Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-29T23:29:23.391Z Has data issue: false hasContentIssue false
  • English
  • Français

Varieties and modules with vanishing cohomology

Published online by Cambridge University Press:  24 October 2008

Jon F. Carlson  and
Geoffrey R. Robinson
Jon F. Carlson
Affiliation:
Department of Mathematics, University of Georgia, Athens, Georgia 30602, USA
Geoffrey R. Robinson
Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida, USA
Get access Rights & Permissions [Opens in a new window]

Extract

Several years ago the authors, together with Dave Benson, conducted an investigation into the vanishing of cohomology for modules over group algebras [2]. It was mostly in the context of kG-modules where k is a field of finite characteristic p and G is a finite group whose order is divisible by p. Aside from some general considerations, the main results of [2] related the existence of kG-modules M with H*(G, M) = 0 to the structure of the centralizers of the p-elements in G. Specifically it was shown that there exists a non-projective module M in the principal block of kG with H*(G, M) = 0 whenever the centralizer of some p-element of G is not p-nilpotent. The converse was proved in the special case that the prime p is an odd integer (p > 2). In addition there was some suspicion and much speculation about the structure of the varieties of such modules. However, proofs seemed to be waiting for a new idea.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 116 , Issue 2 , September 1994 , pp. 245 - 251
Copyright
Copyright © Cambridge Philosophical Society 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Benson, D. J.. Representations and Cohomology II: Cohomology of Groups and Modules, Cambridge Studies in Advanced Mathematics 31 (Cambridge University Press, 1991).Google Scholar
[2]Benson, D. J., Carlson, J. F. and Robinson, G. R.. On the vanishing of group cohomology. J. Algebra 131 (1990), 40–73.CrossRefGoogle Scholar
[3]Carlson, J. F.. Decomposition of the trivial module in the complexity quotient category, (preprint).Google Scholar
[4]Carlson, J. F., Donovan, P. W. and Wheeler, W. W.. Complexity and quotient categories for group algebras, J. Pure Appl. Algebra (to appear).Google Scholar
[5]Evens, L.. The Cohomology of Groups (Oxford University Press, 1991).Google Scholar
[6]Gorenstein, D.. Finite groups (Chelsea, New York, 1968).Google Scholar
[7]Griess, R. L.. Finite groups whose involutions lie in the center. Quart. J. of Mathematics (2) 29 (1978), 241–247.CrossRefGoogle Scholar
[8]Nagao, H.. A proof of Brauer's theorem on generalized decomposition numbers. Nagoya Math. J. 22 (1963), 73–77.Google Scholar
[9]Quillen, P.. A cohomological criterion for p-nilpotency. J. Pure Appl. Algebra 1 (1971), 361–372.CrossRefGoogle Scholar