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