David Benson, Henning Krause and Stefan Schwede,
Realizability of modules over Tate cohomology

\begin{abstract}
Let $k$ be a field and let $G$ be a finite group. There is a canonical
element in the Hochschild cohomology of the Tate cohomology
$\gamma_G\in\HH^{3,-1}\hat H^*(G,k)$ with the following property. Given
a graded $\hat H^*(G,k)$-module $X$, the image of $\gamma_G$
in $\Ext^{3,-1}_{\hat H^*(G,k)}(X,X)$ vanishes if and only if $X$ is 
isomorphic to a direct summand of $\hat H^*(G,M)$ 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 $\HH^{3,-1}H^*(A)$ which is
a predecessor for these obstructions.
\end{abstract}