Thematic school of homotopy theory
Lille 2-6 april of 2012
John Greenlees (University of Sheffield) "Duality and local cohomology"
Abstract and references:
There are many well known dualities, falling into two classes.
The first class are contravariant constructions of one object from
another (vector space duality, Alexander duality, Koszul duality,
Matlis duality, Spanier-Whitehead duality, Brown-Comenetz duality,
.....). The second class state that applying such a functor to special
objects give the essentially the same thing back (Poincaré duality,
Lefschetz duality, Atiyah duality, Gorenstein duality, Serre duality,
Tate duality, Tate-Poitou duality, Benson-Carlson duality,
Gross-Hopkins duality, Kontsevich duality, Cohen-Jones duality,
Ausoni-Rognes duality, Mahowald-Rezk duality, .....). It is this second
sort of self-duality that is of particular interest.
Poincaré duality states that the cohomology of an orientable manifold is isomorphic to its dual. We can break down this process.
(0) Start with a nice geometric object (a manifold M),
(1) Take cochains C^*(M;k) to move towards algebra
(2) Take cohomology H^*(M;k) to get a conventional graded ring
(3) This ring has duality.
This can be generalized in several different directions. We can
generalize the class of geometric objects, replace ordinary cohomology
by another cohomology theory (K-theory, bordism, ....), or we can
consider group actions (the manifold has a group action and the
cohomology is equivariant). Alternatively we could permit 0-dimensional
rings (like H^*(M;k)) to be replaced by a ring of higher Krull
dimension. These generalizations overlap, and altogether the message is
that we should work with cochains (ie do the algebraic work in a
homotopy invariant way at level (1)), and replace C^*(M;k)---->k by
a more general map R----->k of commutative rings. The resulting
dualities include most of those listed and lead to a wide variety of
rich and interesting examples.
One example provides an excellent illustration of the techniques:
where the ring in (3) is the group cohomology ring
H^*(BG;k)=Ext_{kG}^*(k,k) for a finite group G. To start with, there
are enormous numbers of explicit examples available from Jon Carlson's
calculations. The duality gives very concrete and accessible
consequences: for example it gives restrictions on the Hilbert series
in good cases (Benson-Carlson duality) and to conditions on
localizations in general. This example is also good for illustrating
the different methods. For example it can be viewed as generalizing the
manifold to the classifying space BG, it can be proved by applying
equivariant techniques to ordinary Borel cohomology, and it can be
proved by applying methods of commutative algebra in a homotopy
invariant way. Furthermore, it can be given proofs in several different
contexts (chain complexes of kG modules, ring spectra or equivariant
ring spectra).
The lectures should cover the following in varying degrees of detail.
Examples. Koszul complexes, stable Koszul complexes, local cohomology
and local homology. The local cohomology theorem for group cohomology.
Symonds's theorem. Various classes of groups. Examples. Proof for
finite groups. Equivariant proof. Other equivariant examples. Morita
theory. Cellularization. Exterior algebras. Proxy-regularity.
Brown-Comenetz-Matlis duality. The Gorenstein condition. Orientability
and Gorenstein duality. Local duality. Implications for Hilbert
series. Non-orientable examples. Stringy examples. Chromatic
examples. Rational homotopy examples.
One good starting place, with a selection of references is
J.P.C.Greenlees
``First steps in brave new commutative algebra''
Proceedings of the 2004 Chicago Summer School
Contemporary Mathematics, 436 (2007) 239-275
Background in the cohomology of groups could come from
D.J.Benson ``Commutative Algebra in the Cohomology of Groups''
Trends in Commutative Algebra MSRI Publications
Volume 51, 2004