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Realizability of modules over Tate cohomology

Authors: David Benson, Henning Krause and Stefan Schwede
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 3621-3668
MSC (2000): Primary 20J06; Secondary 16E40, 16E45, 55S35
Published electronically: December 12, 2003
MathSciNet review: 2055748
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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 ${\text{\rm Ext}}^{3,-1}_{\hat H^*(G,k)}(X,X)$ 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.

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Additional Information

David Benson
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602

Henning Krause
Affiliation: Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
Address at time of publication: Institut für Mathematik, Universität Paderborn, D-33095 Paderborn, Germany

Stefan Schwede
Affiliation: SFB 478 Geometrische Strukturen in der Mathematik, Westfälische Wilhelms-Universität Münster, Hittorfstr. 27, 48149 Münster, Germany

Received by editor(s): April 5, 2002
Received by editor(s) in revised form: April 25, 2003
Published electronically: December 12, 2003
Additional Notes: The first author was partly supported by NSF grant DMS-9988110
Article copyright: © Copyright 2003 American Mathematical Society

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