every α → α(ε, x).

In light of theorem 7.5.19, for every α → β, the subset

X_{α}(ε) := {y ∈ M(X_{α}) | |𝔡_{Y_{α}/X_{α}}^{+}(y)| ≤ ε}

is closed in M(X_{α}), hence compact. From (7.5.24) we see that lim_{α→β}X_{α}(ε) = ∅, therefore one

of the X_{α}(ε) must be empty ([18, Ch.I, §9, n.6, Prop.8(b)]), and the claim follows. □

7.5.25. Let us choose an imbedding ρ : (Γ_{K}, ·, ≤) ↪ (ℝ, +, ≤) as in example 6.1.21(vi). Let f : Y → X be a finite étale morphism of adic spaces of finite type over Spa(K, K^{+}). For every x ∈ X, set A(x) := (f_{∗}𝒪_{Y}^{+})_{x} ⊗_{𝒪_{X,x}^{+}} (f_{∗}𝒪_{Y}^{+})_{x}; we denote by e_{x} ∈ A(x) ⊗_{K^{+}} K the unique idempotent characterized by the conditions of proposition 3.1.4. We define the defect of the morphism f as the real number

def(f) := inf{r ∈ ℝ_{≥0} | ε · e_{x} ∈ A(x) for every x ∈ X and every ε ∈ K^{+} with ρ(|ε|) ≤ −r }.

Clearly def(f) ≥ 0 and def(f) = 0 if and only if (f_{∗}𝒪_{Y}^{+})_{x}^{a} is an étale 𝒪_{X,x}^{+a}-algebra for every x ∈ X. Furthermore we remark that, by proposition 2.4.19, the map 𝒪_{X,x}^{+a} → (f_{∗}𝒪_{Y}^{+})_{x}^{a} is weakly étale if and only if it is étale. Corollary 7.5.26. In the situation of (7.5.22), for every real number r > 0 there exists α ∈ I such that, for every morphism γ → α, we have: def(f_{γ}) < r.

Proof. Let r > 0; according to theorem 7.5.23 and claim 7.4.21, there exists α ∈ I such that ε · e_{x} ∈ A(x) for every γ → α, every x ∈ M(X_{γ}) and every ε ∈ K^{+} with ρ(|ε|) ≤ −r. If now y ∈ X_{γ} is any point, there is a unique generisation x of y in M(X_{γ}). Let ϕ : A(y) → A(x) be the induced specialisation map, and set ϕ_{K} := ϕ ⊗_{K^{+}} 1_{K}. One verifies easily that

ϕ_{K}^{−1}(𝔪 · A(x)) ⊂ A(y).

Since ϕ_{K}(e_{y}) = e_{x}, the claim follows easily. □

7.5.27. To conclude, we want to explain briefly what kind of Galois cohomology calculations are enabled by the results of this section. Let f : Y → X be a finite étale Galois morphism of Spa(K, K^{+})-adic spaces of finite type, and let G denote the group of X-automorphisms of Y. Denote by f_{∗}𝒪_{Y}^{+}[G]-Mod the category of f_{∗}𝒪_{Y}^{+}-modules on X, endowed with a semilinear action of G. Let \underline{Γ}^{G} : f_{∗}𝒪_{Y}^{+}[G]-Mod → 𝒪_{X}^{+}-Mod be the functor that associates to an f_{∗}𝒪_{Y}^{+}[G]-module the sheaf of its G-invariant local sections. A standard argument shows that, for every f_{∗}𝒪_{Y}^{+}[G]-modules ℱ, the cone of the natural morphism in D(f_{∗}𝒪_{Y}^{+}-Mod)

(7.5.28)  𝒪_{Y}^{+} ⊗_{𝒪_{X}^{+}} R\underline{Γ}^{G}ℱ → ℱ

is annihilated by all ε ∈ 𝔪 such that ρ(ε) < −def(f). However, for applications one is rather more interested in understanding the Galois cohomology groups H^{i} := H^{i}(G, H^{0}(X, ℱ)). One can try to study H^{i} via (7.5.28); indeed, a bridge between these two objects is provided by the higher derived functors of the related functor Γ^{G} : f_{∗}𝒪_{Y}^{+}[G]-Mod → Γ(X, 𝒪^{+})-Mod, defined by ℱ ↦ Γ(X, \underline{Γ}^{G}ℱ) = Γ(X, ℱ)^{G}. We have two spectral sequences converging to RΓ^{G}ℱ, namely

\begin{matrix}E_{2}^{pq} : H^{p}(X, R^{q}\underline{Γ}^{G}ℱ) ⇒ & R^{p+q}Γ^{G}ℱ \\ F_{2}^{pq} : H^{p}(G, H^{q}(X, ℱ)) ⇒ & R^{p+q}Γ^{G}ℱ . \end{matrix}

Using (7.5.28) one deduces that E_{2}^{pq} degenerates up to some torsion, which can be estimated precisely in terms of the defect of the morphism f. However, the spectral sequence F_{2}^{pq} contains the terms H^{q}(X, ℱ), about which not much is currently known. In this direction, the only results that we could found in the literature concern the calculation of H^{i}(Y, 𝒪_{Y}^{+}), for an affinoid space, under some very restrictive assumptions : in [7] these groups are shown to be almost zero modules for i > 0, in case Y admits a smooth formal model over K^{+}; in [65] the case of generalized polydiscs is taken up, and the same kind of almost vanishing is proven.

8. Appendix

In this appendix we have gathered a few miscellaneous results that were found in the course of our investigation, and which may be useful for other applications.

8.1. Simplicial almost algebras. We need some preliminaries on simplicial objects : first of all, a simplicial almost algebra is just an object in the category s.(V^{a}-Alg). Then for a given simplicial almost algebra A we have the category A-Mod of A-modules : it consists of all simplicial almost V -modules M such that M[n] is an A[n]-module and such that the face and degeneracy morphisms d_{i} : M[n] → M[n − 1] and s_{i} : M[n] → M[n + 1] (i = 0, 1, ..., n) are A[n]-linear.

8.1.1. We will need also the derived category of A-modules; it is defined as follows. A bit more generally, let 𝒞 be any abelian category. For an object X of s.𝒞 let N(X) be the normalized chain complex (defined as in [46, I.1.3]). By the theorem of Dold-Kan ([67, Th. 8.4.1]) X ↦ N(X) induces an equivalence N : s.𝒞 → C_{•}(𝒞) with the category C_{•}(𝒞) of chain complexes of object of 𝒞 that vanish in positive degrees. Now we say that a morphism X → Y in s.𝒞 is a quasi-isomorphism if the induced morphism N(X) → N(Y) is a quasiisomorphism of chain complexes.

8.1.2. In the following we fix a simplicial almost algebra A. Definition 8.1.3. We say that A is exact if the almost algebras A[n] are exact for all n ∈ ℕ. A morphism ϕ : M → N of A-modules (or A-algebras) is a quasi-isomorphism if the morphism ϕ of underlying simplicial almost V -modules is a quasi-isomorphism. We define the category D(A-Mod) (resp. the category D(A-Alg)) as the localization of the category A-Mod (resp. A-Alg) with respect to the class of quasi-isomorphisms.

8.1.4. As usual, the morphisms in D(A-Mod) can be computed via a calculus of fraction on the category Hot(A) of simplicial complexes up to homotopy. Moreover, if A_{1} and A_{2} are two simplicial almost algebras, then the “extension of scalars” functors define equivalences of categories

D(A_{1} × A_{2}-Mod) ⟶^{∼} D(A_{1}-Mod) × D(A_{2}-Mod)
D(A_{1} × A_{2}-Alg) ⟶^{∼} D(A_{1}-Alg) × D(A_{2}-Alg).

Proposition 8.1.5. Let A be a simplicial V^{a}-algebra.

(i) The functor on A-algebras given by B ↦ (s.V^{a} × B)_{!!} preserves quasi-isomorphisms and therefore induces a functor D(A-Alg) → D((s.V^{a} × A)_{!!}-Alg).

(ii) The localisation functor R ↦ R^{a} followed by “extension of scalars” via s.V^{a}×A → A induces a functor D((s.V^{a} × A)_{!!}-Alg) → D(A-Alg) and the composition of this and the above functor is naturally isomorphic to the identity functor on D(A-Alg).

