Cohomology of Modules in the Principal Block of a Finite Group
Cohomology of Modules in the Principal Block of a Finite Group
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有限群主块中模的上同调
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发表时间：
1995
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通讯作者：
D. Benson
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文献类型：
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作者：
D. Benson
本文证明了作者与Carlson和Robinson联合发表的关于nite群G的上同调为零的猜想。特别地，我们证明了：如果k是特征p的域，则主块中的每个非投射kg-模M具有H(G；M)60意义下的非平凡上同调，当且仅当每个p阶元在G中的中心化子是p-幂零的(文中证明了这一点是p奇数的，但这里的证明与p无关)。证明了无论这些条件是否成立，没有上同调的主块的模的变元的并与中心化子不是p-幂零的初等交换p-子群(即核)的变元的并重合。证明涉及到Rickard的新的幂等函子机器。
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 nite group G. In particular, we prove that if k is a eld of characteristic p, then every non-projective kG-module M in the principal block has nontrivial cohomology in the sense that H(G;M)6 0, if and only if the centralizer in G of every element of order p is p-nilpotent (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 p-subgroups whose centralizers are not p-nilpotent (i.e., the nucleus). The proofs involve the new idempotent functor machinery of Rickard.