Finite generation of Tate cohomology
Authors:
Jon F. Carlson, Sunil K. Chebolu and Ján Mináč
Journal:
Represent. Theory 15 (2011), 244257
MSC (2010):
Primary 20C20, 20J06; Secondary 55P42
Posted:
March 14, 2011
MathSciNet review:
2781019
Fulltext PDF Free Access
Abstract 
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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 AuslanderReiten 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, Almostsplit sequences and group rings, J. Algebra
103 (1986), no. 1, 122–140. MR 860693
(88a:16054), http://dx.doi.org/10.1016/00218693(86)901730
 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
(92m:20005)
 3.
D.
J. Benson and Jon
F. Carlson, Products in negative cohomology, J. Pure Appl.
Algebra 82 (1992), no. 2, 107–129. MR 1182934
(93i:20058), http://dx.doi.org/10.1016/00224049(92)90116W
 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
(98g:20021)
 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
(91c:20073), http://dx.doi.org/10.1016/00218693(90)90165K
 6.
W.
Burnside, Theory of groups of finite order, Dover Publications
Inc., New York, 1955. 2d ed. MR 0069818
(16,1086c)
 7.
Jon
F. Carlson, Modules and group algebras, Lectures in
Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1996. Notes by
Ruedi Suter. MR
1393196 (97c:20013)
 8.
Jon
F. Carlson, Lisa
Townsley, Luis
ValeriElizondo, 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, ValeriElizondo and Zhang. MR 2028960
(2004k:20110)
 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
(2000h:18022)
 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
(2008k:20017), http://dx.doi.org/10.1090/S0002993907090582
 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
(2008m:20018), http://dx.doi.org/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
(99g:18007), http://dx.doi.org/10.1006/aima.1998.1735
 13.
Richard
G. Swan, Groups with periodic
cohomology, Bull. Amer. Math. Soc. 65 (1959), 368–370. MR 0115175
(22 #5977), http://dx.doi.org/10.1090/S000299041959103785
 14.
John
Tate, The higher dimensional cohomology groups of class field
theory, Ann. of Math. (2) 56 (1952), 294–297.
MR
0049950 (14,252b)
 1.
 M. Auslander and J. F. Carlson,
Almostsplit sequences and group rings. J. Algebra, 103(1):122140, 1986. MR 860693 (88a:16054)
 2.
 D. J. Benson, Representations and Cohomology, I, II, Cambridge Univ. Press, Cambridge, 1991. MR 1110581 (92m:20005)
 3.
 D. J. Benson and J. F. Carlson, Products in negative cohomology, J. Pure Appl. Algebra, 82(1992), 107129. MR 1182934 (93i:20058)
 4.
 D. J. Benson, J. F. Carlson and J. Rickard, Thick subcategories of the stable category, Fund. Math. 153(1997), 5980. MR 1450996 (98g:20021)
 5.
 D. J. Benson, J. F. Carlson, and G. R. Robinson,
On the vanishing of group cohomology, J. Algebra, 131(1):4073, 1990. MR 1054998 (91c:20073)
 6.
 W. Burnside,
Theory of groups of finite order, Dover Publications Inc., New York, 1955. 2nd ed. MR 0069818 (16:1086c)
 7.
 J. F. Carlson,
Modules and group algebras. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1996. Notes by Ruedi Suter. MR 1393196 (97c:20013)
 8.
 J. Carlson, L. Townsley, L. ValeroElizondo and M. Zhang, Cohomology Rings of Finite Groups, Kluwer, Dordrecht, 2003. MR 2028960 (2004k:20110)
 9.
 H. Cartan and S. 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 (2000h:18022)
 10.
 S. K. Chebolu, J. D. Christensen, and J. Mináč,
Groups which do not admit ghosts. Proc. Amer. Math. Soc., 136:11711179, 2008. MR 2367091 (2008k:20017)
 11.
 S. K. Chebolu, J. D. Christensen, and J. Mináč,
Ghosts in modular representation theory, Advances in Mathematics, 217:27822799, 2008. MR 2397466 (2008m:20018)
 12.
 J. D. Christensen,
Ideals in triangulated categories: phantoms, ghosts and skeleta, Adv. Math., 136(2):284339, 1998. MR 1626856 (99g:18007)
 13.
 R. G. Swan,
Groups with periodic cohomology. Bull. Amer. Math. Soc., 65:368370, 1959. MR 0115175 (22:5977)
 14.
 J. Tate,
The higher dimensional cohomology groups of class field theory. Ann. of Math. (1), 56:294297, 1952. MR 0049950 (14:252b)
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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:
http://dx.doi.org/10.1090/S10884165201100385X
PII:
S 10884165(2011)00385X
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
Posted:
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.
