Realizability of modules over Tate cohomology

Abstract

Let k k be a field and let G G be a finite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology γ G ∈ H H 3 , − 1 H ^ ∗ ( G , k ) \gamma _G\in H\!H^{3,-1}\hat H^*(G,k) with the following property. Given a graded H ^ ∗ ( G , k ) \hat H^*(G,k) -module X X , the image of γ G \gamma _G in Ext H ^ ∗ ( G , k ) 3 , − 1 ⁡ ( X , X ) \operatorname {Ext}^{3,-1}_{\hat H^*(G,k)}(X,X) vanishes if and only if X X is isomorphic to a direct summand of H ^ ∗ ( G , M ) \hat H^*(G,M) for some k G kG -module M M . The description of the realizability obstruction works in any triangulated category with direct sums. We show that in the case of the derived category of a differential graded algebra A A , there is also a canonical element of Hochschild cohomology H H 3 , − 1 H ∗ ( A ) H\!H^{3,-1}H^*(A) which is a predecessor for these obstructions.