## Finite generation of Tate cohomology

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- by Jon F. Carlson, Sunil K. Chebolu and Ján Mináč
- Represent. Theory
**15**(2011), 244-257 - DOI: https://doi.org/10.1090/S1088-4165-2011-00385-X
- Published electronically: March 14, 2011
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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.## References

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## Bibliographic Information

**Jon F. Carlson**- Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
- MR Author ID: 45415
- Email: jfc@math.uga.edu
**Sunil K. Chebolu**- Affiliation: Department of Mathematics, Illinois State University, Campus box 4520, Normal, Illinois 61790
- Email: schebol@ilstu.edu
**Ján Mináč**- Affiliation: Department of Mathematics, University of Western Ontario, London, ON N6A 5B7, Canada
- Email: minac@uwo.ca
- 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
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory
**15**(2011), 244-257 - MSC (2010): Primary 20C20, 20J06; Secondary 55P42
- DOI: https://doi.org/10.1090/S1088-4165-2011-00385-X
- MathSciNet review: 2781019

Dedicated: Dedicated to Professor Luchezar Avramov on his sixtieth birthday.