Proof. (i) : let B → C be a quasi-isomorphism of A-algebras. Clearly the induced morphism s.V^{a} × B → s.V^{a} × C is still a quasi-isomorphism of V -algebras. But by remark 2.2.26, s.V^{a} × B and s.V^{a} × C are exact simplicial almost V -algebras; moreover, it follows from corollary 2.2.22 that (s.V^{a} × B)_{!} → (s.V^{a} × C)_{!} is a quasi-isomorphism of V -modules. Then the claim follows easily from the exactness of the sequence (2.2.24). Now (ii) is clear. □

Remark 8.1.6. In case 𝔪 is flat, then all A-algebras are exact, and the same argument shows that the functor B ↦ B_{!!} induces a functor D(A-Alg) → D(A_{!!}-Alg). In this case, composition with localisation is naturally isomorphic to the identity functor on D(A-Alg). Proposition 8.1.7. Let f : R → S be a map of V -algebras. We have:

(i) If f^{a} : R^{a} → S^{a} is an isomorphism, then 𝕃_{S/R}^{a} ≃ 0 in D(s.S^{a}-Mod).

(ii) The natural map 𝕃_{S^{a}/R^{a}}^{a} → 𝕃_{S/R}^{a} is an isomorphism in D(s.S^{a}-Mod).

Proof. (ii) is an easy consequence of (i) and of transitivity. To show (i) we prove by induction on q that

VAN(q; S/R) H_{q}(𝕃_{S/R}^{a}) = 0.

For q = 0 the claim follows immediately from [46, II.1.2.4.2]. Therefore suppose that q > 0 and that VAN(j; D/C) is known for all almost isomorphisms of V -algebras C → D and all j < q. Let \overline{R} := f(R). Then by transitivity ([46, II.2.1.2]) we have a distinguished triangle in D(s.S^{a}-Mod)

(S ⊗\overline{R} 𝕃_{\overline{R}/R})^{a} →^{u} 𝕃_{S/R}^{a} →^{v} 𝕃_{S/\overline{R}}^{a} → σ(S ⊗\overline{R} 𝕃_{\overline{R}/R})^{a}.

We deduce that VAN(q; \overline{R}/R) and VAN(q; S/\overline{R}) imply VAN(q; S/R), thus we can assume that f is either injective or surjective. Let S_{•} → S be the simplicial V -algebra augmented over S defined by S_{•} := P_{V} (S). It is a simplicial resolution of S by free V -algebras, in particular the augmentation is a quasi-isomorphism of simplicial V -algebras. Set R_{•} := S_{•} ×_{S} R. This is a simplicial V -algebra augmented over R via a quasi-isomorphism. Moreover, the induced morphisms R[n]^{a} → S[n]^{a} are isomorphisms. By [46, II.1.2.6.2] there is a quasi-isomorphism 𝕃_{S/R} ≃ 𝕃_{S_{•}/R_{•}}^{Δ}. On the other hand we have a spectral sequence

E_{ij}^{1} := H_{j}(𝕃_{S[i]/R[i]}) ⇒ H_{i+j}(𝕃_{S_{•}/R_{•}}^{Δ}).

It follows easily that VAN(j; S[i]/R[i]) for all i ≥ 0, j ≤ q implies VAN(q; S/R). Therefore we are reduced to the case where S is a free V -algebra and f is either injective or surjective. We examine separately these two cases. If f : R → V [T] is surjective, then we can find a right inverse s : V [T] → R for f. By applying transitivity to the sequence V [T] → R → V [T] we get a distinguished triangle

(V [T] ⊗_{R} 𝕃_{R/V [T]})^{a} →^{u} 𝕃_{V [T]/V [T]}^{a} →^{v} 𝕃_{V [T]/R}^{a} → σ(V [T] ⊗_{R} 𝕃_{R/V [T]})^{a}.

Since 𝕃_{V [T]/V [T]}^{a} ≃ 0 there follows an isomorphism : H_{q}(𝕃_{V [T]/R})^{a} ≃ H_{q−1}(V [T]⊗_{R}𝕃_{R/V [T]})^{a}. Furthermore, since f^{a} is an isomorphism, s^{a} is an isomorphism as well, hence by induction (and by a spectral sequence of the type [46, I.3.3.3.2]) H_{q−1}(V [T]⊗_{R} 𝕃_{R/V [T]})^{a} ≃ 0. The claim follows in this case.

Finally suppose that f : R → V [T] is injective. Write V [T] = Sym(F), for a free V -module F and set F̃ = 𝔪̃ ⊗_{V} F; since f^{a} is an isomorphism, Im(Sym(F̃) → Sym(F)) ⊂ R. We apply transitivity to the sequence Sym(F̃) → R → Sym(F). By arguing as above we are reduced to showing that 𝕃_{Sym(F)/Sym(F̃)}^{a} ≃ 0. We know that H_{0}(𝕃_{Sym(F)/Sym(F̃)}^{a}) ≃ 0 and we will show that H_{q}(𝕃_{Sym(F)/Sym(F̃)}^{a}) ≃ 0 for q > 0. To this purpose we apply transitivity to the sequence V →

Sym(F̃) → Sym(F). As F and F̃ are flat V -modules, [46, II.1.2.4.4] yields H_{q}(𝕃_{Sym(F)/V}) ≃ H_{q}(𝕃_{Sym(F̃)/V}) ≃ 0 for q > 0 and H_{0}(𝕃_{Sym(F̃)/V}) is a flat Sym(F̃)-module. In particular H_{j}(Sym(F) ⊗_{Sym(F̃)} 𝕃_{Sym(F̃)/V}) ≃ 0 for all j > 0. Consequently H_{j+1}(𝕃_{Sym(F)/Sym(F̃)}) ≃ 0 for all j > 0 and H_{1}(𝕃_{Sym(F)/Sym(F̃)}) ≃ Ker(Sym(F) ⊗_{Sym(F̃)} Ω_{Sym(F̃)/V} → Ω_{Sym(F)/V}). The latter module is easily seen to be almost zero. □

Theorem 8.1.8. Let ϕ : R → S be a map of simplicial V -algebras inducing an isomorphism R^{a} →^{∼} S^{a} in D(R^{a}-Mod). Then (𝕃_{S/R}^{Δ})^{a} ≃ 0 in D(S^{a}-Mod).

Proof. Apply the base change theorem [46, II.2.2.1] to the (flat) projections of s.V × R onto R and respectively s.V to deduce that the natural map 𝕃_{s.V ×S/s.V ×R}^{Δ} → 𝕃_{S/R}^{Δ} ⊕ 𝕃_{s.V/s.V}^{Δ} → 𝕃_{S/R}^{Δ} is a quasi-isomorphism in D(s.V × S-Mod). By proposition 8.1.5 the induced morphism (s.V × R)_{!!}^{a} → (s.V × S)_{!!}^{a} is still a quasi-isomorphism. There are spectral sequences

\begin{matrix}E_{ij}^{1} := H_{j}(𝕃_{(V ×R[i])/(V ×R[i])_{!!}^{a}}) ⇒ H_{i+j}(𝕃_{(s.V ×R)/(s.V ×R)_{!!}^{a}}^{Δ}) \\ F_{ij}^{1} := H_{j}(𝕃_{(V ×S[i])/(V ×S[i])_{!!}^{a}}) ⇒ H_{i+j}(𝕃_{(s.V ×S)/(s.V ×S)_{!!}^{a}}^{Δ}). \end{matrix}

On the other hand, by proposition 8.1.7(i) we have 𝕃_{(V ×R[i])/(V ×R[i])_{!!}^{a}}^{a} ≃ 0 ≃ 𝕃_{(V ×S[i])/(V ×S[i])_{!!}^{a}}^{a} for all i ∈ ℕ. Then the theorem follows directly from [46, II.1.2.6.2(b)] and transitivity. □

8.1.9. In view of proposition 8.1.7(i) we have 𝕃_{(V^{a}×A)_{!!}/V ×A_{!!}}^{a} ≃ 0 in D(V^{a} × A-Mod). By this, transitivity and localisation ([46, II.2.3.1.1]) we derive that 𝕃_{B/A}^{a} → 𝕃_{B_{!!}/A_{!!}}^{a} is a quasiisomorphism for all A-algebras B. If A and B are exact (e.g. if 𝔪 is flat), we conclude from proposition 2.5.42 that the natural map 𝕃_{B/A} → 𝕃_{B_{!!}/A_{!!}} is a quasi-isomorphism.

8.1.10. Finally we want to discuss left derived functors of (the almost version of) some notable non-additive functors that play a role in deformation theory. Let R be a simplicial V - algebra. Then we have an obvious functor G : D(R-Mod) → D(R^{a}-Mod) obtained by applying dimension-wise the localisation functor. Let Σ be the multiplicative set of morphisms of D(R-Mod) that induce almost isomorphisms on the cohomology modules. An argument as in section 2.4 shows that G induces an equivalence of categories Σ^{−1}D(R-Mod) → D(R^{a}-Mod).

