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Representation Theory

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Finite generation of Tate cohomology

Authors: Jon F. Carlson, Sunil K. Chebolu and Ján Mináč
Journal: Represent. Theory 15 (2011), 244-257
MSC (2010): Primary 20C20, 20J06; Secondary 55P42
Published electronically: March 14, 2011
MathSciNet review: 2781019
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Abstract: Let $ G$ be a finite group and let $ k$ be a field of characteristic $ p$. Given a finitely generated indecomposable nonprojective $ kG$-module $ M$, we conjecture that if the Tate cohomology $ \hat{H}^*(G, M)$ of $ G$ with coefficients in $ M$ is finitely generated over the Tate cohomology ring $ \hat{H}^*(G, k)$, then the support variety $ V_G(M)$ of $ M$ is equal to the entire maximal ideal spectrum $ V_G(k)$. We prove various results which support this conjecture. The converse of this conjecture is established for modules in the connected component of $ k$ in the stable Auslander-Reiten quiver for $ kG$, but it is shown to be false in general. It is also shown that all finitely generated $ kG$-modules over a group $ G$ have finitely generated Tate cohomology if and only if $ G$ has periodic cohomology.

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Additional Information

Jon F. Carlson
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602

Sunil K. Chebolu
Affiliation: Department of Mathematics, Illinois State University, Campus box 4520, Normal, Illinois 61790

Ján Mináč
Affiliation: Department of Mathematics, University of Western Ontario, London, ON N6A 5B7, Canada

Keywords: Tate cohomology, finite generation, periodic modules, support varieties, stable module category, almost split sequence
Received by editor(s): August 17, 2009
Received by editor(s) in revised form: March 9, 2010
Published electronically: March 14, 2011
Additional Notes: The first author is partially supported by a grant from NSF and the third author is supported from NSERC
Dedicated: Dedicated to Professor Luchezar Avramov on his sixtieth birthday.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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