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

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.

**1.**Maurice Auslander and Jon F. Carlson,*Almost-split sequences and group rings*, J. Algebra**103**(1986), no. 1, 122–140. MR**860693**, 10.1016/0021-8693(86)90173-0**2.**D. J. Benson,*Representations and cohomology. I*, Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1991. Basic representation theory of finite groups and associative algebras. MR**1110581****3.**D. J. Benson and Jon F. Carlson,*Products in negative cohomology*, J. Pure Appl. Algebra**82**(1992), no. 2, 107–129. MR**1182934**, 10.1016/0022-4049(92)90116-W**4.**D. J. Benson, Jon F. Carlson, and Jeremy Rickard,*Thick subcategories of the stable module category*, Fund. Math.**153**(1997), no. 1, 59–80. MR**1450996****5.**D. J. Benson, J. F. Carlson, and G. R. Robinson,*On the vanishing of group cohomology*, J. Algebra**131**(1990), no. 1, 40–73. MR**1054998**, 10.1016/0021-8693(90)90165-K**6.**W. Burnside,*Theory of groups of finite order*, Dover Publications, Inc., New York, 1955. 2d ed. MR**0069818****7.**Jon F. Carlson,*Modules and group algebras*, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1996. Notes by Ruedi Suter. MR**1393196****8.**Jon F. Carlson, Lisa Townsley, Luis Valeri-Elizondo, and Mucheng Zhang,*Cohomology rings of finite groups*, Algebras and Applications, vol. 3, Kluwer Academic Publishers, Dordrecht, 2003. With an appendix: Calculations of cohomology rings of groups of order dividing 64 by Carlson, Valeri-Elizondo and Zhang. MR**2028960****9.**Henri Cartan and Samuel Eilenberg,*Homological algebra*, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1999. With an appendix by David A. Buchsbaum; Reprint of the 1956 original. MR**1731415****10.**Sunil K. Chebolu, J. Daniel Christensen, and Ján Mináč,*Groups which do not admit ghosts*, Proc. Amer. Math. Soc.**136**(2008), no. 4, 1171–1179. MR**2367091**, 10.1090/S0002-9939-07-09058-2**11.**Sunil K. Chebolu, J. Daniel Christensen, and Ján Mináč,*Ghosts in modular representation theory*, Adv. Math.**217**(2008), no. 6, 2782–2799. MR**2397466**, 10.1016/j.aim.2007.11.008**12.**J. Daniel Christensen,*Ideals in triangulated categories: phantoms, ghosts and skeleta*, Adv. Math.**136**(1998), no. 2, 284–339. MR**1626856**, 10.1006/aima.1998.1735**13.**Richard G. Swan,*Groups with periodic cohomology*, Bull. Amer. Math. Soc.**65**(1959), 368–370. MR**0115175**, 10.1090/S0002-9904-1959-10378-5**14.**John Tate,*The higher dimensional cohomology groups of class field theory*, Ann. of Math. (2)**56**(1952), 294–297. MR**0049950**

Retrieve articles in *Representation Theory of the American Mathematical Society*
with MSC (2010):
20C20,
20J06,
55P42

Retrieve articles in all journals with MSC (2010): 20C20, 20J06, 55P42

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:
http://dx.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.