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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
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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

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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