8.1.11. Now let R be a V -algebra and ℱ_{p} one of the functors ⊗^{p}, Λ^{p}, Sym^{p}, Γ^{p} defined in [46, I.4.2.2.6]. Lemma 8.1.12. Let ϕ : M → N be an almost isomorphism of R-modules. Then ℱ_{p}(ϕ) : ℱ_{p}(M) → ℱ_{p}(N) is an almost isomorphism.

Proof. Let ψ : 𝔪̃ ⊗_{V} N → M be the map corresponding to (ϕ^{a})^{−1} under the bijection (2.2.4). By inspection, the compositions ϕ ∘ ψ : 𝔪̃ ⊗_{V} N → N and ψ ∘ (1𝔪̃ ⊗ ϕ) : 𝔪̃ ⊗_{V} M → M are induced by scalar multiplication. Pick any s ∈ 𝔪 and lift it to an element s̃ ∈ 𝔪̃; define ψ_{s} : N → M by n ↦ ψ(s̃⊗ n) for all n ∈ N. Then ϕ ∘ ψ_{s} = s · 1_{N} and ψ_{s} ∘ ϕ = s · 1_{M}. This easily implies that s^{p} annihilates Ker ℱ_{p}(ϕ) and Coker ℱ_{p}(ϕ). In light of proposition 2.1.7(ii), the claim follows. □

8.1.13. Let B be an almost V -algebra. We define a functor ℱ_{p}^{a} on B-Mod by letting M ↦ (ℱ_{p}(M_{!}))^{a}, where M_{!} is viewed as a B_{!!}-module or a B_{∗}-module (to show that these choices define the same functor it suffices to observe that B_{∗} ⊗_{B_{!!}} N ≃ N for all B_{∗}-modules N such that N = 𝔪 · N). For all p > 0 we have diagrams :

(8.1.14)  \xymatrix{R-Mod \ar[r]^{ℱ_{p}} \ar[d] & R-Mod \ar[d] \\ R^{a}-Mod \ar[r]^{ℱ_{p}^{a}} \ar[u] & R^{a}-Mod \ar[u]}

where the downward arrows are localisation and the upward arrows are the functors M ↦ M_{!}. Lemma 8.1.12 implies that the downward arrows in the diagram commute (up to a natural isomorphism) with the horizontal ones. It will follow from the following proposition 8.1.16 that the diagram commutes also going upward.

8.1.15. For any V -module N we have an exact sequence Γ^{2}N → ⊗^{2}N → Λ^{2}N → 0. As observed in the proof of proposition 2.1.7, the symmetric group S_{2} acts trivially on ⊗^{2}𝔪̃ and Γ^{2}𝔪̃ ≃ ⊗^{2}𝔪̃, so Λ^{2}𝔪̃ = 0. Also we have natural isomorphisms Γ^{p}𝔪̃ ≃ 𝔪̃ for all p > 0. Proposition 8.1.16. Let R be a commutative ring and L a flat R-module with Λ^{2}L = 0. Then for p > 0 and for all R-modules N we have natural isomorphisms

Γ^{p}(L) ⊗_{R} ℱ_{p}(N) →^{∼} ℱ_{p} (L ⊗_{R} N).

Proof. Fix an element x ∈ ℱ_{p}(N). For each R-algebra R′ and each element l ∈ R′ ⊗_{R} L we get a map ϕ_{l} : R′ ⊗_{R} N → R′ ⊗_{R} L ⊗_{R} N by y ↦ l ⊗ y, hence a map ℱ_{p}(ϕ_{l}) : R′ ⊗_{R} ℱ_{p}(N) ≃ ℱ_{p}(R′ ⊗_{R} N) → ℱ_{p}(R′ ⊗_{R} L ⊗_{R} N) ≃ R′ ⊗_{R} ℱ_{p}(L ⊗_{R} N). For varying l we obtain a map of sets ψ_{R′,x} : R′ ⊗_{R} L → R′ ⊗_{R} ℱ_{p}(L ⊗_{R} N) : l ↦ ℱ_{p}(ϕ_{l})(1 ⊗ x). According to the terminology of [61], the system of maps ψ_{R′,x} for R′ ranging over all R-algebras forms a homogeneous polynomial law of degree p from L to ℱ_{p}(L ⊗_{R} N), so it factors through the universal homogeneous degree p polynomial law γ_{p} : L → Γ^{p}(L) . The resulting R-linear map \overline{ψ}_{x} : Γ^{p}(L) → ℱ_{p}(L ⊗_{R} N) depends R-linearly on x, hence we derive an R-linear map ψ : Γ^{p}(L)⊗_{R}ℱ_{p}(N) → ℱ_{p} (L⊗_{R}N). Next notice that by hypothesis S_{2} acts trivially on ⊗^{2}L so S_{p} acts trivially on ⊗^{p}L and we get an isomorphism β : Γ^{p}(L) ⟶^{∼} ⊗^{p}L. We deduce a natural map (⊗^{p}L) ⊗_{R} ℱ_{p}(N) → ℱ_{p}(L ⊗_{R} N). Now, in order to prove the proposition for the case ℱ_{p} = ⊗^{p}, it suffices to show that this last map is just the natural isomorphism that “reorders the factors”. Indeed, let x_{1}, ..., x_{n} ∈ L and q := (q_{1}, ..., q_{n}) ∈ ℕ^{n} such that |q| := ∑_{i} q_{i} := p; then β sends the generator x_{1}^{[q_{1}]} · ... · x_{n}^{[q_{n}]} to \binom{p}{q_{1},...,q_{n}} · x_{1}^{⊗q_{1}} ⊗ ... ⊗ x_{n}^{⊗q_{n}}. On the other hand, pick any y ∈ ⊗^{p}N and let R[T] := R[T_{1}, ..., T_{n}] be the polynomial R-algebra in n variables; write (T_{1} ⊗ x_{1} + ... + T_{n} ⊗ x_{n})^{⊗p} ⊗ y = ψ_{R[T],y}(T_{1} ⊗ x_{1} + ... + T_{n} ⊗ x_{n}) = ∑_{r∈ℕ^{n}} T^{r} ⊗ w_{r} with w_{r} ∈ ⊗^{p}(L ⊗_{R} N). Then ψ((x_{1}^{[q_{1}]} · ... · x_{n}^{[q_{n}]}) ⊗ y) = w_{q} (see [61, pp.266-267]) and the claim follows easily. Next notice that Γ^{p}(L) is flat, so that tensoring with Γ^{p}(L) commutes with taking coinvariants (resp. invariants) under the action of the symmetric group; this implies the assertion for ℱ_{p} := Sym^{p} (resp. ℱ_{p} := TS^{p}). To deal with ℱ_{p} := Λ^{p} recall that for any V -module M and p > 0 we have the antisymmetrizer operator a_{M} := ∑_{σ∈S_{p}} sgn(σ)·σ : ⊗^{p}M → ⊗^{p}M and a surjection Λ^{p}(M) → Im(a_{M}) which is an isomorphism for M free, hence for M flat. The result for ℱ_{p} = ⊗^{p} (and again the flatness of Γ^{p}(L)) then gives Γ^{p}(L)⊗Im(a_{N}) ≃ Im(a_{L⊗_{R}N}), hence the assertion for ℱ_{p} = Λ^{p} and N flat. For general N let F_{1} →^{∂} F_{0} →^{ε} N → 0 be a presentation with F_{i} free. Define j_{0}, j_{1} : F_{0} ⊕ F_{1} → F_{0} by j_{0}(x, y) := x + ∂(y) and j_{1}(x, y) := x. By functoriality we derive an exact sequence

\xymatrix{Λ^{p}(F_{0} ⊕ F_{1}) \ar[r] \ar[r] & Λ^{p}(F_{0}) \ar[r] & Λ^{p}(N) \ar[r] & 0}

which reduces the assertion to the flat case. For ℱ_{p} := Γ^{p} the same reduction argument works as well (cp. [61, p.284]) and for flat modules the assertion for Γ^{p} follows from the corresponding assertion for TS^{p}. □

Lemma 8.1.17. Let A be a simplicial almost algebra, L, E and F three A-modules, f : E → F a quasi-isomorphism. If L is flat or E, F are flat, then L ⊗_{A} f : L ⊗_{A} E → L ⊗_{A} F is a quasiisomorphism.

