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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Finite generation of Tate cohomology
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by Jon F. Carlson, Sunil K. Chebolu and Ján Mináč PDF
Represent. Theory 15 (2011), 244-257 Request permission

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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Additional 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

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