Secondary multiplication in Tate cohomology of generalized quaternion groups

SECONDARY MULTIPLICATION IN TATE COHOMOLOGY OF CERTAIN p-GROUPS

Let k be a field and let G be a finite group. By a theorem of D. Benson, H. Krause and S. Schwede, there is a canonical element in the Hochschild cohomology of the Tate cohomology γG ∈ HH 3,−1Ĥ∗(G) with the following property: Given any graded Ĥ∗(G)-module X, the image of γG in Ext 3,−1 Ĥ∗(G) (X,X) is zero if and only if X is isomorphic to a direct summand of Ĥ(G,M) for some kG-module M . Suppo...

Hochschild Cohomology of Algebras of Quaternion Type, I: Generalized Quaternion Groups

In terms of generators and defining relations, a description is given of the Hochschild cohomology algebra for one of the series of local algebras of quaternion type. As a corollary, the Hochschild cohomology algebra is described for the group algebras of generalized quaternion groups over algebraically closed fields of characteristic 2. Introduction Let R be a finite-dimensional algebra over a...

Tate Cohomology for Arbitrary Groups via Satellites

We define cohomology groups Ĥn(G;M), n ∈ Z, for an arbitrary group G and G-module M , using the concept of satellites. These cohomology groups generalize the Farrell-Tate groups for groups of finite virtual cohomological dimension and form a connected sequence of functors, characterized by a natural universal property. The classical Tate cohomology groups of finite groups have been generalized ...

A Tate cohomology sequence for generalized Burnside rings

We generalize the fundamental theorem for Burnside rings to the mark morphism of plus constructions defined by Boltje. The main observation is the following: If D is a restriction functor for a finite group G, then the mark morphism φ : D+ → D is the same as the norm map of the Tate cohomology sequence (over conjugation algebra for G) after composing with a suitable isomorphism of D. As a conse...

Tate Cohomology in Commutative Algebra.