Proof. It is deduced directly from [46, I.3.3.2.1] by applying M ↦ M_{!}. □

8.1.18. As usual, this allows one to show that ⊗ : Hot(A) × Hot(A) → Hot(A) admits a left

derived functor ⊗^{L} : D(A-Mod) × D(A-Mod) → D(A-Mod). If R is a simplicial V -algebra then we have essentially commutative diagrams

\xymatrix{D(R-Mod) × D(R-Mod) \ar[r]^{⊗^{L}} \ar[d] & D(R-Mod) \ar[d] \\ D(R^{a}-Mod) × D(R^{a}-Mod) \ar[r]^{⊗^{L}} \ar[u] & D(R^{a}-Mod) \ar[u]}

where again the downward (resp. upward) functors are induced by localisation (resp. by M ↦ M_{!}).

8.1.19. We mention the derived functors of the non-additive functor ℱ_{p} defined above in the simplest case of modules over a constant simplicial ring. Let A be a (commutative) V^{a}-algebra. Lemma 8.1.20. If u : X → Y is a quasi-isomorphism of flat s.A-modules then ℱ_{p}^{a}(u) : ℱ_{p}^{a}(X) → ℱ_{p}^{a}(Y) is a quasi-isomorphism.

Proof. This is deduced from [46, I.4.2.2.1] applied to N(X_{!}) → N(Y_{!}) which is a quasiisomorphism of chain complexes of flat A_{!!}-modules. We note that loc. cit. deals with a more general mixed simplicial construction of ℱ_{p} which applies to bounded above complexes, but one can check that it reduces to the simplicial definition for complexes in 𝒞_{•}(A_{!!}). □

8.1.21. Using the lemma one can construct Lℱ_{p}^{a} : D(s.A-Mod) → D(s.A-Mod). If R is a V -algebra we have the derived category version of the essentially commutative squares (8.1.14), relating Lℱ_{p} : D(s.R-Mod) → D(s.R-Mod) and Lℱ_{p}^{a} : D(s.R^{a}-Mod) → D(s.R^{a}-Mod).

8.2. Fundamental group of an almost algebra. We will need some generalities from [40, Exp.V] and [5, Exp.VI]. In the following we fix a universe U and suppose that all our categories are U-categories and all our topoi are U-topoi in the sense of [4, Exp.I, Déf.1.1 and Exp.IV, Déf.1.1]. No further mention of universes will be necessary.

8.2.1. Let 𝒞 be a site. Recall ([5, Exp.VI, Déf.1.1]) that an object X of 𝒞 is called quasicompact if, for every covering family (X_{i} → X | i ∈ I) there is a finite subset J ⊂ I such that the subfamily (X_{j} → X | j ∈ J) is still covering.

8.2.2. Let E be a topos; in the following we always endow E with its canonical topology ([4, Exp.II, Déf 2.5]), so E is a site in a natural way and the terminology of (8.2.1) applies to the objects of E. Moreover, if 𝒞 is any site and ε : 𝒞 → 𝒞^{∼} the natural functor to the category 𝒞^{∼} of sheaves on 𝒞, then an object X of 𝒞 is quasi-compact in 𝒞 if and only if ε(X) is quasi-compact in 𝒞^{∼} ([5, Exp.VI, Prop.1.2]).

Furthermore, since in E all finite limits are representable, we can make the following further definitions ([5, Exp.VI, Déf.1.7]). A morphism f : X → Y in E is called quasi-compact if, for every morphism Y′ → Y in E with quasi-compact Y′, the object X ×_{Y} Y′ is quasi-compact. We say that f is quasi-separated if the diagonal morphism X → X ×_{Y} X is quasi-compact. We say that f is coherent if it is quasi-compact and quasi-separated.

8.2.3. Let X be an object of a topos E. We say that X is quasi-separated if, for every quasicompact object S of E, every morphism S → X is quasi-compact. We say that X is coherent if it is quasi-compact and quasi-separated ([5, Exp.VI, Déf.1.13]). We denote by E_{coh} the full subcategory of E consisting of all the coherent objects.

Suppose that the object Y of E is coherent and let f : X → Y be a coherent morphism; by [5, Exp.VI, Prop.1.14(ii)], it follows that X is coherent. Definition 8.2.4. (cp. [5, Exp.VI, Déf.2.3]) We say that a topos E is coherent if it satisfies the following conditions:

(i) E admits a full generating subcategory 𝒞 consisting of coherent objects.

(ii) Every object X of 𝒞 is quasi-separated over the final object of E, i.e. the diagonal

morphism X → X × X is quasi-compact. (iii) The final object of E is quasi-separated.

8.2.5. If E is a coherent topos, then E_{coh} is stable under arbitrary finite limits (of E) ([5, Exp.VI, 2.2.4]). Moreover, a topos E is coherent if and only if it is equivalent to a topos of the form 𝒞^{∼}, where 𝒞 is a site whose objects are quasi-compact and whose finite limits are representable.

It is possible to characterize the categories of the form E_{coh} arising from a coherent topos : this leads to the following definition.

Definition 8.2.6. A pretopos is a small category 𝒞 satisfying the following conditions ([5, Exp.VI, Exerc.3.11]). (PT1) All finite limits are representable in 𝒞. (PT2) All finite sums are representable in 𝒞 and they are universal and disjoint. (PT3) Every equivalence relation in 𝒞 is effective, and every epimorphism is universally

effective.

8.2.7. As in [5, Exp.VI, Exerc.3.11], we leave to the reader the verification that, for every coherent topos E, the subcategory E_{coh} is a pretopos, and E induces on E_{coh} the precanonical topology, i.e. the topology whose covering families (X_{i} → X | i ∈ I) are those admitting a finite subfamily which is covering for the canonical topology of E_{coh}. One deduces that E is equivalent to (E_{coh})^{∼}, the topos of sheaves on the precanonical topology of E_{coh}.

Conversely, if 𝒞 is a pretopos, let E := 𝒞^{∼} be the topos of sheaves on the precanonical topology of 𝒞; then E is a coherent topos and the natural functor ε : 𝒞 → E induces an equivalence of 𝒞 with E_{coh}.

8.2.8. Furthermore, if 𝒞 is a pretopos (endowed with the precanonical topology), the natural functor 𝒞 → 𝒞^{∼} commutes with finite sums, with quotients under equivalence relations, and it is left exact (i.e. commutes with finite limits) ([5, Exp.VI, Exerc.3.11]).

8.2.9. Recall ([4, Exp.IV, Déf.6.1]) that a point of a topos E is a morphism of topoi p : Set → E (where one views the category Set as the topos of sheaves on the one-point topological space). By [4, Exp.IV, Cor.1.5], the assignment p ↦ p^{∗} defines an equivalence from the category of points of E to the opposite of the category of all functors F : E → Set that commute with all colimits and are left exact. A functor F : E → Set with these properties is called a fibre functor for E. By [4, Exp.IV, Cor.1.7], a functor E → Set is a fibre functor if and only if it is left exact and it takes covering families to surjective families.

8.2.10. By a theorem of Deligne ([5, Exp.VI, Prop.9.0]) every coherent non-empty topos admits at least a fibre functor. (Actually Deligne’s result is both more precise and more general, but for our purposes, the foregoing statement will suffice).

In several contexts, it is useful to attach fibre functors to categories that are not quite topoi. These situations are axiomatized in the following definition. Definition 8.2.11. A Galois category ([40, Exp.V, §5]) is the datum of a category 𝒞 and a functor F from 𝒞 to the category f.Set of finite sets, satisfying the following conditions:

(G1) all finite limits exist in 𝒞 (in particular 𝒞 has a final object).

(G2) Finite sums exist in 𝒞 (in particular 𝒞 has an initial object). Also, for every object X

of 𝒞 and every finite group G of automorphisms of X, the quotient X/G exists in 𝒞. (G3) Every morphism u : X → Y in 𝒞 factors as a composition X →^{u′} Y′ →^{u′′} Y, where u′ is

a strict epimorphism and u′′ is both a monomorphism and an isomorphism on a direct summand of Y.

(G4) The functor F is left exact.

(G5) F commutes with finite direct sums and with quotients under actions of finite groups

of automorphisms. Moreover F takes strict epimorphisms to epimorphisms. (G6) Let u : X → Y be a morphism of 𝒞. Then u is an isomorphism if and only if F (u) is.

8.2.12. Given a Galois category (𝒞 , F), one says that F is a fibre functor of 𝒞. It is shown in [40, Exp.V, §4] that for any Galois category (𝒞 , F) the functor F is pro-representable and its automorphism group is a profinite group π in a natural way. Furthermore, F factors naturally through the forgetful functor π-f.Set → f.Set from the category π-f.Set of (discrete) finite sets with continuous π-action, and the resulting functor 𝒞 → π-f.Set is an equivalence.

