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 (PoincarĂ© 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.

PoincarĂ© duality states that the cohomology of an orientable manifold is isomorphic to its dual. We can break down this process.
(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