 

D. J. Benson
Cohomology of Modules in the Principal Block of a Finite Group


Published: 
December 21, 1995 
Keywords: 
finite group, representations, cohomology, nucleus, idempotent functor 
Subject: 
20C20, 20J06 


Abstract
In this paper, we prove the conjectures made in a joint paper of the
author with Carlson and Robinson, on the vanishing of cohomology of
a finite group G.
In particular, we prove that if k is a field of characteristic
p, then every nonprojective kGmodule
M in the principal block has nontrivial
cohomology in the sense that H*(G,M) ≠ 0,
if and only if the centralizer in G of every element of
order p is pnilpotent (this was proved for p odd in
the above mentioned paper, but the proof here is independent
of p). We prove the
stronger statement that whether or not these conditions hold, the union
of the varieties of the modules in the principal block having no
cohomology coincides with the union of the varieties of the
elementary abelian psubgroups whose centralizers are not
pnilpotent (i.e., the nucleus). The proofs involve the new
idempotent functor machinery of Rickard.


Acknowledgements
Partially supported by a grant from the NSF


Author information
Department of Mathematics, University of Georgia, Athens GA 30602, USA.
Current address: Department of Mathematics,
University of Aberdeen,
Meston Building,
King's College,
Aberdeen AB24 3UE,
Scotland, UK.
http://www.maths.abdn.ac.uk/~bensondj/html/index.html