8.2.13. The category of étale coverings Cov(A) of an almost algebra A (to be defined in (8.2.22)) is not directly presented as a Galois category, since it does not afford an a priori choice of fibre functor; rather, the existence of a fibre functor is deduced from Deligne’s theorem. The argument only appeals to some general properties of the category Cov(A), which are abstracted in the following definition 8.2.14 and lemma 8.2.15. Definition 8.2.14. A pregalois category is a category 𝒞 satisfying the following conditions. (PG1) Every monomorphism X → Y in 𝒞 induces an isomorphism of X onto a direct sum-

mand of Y. (PG2) 𝒞 admits a final object e which is connected and non-empty (that is, e is not an initial

object). (PG3) For every object X of 𝒞, there exists n ∈ ℕ such that, for every non-empty object Y

of 𝒞, the product X × Y exists in 𝒞 and is not Y -isomorphic to an object of the form

(⨿^{n} Y) ⨿ Z (where Z is any other object of 𝒞). Lemma 8.2.15. Let 𝒞 be a pregalois pretopos. Then there exists a functor F : 𝒞 → f.Set such that (𝒞 , F) is a Galois category.

Proof. (G1) holds because it is the same as (PT1). (G2) follows easily from (PT2) and (PT3). In order to show (G3) we will need the following: Claim 8.2.16. A morphism u : X → Y of 𝒞 is an isomorphism if and only if it is both a monomorphism and an epimorphism. Proof of the claim: One direction is obvious, so we can suppose that u is both a monomorphism and an epimorphism. By (PG1), it follows that, up to composing with an isomorphism, Y = X ⨿ Z for some object Z of 𝒞, and u can be identified with the natural morphism X → X ⨿ Z. Let v : Z → X ⨿ Z be the natural morphism; by (PT3) the induced morphism u ×_{Y} Z : X ×_{Y} Z → Z is an epimorphism and by (PT2) we have X ×_{Y} Z ≃ ∅, the initial object of 𝒞. Since in 𝒞 all epimorphisms are effective, one derives that Z ≃ ∅, and the claim follows.

Now, let u : X → Y be a morphism in 𝒞; the induced morphisms

pr_{1}, pr_{2} : \xymatrix{X ×_{Y} X \ar[r] \ar[r] & X}

define an equivalence relation; by (PT3) there is a corresponding quotient morphism u′ : X → Y′ and moreover u′ is a strict epimorphism. Clearly u factors via a morphism u′′ : Y′ → Y. We need to show that u′′ is a monomorphism, or equivalently, that the induced diagonal morphism δ : Y′ → Y′ ×_{Y} Y′ is an isomorphism. However, there is a natural commutative diagram

\xymatrix{X ×_{Y′} X \ar[r]^{α} \ar[d] & X ×_{Y} X \ar[d] \\ Y′ \ar[r]^{δ} & Y′ ×_{Y} Y′}

where α is an isomorphism by construction and both vertical arrows are epimorphisms. It follows that δ is an epimorphism; since it is obviously a monomorphism as well, we deduce (G3) in view of (PG1) and claim 8.2.16. Let 𝒞^{∼} be the topos of sheaves on the precanonical topology of 𝒞; by (8.2.7) and Deligne’s theorem (8.2.10), there exists a fibre functor 𝒞^{∼} → Set. Composing with the natural functor 𝒞 → 𝒞^{∼} we obtain a functor F : 𝒞 → Set which is left exact, commutes with finite sums and quotients under equivalence relations, in view of (8.2.8), so (G4) and (G5) hold for F. Claim 8.2.17. Let X be a non-empty object of 𝒞. Then F (X) ≠ ∅. Proof of the claim: Since we know already that (G3) holds in 𝒞, we deduce using (PG2) that the unique morphism X → e is an epimorphism. Then (PT3) says that e can be written as the quotient of X under the induced equivalence relation pr_{1}, pr_{2} : \xymatrix{X ×_{e} X \ar[r] \ar[r] & X} . Since F commutes with quotients under such equivalence relations, the claim follows after remarking that F (e) ≠ ∅,

Claim 8.2.18. Let u : X → Y be a morphism in 𝒞 such that F (u) is surjective. Then u is an epimorphism. Proof of the claim: We use (G3) to factor u as an epimorphism u′ : X → Y′ followed by a monomorphism of the form Y′ → Y′ ⨿ Z. We need to show that Z = ∅ or equivalently, in view of claim 8.2.17, that F (Z) = ∅. However, the assumption implies that F (Y′) maps onto F (Y′ ⨿ Z); on the other hand, F commutes with finite sums, so the claim holds.

Claim 8.2.19. Let u : X → Y be a morphism in 𝒞 such that F (u) is injective. Then u is a monomorphism. Proof of the claim: The assumption means that the induced diagonal map F (X) → F (X)×_{F (Y)} F (X) is bijective. Then claim 8.2.18 implies that the diagonal morphism X → X ×_{Y} X is an epimorphism. The latter is also obviously a monomorphism, hence an isomorphism, in view of claim 8.2.16; but this means that u is a monomorphism.

Now, taking into account claims 8.2.16, 8.2.18 and 8.2.19 we deduce that (G6) holds for F. It remains only to show that F takes values in finite sets. So, suppose by contradiction that F (X) is an infinite set for some object X of 𝒞. We define inductively a sequence of objects (Y_{i} | i ∈ ℕ), with morphisms ϕ_{i+1} : Y_{i+1} → Y_{i} for every i ∈ ℕ, as follows. Let Y_{0} := e, Y_{1} := X and ϕ_{1} the unique morphism. Let then i > 0 and suppose that Y_{i} and ϕ_{i} have already been given. Using the diagonal morphism, we can write Y_{i} ×_{Y_{i−1}} Y_{i} ≃ Y_{i} ⨿Z for some object Z; we set Y_{i+1} := Z and let ϕ_{i+1} be the restriction of the projection pr_{1} : Y_{i} ×_{Y_{i−1}} Y_{i} → Y_{i}. Notice that F (Y_{i}) ≠ ∅ for every i ∈ ℕ (indeed all the fibers of the induced map F (Y_{i}) → F (Y_{i}) are infinite whenever i > 0); in particular Y_{i} ≠ ∅ for every i ∈ ℕ. On the other hand, for every

n > 0, X × Y_{n} admits a decomposition of the form (⨿^{n} Y_{n}) ⨿ Y_{n+1}, which is against (PG3). The contradiction concludes the proof of the lemma. □

8.2.20. Let E be a topos. Recall that an object X of E is said to be constant if it is a direct sum of copies of the final object e of E ([6, Exp.IX, §2.0]). The object X is locally constant if there exists a covering (Y_{i} → e | i ∈ I) of e, such that, for every i ∈ I, the restriction of X × Y_{i} is constant on the induced topos E_{/Y_{i}} ([4, Exp.III, §5.1]). If additionally there exists an integer n, such that so that each (X × Y_{i})_{|Y_{i}} is a direct sum of at most n copies of Y_{i}, then we say that X is bounded. Lemma 8.2.21. Let E be a topos. Denote by E_{lcb} the full subcategory of all locally constant bounded objects of E (see (8.2.3)). Then:

(i) E_{lcb} is a pretopos.

(ii) If the final object of e is connected and non-empty, E_{lcb} is a pregalois pretopos.

Proof. Let X be an object of E_{lcb}, and (Y_{i} → e) a covering of the final object of E by nonempty objects, such that X × Y_{i} is constant on E_{/Y_{i}} for every i ∈ I, say (X × Y_{i})_{|Y_{i}} ≃ Y_{i} × S_{i}, where S_{i} is some set. Since X is bounded, there exists n ≥ 0 such that the cardinality of every S_{i} is no greater than n. Since all finite limits are representable in E, axiom (PT1) can be checked locally on E, so we can reduce to the case of a finite inverse system \underline{Z} := (Z_{j} | j ∈ J) of constant bounded objects Z_{j} := e × S_{j}. Furthermore, thanks to [6, Exp.IX, Lemme 2.1(i)] we can assume that the inverse system is induced from an inverse system \underline{S} := (S_{j} | j ∈ J) of maps of sets, in which case it is easy to check that lim_{J} \underline{Z} ≃ lim_{J} \underline{S}, which implies (PT1).

