Realizability of modules over Tate cohomology

Benson D, Krause H, Schwede S. Realizability of modules over Tate cohomology. Transactions of the American Mathematical Society. 2004;356(9):3621-3668.Let k be a field and let G be a finite group. There is a canon- ical element in the Hochschild cohomology of the Tate cohomology γG ∈ HH3,−1 ˆH∗(G, k) with the following property. Given a graded ˆH∗(G, k)-module X, the image of γG in Ext3,−1 ˆH∗(G,k) (X, X) vanishes if and only if X is isomorphic to a direct summand of ˆH∗(G, M ) for some kG-module 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, there is also a canonical element of Hochschild cohomology HH3,−1H∗(A) which is a predecessor for these obstructions