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Varieties and modules with vanishing cohomology

Published online by Cambridge University Press:  24 October 2008

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

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
Copyright
Copyright © Cambridge Philosophical Society 1994

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References

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