(PT2) is immediate. Similarly, if R is a locally constant bounded equivalence relation on an object X of E_{lcb}, then X/R is again in E_{lcb}; indeed, since equivalence relations in E are universally effective, this can be checked locally on a covering (Y_{i} → e | i ∈ I). Then again, by [6, Exp.IX, Lemme 2.1(i)] we can reduce to the case of an equivalence relation on sets, where everything is obvious. This shows that (PT3) holds, and proves (i). Suppose next that e is connected and non-empty; since e is in E_{lcb}, it follows that (PG2) holds in E_{lcb}. To show (PG1), consider a morphism u : X → Y in E_{lcb}. As in the foregoing, we can find a covering (Z_{i} → e | i ∈ I) such that (X ×Z_{i})_{|Z_{i}} ≃ Z_{i} ×S_{i}, (Y ×Z_{i})_{|Z_{i}} ≃ Z_{i} ×T_{i} for some sets S_{i}, T_{i}, and u_{|Z_{i}} is induced by a map u_{i} : S_{i} → T_{i}. Let S′_{i} := T_{i} \ u_{i}(S_{i}). Clearly we have an isomorphism (Y × Z_{i})_{|Z_{i}} ≃ (X × Z_{i})_{|Z_{i}} ⨿ (Z_{i} × S′_{i}) for every i ∈ I. Since the induced decompositions agree on Z_{i} × Z_{j} for every i, j ∈ I, the constant objects Z_{i} × S′_{i} glue to a locally constant object X′ of E, and u induces an isomorphism Y ≃ X ⨿ X′. Finally, if X is in E_{lcb}, find sets (S_{i} | i ∈ I) and a covering (Y_{i} → e | i ∈ I) such that (X × Y_{i})_{|Y_{i}} ≃ S_{i} × Y_{i}, and let m be the maximum of the cardinalities of the sets S_{i}. Clearly (PG3) holds for X, if one chooses n := m + 1. □

8.2.22. Let A be an almost algebra. We consider the site 𝒮_{A} := (A-Alg)_{fpqc}^{o} obtained by endowing the category of affine A-schemes with the fpqc topology (see (4.4.23)). Moreover, the category of étale coverings of Spec A is defined as the full subcategory Cov(Spec A) of the category of affine A-schemes consisting of all étale A-schemes of finite rank. Proposition 8.2.23. The natural functor ε : 𝒮_{A} → 𝒮_{A}^{∼} induces an equivalence of the category Cov(Spec A) onto the category of locally constant bounded sheaves on 𝒮_{A}.

Proof. Let B be an étale A-algebra of rank r. By proposition 4.3.27 there is an isomorphism of almost algebras A ≃ ∏_{i=0}^{r} A_{i} such that B_{i} := B ⊗_{A} A_{i} is an A_{i}-algebra of constant rank i for every i = 0, ..., r. In particular, B_{0} = 0 and B_{i} is faithfully flat étale A_{i}-algebra for every i > 0. We use the diagonal morphism to obtain a decomposition B_{i} ⊗_{A_{i}} B_{i} ≃ B_{i} × C_{i}, where C_{i} is again an étale B_{i}-algebra of constant rank i − 1. Iterating this procedure we find faithfully flat A_{i}-algebras D_{i} such that B ⊗_{A} D_{i} is D_{i}-isomorphic to a direct product of i copies of D_{i}, for every i > 0. Setting D_{0} := A_{0}, we obtain a covering (ε(Spec D_{i} → Spec A) | i = 0, ..., r) of ε(Spec A) in 𝒮_{A}^{∼} such that the restriction of ε(Spec B) to each ε(Spec D_{i}) is a bounded constant sheaf. This show that the restriction of ε to Cov(Spec A) lands in the category of locally constant bounded sheaves. Since the fpqc topology is coarser than the canonical topology, ε is fully faithful. To show that ε is essentially surjective amounts to an exercise in faithfully flat descent : clearly every constant sheaf is represented by an étale A-scheme; then one uses [6, Exp.IX, Lemme 2.1(i)] to show that any descent datum of bounded constant sheaves is induced by a cocycle system of morphisms for the corresponding representing algebras, and one can descend the latter. We leave the details to the reader. □

8.2.24. Let us say that an affine almost scheme S is connected if ε(S) is a connected object of the category 𝒮_{𝒪_{S}}^{∼}, which simply means that the only non-zero idempotent of 𝒪_{S∗} is the identity. In this case, proposition 8.2.23 and lemma 8.2.21 show that Cov(S) is a pregalois pretopos, hence it admits a fibre functor F : Cov(S) → f.Set by lemma 8.2.15. Definition 8.2.25. Suppose that the affine almost scheme S is connected. The fundamental group of S is the group π_{1}(S) defined as the automorphism group of any fibre functor F : Cov(S) → f.Set, endowed with its natural profinite topology (see (8.2.12)).

8.2.26. It results from the general theory ([40, Exp.V, §5]) that π_{1}(S) is independent (up to isomorphism) on the choice of fibre functor. We refer to loc.cit. for a general study of fundamental groups of Galois categories. In essence, several of the standard results for schemes admit adequate almost counterpart. We conclude this section with a sample of such statements. Definition 8.2.27. Let A ⊂ B be a pair of V^{a}-algebras; the integral closure of A in B is the subalgebra i.c.(A, B) := i.c.(A_{∗}, B_{∗})^{a}, where for a pair of rings R ⊂ S we let i.c.(R, S) be the integral closure of R in S.

Lemma 8.2.28. If R ⊂ S is any pair of V -algebras, then i.c.(R^{a}, S^{a}) = i.c.(R, S)^{a}.

Proof. It suffices to show the following: Claim 8.2.29. Given a commutative diagram of V -algebras

\xymatrix{R \ar[r] \ar[d] & S \ar[d] \\ R_{1} \ar[r] & S_{1}}

whose vertical arrows are almost isomorphism, the induced map i.c.(R, S) → i.c.(R_{1}, S_{1}) is an almost isomorphism. Proof of the claim: Clearly the kernel of the map is almost zero; it remains to show that for every b ∈ i.c.(R_{1}, S_{1}) and ε ∈ 𝔪, the element εb lifts to i.c.(R, S). By assumption we have a relation b^{N} + ∑_{i=1}^{N} a_{i}b^{N−i} = 0, with a_{i} ∈ R_{1}, so (εb)^{N} + ∑_{i=1}^{N} ε^{i}a_{i}(εb)^{N−i} = 0. By lifting ε^{i}a_{i} to some \overline{a}_{i} ∈ R, we deduce a monic polynomial P (T) over R such that P (εb) = 0, so if \overline{b} is a lifting of εb, we have 𝔪 · P (\overline{b}) = 0. Since the restriction 𝔪S → 𝔪S_{1} is surjective, we can choose \overline{b} ∈ 𝔪S, so \overline{b} · P (\overline{b}) = 0. □

Remark 8.2.30. (i) If A ⊂ B are V^{a}-algebras, then A is integrally closed in B if and only if A_{∗} is integrally closed in B_{∗}. Indeed, by lemma 8.2.28 we know that the integral closure of A_{∗} in B_{∗} is almost isomorphic to A_{∗} and any such V -algebra must be contained in A_{∗}.

(ii) If (A_{i} ⊂ B_{i} | i ∈ I) is a (possibly infinite) family of pairs of V^{a}-algebras, then ∏_{i∈I} A_{i} is integrally closed in ∏_{i∈I} B_{i} if and only if A_{i} is integrally closed in B_{i} for every i ∈ I.

The following proposition is an analogue of [30, Prop.18.12.15]. Proposition 8.2.31. Let A ⊂ B be a pair of V^{a}-algebras such that A = i.c.(A, B).

(i) For any étale almost finite projective A-algebra A_{1} of almost finite rank we have A_{1} = i.c.(A_{1}, A_{1} ⊗_{A} B).

(ii) Suppose that A and B are connected, and choose a fibre functor F for the category Cov(B). Then the functor Cov(A) → f.Set : C ↦ F (C ⊗_{A} B) is a fibre functor, and the induced group homomorphism: π_{1}(Spec B) → π_{1}(Spec A) is surjective.

Proof. (i): using remark 8.2.30(ii) we reduce to the case where A_{1} has constant finite rank over A. Set B_{1} := A_{1} ⊗_{A} B, and suppose that x ∈ B_{1∗} is integral over A_{1∗}, or equivalently, over A_{∗}. Consider the element e ∈ (A_{1} ⊗_{A} A_{1})_{∗} provided by proposition 3.1.4; for given ε ∈ 𝔪 write ε · e = ∑_{i}^{k} c_{i} ⊗ d_{i} for some c_{i}, d_{i} ∈ A_{1∗}. According to remark 4.1.17 we have: ∑_{i}^{k} c_{i} · Tr_{B_{1}/B}(xd_{i}) = ε · x for every x ∈ B_{1∗}. Corollary 4.4.31 implies that Tr_{B_{1}/B}(xd_{i}) is integral over A_{∗} for every i ≤ k, hence it lies in A_{∗}, by remark 8.2.30(i). Hence ε · x ∈ A_{1∗}, as claimed.

