Subject: equivariant cohomology Date: Wed, 17 Jul 2002 18:03:45 -0400 From: Murray Gerstenhaber To: Don Davis Dear Don, In reply to an old queston of Jim Stasheff concerning the equivariant cohomology of an algebra acted upon by a group: An algebra with a group action is just a special case of a presheaf of algebras over small category (called a "diagram of algebras in Algebraic cohomology and deformation theory, M. Gerstenhaber and S.D.Schack, in "Deformation Theory of Algebras and Structures and Applications", M. Hazewinkel and M. Gerstenhaber, eds, Kluwer 1988, pp11-264). A group is just a category with but a single object and all morphisms being isomorphisms. There is a natural cohomology for any such presheaf which combines the cohomology of the nerve of the category and the Hochschild cohomology of the algebras. (In the special case where the category conisists of a single object with only the identity morphism one just gets the cohomology of the algebra; if the algebras are all reduced to the coefficient ring with identity morphism one gets the simpicial cohomology of the nerve.) For any presheaf of algebras, the Cohomology Comparison Theorem of S.D.Schack and myself then asserts that there is a single algebra whose cohomology is identical to the natural cohomology of the presheaf of algebras. (Unfortunately the theory is not well known. Proof for the special case where the category is a poset is in The cohomology of presheaves of algebras. I, Trans. Amer. Math. Soc.310(1988)135--165. The complete proof in the general case was rejected as too long to print.) M. Gerstenhaber