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 . Suppose that the characteristic of k is p 6= 3. We show that γG is trivial if G is a (finite) abelian p-group of p-rank at least 3. Furthermore, γG is non-trivial if G is an abelian p-group of p-rank 2, or if p = 2 and G is the quaternion group with 8 elements.