(ii): is an immediate consequence of (i) and of the general theory of [40, Exp.V, §5]. □

8.2.32. As a special case, let R be a V -algebra whose spectrum is connected; we suppose that there exists an element ε ∈ 𝔪 which is regular in R. Suppose moreover that R is integrally closed in R[ε^{−1}], consequently Spec R[ε^{−1}] is connected and π_{1}(Spec R[ε^{−1}]) is well defined. It follows as well that Spec R^{a} is connected. Indeed, if Spec R^{a} were not connected, neither would be Spec R_{∗}^{a}; but since R_{∗}^{a} ⊂ R[ε^{−1}], this is absurd. Then π_{1}(Spec R^{a}) is also well defined and, after a fibre functor for Cov(R[ε^{−1}]) is chosen, the functors Cov(Spec R) → Cov(Spec R^{a}) → Cov(Spec R[ε^{−1}]) : B ↦ B^{a} ↦ B_{∗}^{a}[ε^{−1}] induce continuous group homomorphisms ([40, Exp.V, Cor.6.2])

(8.2.33)  π_{1}(Spec R[ε^{−1}]) → π_{1}(Spec R^{a}) → π_{1}(Spec R).

Proposition 8.2.34. Under the assumptions of (8.2.32), we have:

(i) R_{∗}^{a} is integrally closed in R[ε^{−1}]. (ii) The maps (8.2.33)are surjective.

Proof. (i): let x ∈ R[ε^{−1}] such that x^{n}+a_{1}·x^{n−1}+...+a_{n} = 0 for some elements a_{1}, ..., a_{n} ∈ R_{∗}^{a}. For every δ ∈ 𝔪 we have (δ·x)^{n}+(δ·x)·(δ·x)^{n−1}+...+δ^{n}·a_{n} = 0, which shows that δ·x ∈ R, since by assumption R is integrally closed in R[ε^{−1}] and δ^{i} · a_{i} ∈ R for every i = 1, ..., n. The assertion follows.

(ii): it suffices to show the assertion for the leftmost map and for the composition of the two maps. However, the composition of the two maps is actually a special case of the leftmost map (for the classical limit V = 𝔪), so we need only consider the leftmost map. Then the assertion follows from (i) and proposition 8.2.31(ii). □

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Index

(V, 𝔪), 𝔪̃ : Basic setup, 2.1.1 (t, I)-adic topology on a module, 5.4.10 (t, I)-adic topology on a scheme, 5.4.15 A_{∞}-Mod_{top} : topological modules on a complete

almost algebra, 5.3.14 B_{(m)}, 3.5.7 I_{B/A}, 2.5.22 K^{+} : Valuation ring(s), 6.1.11

K^{+h}, K^{+sh}, K^{sh} : (strict) henselization of a, 6.1.12,

6.2.16 standard setup attached to a, 6.1.15 total log structure on a, 6.4.14

M^{a}, 2.2.2 R-Mod, R-Alg (for a ring R), 2.2.1 V -Mon : category of V -monoids, 3.5.2 V^{a}-Alg, A-Mod, A-Alg (for an almost algebra A)

, 2.2.7 V^{a}-Alg.Morph, V^{a}-Alg.Mod, 2.5.22 V^{a}-Mod, (V, 𝔪)^{a}-Mod, 2.2.2 alHom_{A}(M, N) : Almost homomorphisms, 2.2.10 B_{!!}, 2.2.23, 3.3.5 Desc(𝒞 , Y/X), 3.4.23 Idemp(R) : idempotent elements of a ring, 3.4.42 I_{!∗}, 2.5.19 𝕃_{X/Y,k}^{G}, 52 𝕃_{B/A}^{an} : Analytic cotangent complex for topological

algebras, 7.1.23 M_{!}, 2.2.19 Ω_{P/K}^{na} : non-analytic differentials, 7.3.28 Ω_{B/A}^{an} : module of analytic differentials for topolog-

ical algebras, 7.1.23, 7.2.34 Proj(A) : projective almost scheme associated to a

graded almost algebra, 5.7.11 Set : category of sets, 2.2.1 Tor_{i}^{A}(M, N), alExt_{A}^{i}(M, N), 2.4.10 Exal_{A}, A^{†}-Uni.Mod, 2.5.7, 2.5.8 μ(F), 6.2.3 ℬ : category of basic setups, 3.5 ℬ-monoid(s), 3.5.2

Φ_{A} : Frobenius endomorphism of, 3.5.6

ℬ-Mon : category of, 3.5.2

Frobenius nilpotent, 3.5.8

invertible morphism up to Φ^{m} of , 3.5.8 ℬ/𝔽_{p}, ℬ-Alg/𝔽_{p}, ℬ-Mon/𝔽_{p}, ℬ^{a}-Alg/𝔽_{p}, 3.5.5 ℬ-Alg : category of ℬ-algebras, 3.5.2 ℬ-Mod : category of ℬ-modules, 3.5.1 ℬ^{a}-Mod, ℬ^{a}-Alg, ℬ^{a}-Mon, 3.5.3 𝒞 : Category(ies)

𝒞 ^{o} : opposite of a, 2.2

1_{X} : identity morphism of an object in a, 2.2 D(𝒞), D^{+}(𝒞), D^{−}(𝒞) : derived categories of a,

2.2.1 s.𝒞 : simplicial objects of a, 2.2 Galois, 8.2.11

fibre functor of a, 8.2.12 Pregalois, 8.2.14 tensor, 2.2.5

X ∗ϕ, ψ ∗X : push-out, pull-back of extensions

in a, 2.5.5 Exmon_{𝒞} (B, I), Exal_{𝒞} (B, I) : isomorphism classes

of extensions in a, 2.5.5, 2.5.7 Exmon_{𝒞} (B, I), Exun_{𝒞} , Exal_{𝒞} : extensions of a monoid (or an algebra) in a , 2.5.2, 2.5.4

μ_{A} : multiplication of a monoid in a, 2.2.5 σ_{M/A} : scalar multiplication of a module in a,

2.2.5 (left, right, unitary) module in a , 2.2.5 θ_{M|N}: switch operator for two modules in a,

2.2.5 monoid, algebra in a, 2.2.5

𝒟-Mod : modules over a diagram 𝒟 of almost al-

gebras, 3.4.7 𝒢_{n}, 𝒢_{∞}, 4.2.6, 4.3.9 ℐ_{A}(M), E_{M}(𝔪_{0}), E_{ℳ} (𝔪_{0}), 2.3.1 𝒪_{X}^{a}-Mod, 𝒪_{X}^{a}-Alg : sheaves of 𝒪_{X}^{a}-modules, resp.

𝒪_{X}^{a}-algebras on a scheme X, 5.5.1 𝒪_{X}^{a}-Mod_{qcoh}, 𝒪_{X}^{a}-Alg_{qcoh} : quasi-coherent, 5.5.1 ℱ_{!}, ℱ_{∗} (for an 𝒪_{X}^{a}-module ℱ), 5.5.3

χ_{P} (X), ψ_{P} (X), 4.3.7 det(1_{P} + ϕ), 4.3.1 η_{B,C}, 4.1.13 u.Ét^{o}, 3.4.44 𝔼xt_{R}^{p}(E, F), 2.5.21 Hom_{𝒪_{X}} (ℱ , ∗), 6.4.10 Mnd_{X}, ℤ-Mod_{X}, 6.4.1 ω(R, u′) : obstruction class, 3.2.20 ω(B̃, f_{0}, u), ω(B̃, f_{0}) : obstruction classes, 3.2.7,

3.2.10 ω_{P/A}, 2.4.26 ϕ^{♯} (for a morphism ϕ of almost schemes), 3.3.3 ρ_{E^{+}/K^{+}}, 6.5.1 D^{−}(A-Mod)^{∧}, 7.1.13 H_{R}(F, J), 5.4.1 j: morphism of fpqc-sites, 5.7 τ_{e}, τ_{w} : topologies on V^{a}-Alg, 3.4.42 ζ_{P}, 4.1.1 u_{M}, 4.4.32 Bud(n, d, R) : n-buds of a formal group, 4.2.2

𝔾_{m,R}(n) : n-bud of 𝔾̂_{m}, 4.2.1 N (A) : Spectrum of seminorms of a ring, 7.5.11 (n-)pseudo-coherent complex(es), 7.1.15

D(R-Mod)_{coh} : derived category of, 7.1.16

Adic space(s), 7.2.23 Exan_{Y} (X, ℱ) : analytic deformations of an, 7.3.12 𝕃_{X/Y}^{+} , 𝕃_{X/Y}^{an} : analytic cotangent complex of a

morphism of, 7.2.31 Ω_{X/K}^{+} : sheaf of analytic differentials of an, 7.4 d(𝔛) : adic space associated to a formal scheme,

