In this paper, we prove the conjectures made in a joint paper of theauthor with Carlson and Robinson, on the vanishing of cohomology ofa finite group G. In particular, we prove that if k is a field of characteristicp, then every non-projective kG-module M in the principal block has nontrivialcohomology in the sense that H*(G,M) ≠ 0, if and only if the centralizer in G of every element oforder p is p-nilpotent (this was proved for p odd inthe above mentioned paper, but the proof here is independentof p). We prove thestronger 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 theelementary abelian p-subgroups whose centralizers are notp-nilpotent (i.e., the nucleus). The proofs involve the newidempotent functor machinery of Rickard.
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