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

DOI:
https://doi.org/10.1090/S1088-4165-2011-00385-X

Published electronically:
March 14, 2011

MathSciNet review:
2781019

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a finite group and let be a field of characteristic . Given a finitely generated indecomposable nonprojective -module , we conjecture that if the Tate cohomology of with coefficients in is finitely generated over the Tate cohomology ring , then the support variety of is equal to the entire maximal ideal spectrum . We prove various results which support this conjecture. The converse of this conjecture is established for modules in the connected component of in the stable Auslander-Reiten quiver for , but it is shown to be false in general. It is also shown that all finitely generated -modules over a group have finitely generated Tate cohomology if and only if has periodic cohomology.

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Additional Information

**Jon F. Carlson**

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

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

DOI:
https://doi.org/10.1090/S1088-4165-2011-00385-X

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.