7.2.26 étale morphism of, 7.2.25

𝔡_{Y/X}^{+} : discriminant of an, 7.5.10

def(f) : defect of an, 7.5.25 affinoid, 7.2.23 deeply ramified system of, 7.4.10 morphism locally of finite type of, 7.2.24 smooth, unramified morphism of, 7.2.25

Affine almost group scheme(s), 3.3.3 ℓ_{G/S} : colie complex of an, 3.3.33 (right, left) action on an almost scheme by a, 3.3.6 (right, left) action on an almost scheme of a

G_{X}^{•} : nerve of an, 3.3.6 action on a quasi-coherent module, equivariant quasi-

coherent module, 3.3.6 torsor over an, 3.3.28

Affinoid ring(s), 7.2.17 Spa A : adic spectrum of an, 7.2.22 completion of, 7.2.18

Almost algebra(s), 2.2.5, 2.2.7 A-Ét_{afp} : étale almost finitely presented, 3.2.27 Cov(Spec A) : étale covering of an, 8.2.22 Der_{A}(B, M) : derivations of an, 2.5.22 𝕃_{B/A} : cotangent complex of an, 2.5.20

transitivity of the , 2.5.32 Ω_{B/A} : relative differentials of an , 2.5.22 Spec A : spectrum of an, 3.3.3 nil(A) : nilradical of an, 4.1.18 rad(A) : Jacobson radical, 5.1.1 (faithfully) flat, 3.1.1 (tight) henselian pair, 5.1.8 (uniformly) almost finite, 3.1.1 (weakly) unramified, 3.1.1

e_{B/A} : idempotent associated to an, 3.1.4 (weakly) étale, 3.1.1

A-w.Ét, A-Ét : category of, 3.2.17 almost finite projective, 3.1.1 almost projective, 3.1.1

Tr_{B/A} : trace morphism of an , 4.1.7

𝒟_{B/A} : different ideal of an , 4.1.22

𝔡_{B/A} : discriminant of an, 7.5.2

t_{B/A}, τ_{B/A} : trace form of an, 4.1.12 connected, 8.2.24

π_{1}(Spec A) : fundamental group of a, 8.2.25 exact, 2.2.25 henselization of, 5.1.13 ideal in an, 2.2.5

tight, 5.1.5 integral closure of an, 8.2.27 projective topology on a complete linearly topolo-

gized, 5.3.4 reduced, 4.1.18 strictly finite, 3.1.1

Almost isomorphism, 2.1.3 Almost module(s), 2.2.2

A-Mod_{afpr} : almost finitely generated projective,

2.4.4 M ⊗_{A} N : tensor product of, 2.2.11 M^{∗} : dual of an, 2.4.23 M_{∗} : almost element(s) of an, 2.2.8, 2.2.9

M-regular, 4.1.21 Ann_{A}(M) : annihilator of an, 2.2.5 Λ-nilpotent endomorphism of an , 4.3.1 ℰ_{M/A} : evaluation ideal, 2.4.23 ℰ_{M/A}(f) : evaluation ideal of an almost element

in an, 2.4.23 ℳ_{c}(A) : A-modules generated by at most c al-

most elements, 2.3 𝒰_{n}(A) : uniformly almost finitely generated, 2.3.8

F_{i}(M) : Fitting ideals of a, 2.3.23 ev_{M/A} : evaluation morphism of an, 2.4.23 (almost) finitely generated , 2.3.8 (almost) finitely presented , 2.3.8 (faithfully) flat, 2.4.4 almost projective, 2.4.4

Split(A, P) : splitting algebra of an, 4.4.1 𝔡_{A}(P, b) : discriminant of a bilinear form on a,

7.5.1 f.rk_{A}(P) : formal rank of an , 4.3.9 f.rk_{A}(P, 𝔭) : localization at a prime ideal of the

formal rank of an, 4.4.13 tr_{P/A} : trace morphism of an , 4.1.1 of almost finite rank , 4.3.9 of constant rank , 4.3.9 of finite rank , 4.3.9

invertible, 2.4.23

linear topology on an, 5.3.1 I-preadic, I-c-preadic, I-adic, 5.3.1 complete, 5.3.1 completion of a, 5.3.2 continuous morphism of, 5.3.2 generated by a family of submodules, 5.3.1 ideal of definition of a, 5.3.1 induced topology on a submodule, 5.3.2 topological closure of a submodule in a, 5.3.1

reflexive, 2.4.23

strictly invertible , 4.4.32

that admits infinite splittings , 4.4.15 Almost scheme(s)

X_{sm} : smooth locus of an, 5.6.1, 5.7.21 𝕃_{X/Y} : cotangent complex of an, 3.3.13 ℙ_{R^{a}}(E), Grass_{R^{a}}^{r}(E) : projective space, Grass-

mannian, 5.7.7 Exal_{Y} (X/G, ℐ) : equivariant square zero defor-

mation of an, 3.3.14 affine, morphism of affine, 3.3.3 almost finite, étale, flat, almost projective morphism

of, 4.5.5 almost finitely presented, 5.6.3 closed imbedding of, 4.5.5 open and closed imbedding of, 4.5.5 quasi-affine, 5.7.1 quasi-coherent module on an, 3.3.3 square zero deformation of an, 3.3.13

Almost zero module, 2.1.3

bicovering morphism in a site, 3.5.16 Cauchy product, 2.3.4 Classical limit, 2.1.2 Closed imbedding, difference of sheaves on (R^{a}-Alg)_{fpqc}^{o},

5.7.5 Conditions A and B, 2.1.6

Decomposition subgroup, 6.2.3

f-adic ring(s), 7.2.15 Cont(A) : continuous valuations of an, 7.2.21 topologically finite type map of, 7.2.16 ideal of definition of an, 7.2.15 ring of definition of an, 7.2.15

Formal scheme(s)

𝕃_{𝔛/𝔜}^{an} : Cotangent complex of a morphism of, 7.2.3 Ω_{𝔛/𝔜}^{an} : sheaf of analytic relative differentials of a

morphism of, 7.2.8 Deeply ramified system of, 7.4.5

Groupoid(s), 4.5.1 X_{0}/G : quotient by an action of a, 4.5.5 closed equivalence relation, flat, étale, almost fi-

nite, almost projective, of finite rank, 4.5.5 with trivial automorphisms, 4.5.1

Inertia subgroup, 6.2.3

Log structure(s), 6.4.4 log_{X} : category of, 6.4.4 regular, 6.4.9

Monoid(s), 6.1.26 M^{gp} : group associated to an abelian, 6.1.29 Spec M : spectrum of a, 6.1.27 Mnd : category of commutative, 6.1.29 convex submonoid of a, 6.1.26 free, 6.1.28

basis of a, 6.1.28 ideal, prime ideal in a, 6.1.27 integral, 6.1.28

Ordered abelian group(s), 6.1.20 Γ^{+} : submonoid of positive elements in an, 6.1.20 Spec Γ : spectrum of an, 6.1.20 c.rk(Γ), rk(Γ) : convex and rational rank of an,

6.1.20 convex subgroup of an, 6.1.20

Pre-log structure(s), 6.4.2 Ω_{X/ℤ}(log \underline{M}) : logarithmic differentials of a, 6.4.11 pre-log_{X} : category of, 6.4.2

Pretopos, 8.2.6 Prüfer Domain, 6.1.12

Simplicial almost algebra(s), 8.1 exact, 8.1.2 modules over a, 8.1

derived category of , 8.1.1

quasi-isomorphism of , 8.1.2

Topos, 8.2 coherent, 8.2.4 constant, locally constant, bounded object in a, 8.2.20 fibred, 3.3.15

Top(X) : topos associated to a, 3.3.15 point of a, 8.2.9 quasi-compact object of a, 8.2.1 quasi-compact, quasi-separated, coherent morphism

in a, 8.2.2 quasi-separated, coherent object of a, 8.2.3

Valuation(s), 6.1.1 Gauss, 6.1.4 rank of a, 6.1.23 Valued field(s), 6.1.1

K^{s}, K^{a} : separable, algebraic closure of a, 6.2.16 K^{t} : maximal tame extension of a, 6.2.16 Γ_{K} : value group of a, 6.1.1 Div(K^{+a}) : fractional ideals of a, 6.1.16 deeply ramified, 6.6.1 extension of a, 6.1.3

𝔡_{L/K}^{+} : discriminant of an algebraic, 7.5.6 valuation topology of a, 6.1.21