Centralizers of elementary Abelian \(p\)-subgroups and mod-\(p\) cohomology of profinite groups
Let \(G\) be a profinite group and \({\mathcal A}_*(G)\) is the category whose objects are all elementary \(p\)-subgroups of \(G\) and morphisms are homomorphisms induced by conjugation in \(G\) (i.e., if \(E_1\) and \(E_2\) are elementary Abelian subgroups of \(G\), then the set of morphisms from \(E_1\) to \(E_2\) consists of those homomorphisms \(\alpha\colon E_1\to E_2\) for which there exists an element \(g\in G\) with \(\alpha(e)=geg^{-1}\) for all \(e\in E_1\)). Let \(C_G(E)\) be the centralizer of \(E\) in \(G\). The assignment \(E\to H^*(C_G(E),\mathbb{F}_p)\) extends to a functor from \({\mathcal A}_*(G)\) to graded \(\mathbb{F}_p\)-algebras, and the restriction homomorphisms \(H^*(G(E),\mathbb{F}_p)\to H^*(C_G(E),\mathbb{F}_p)\) induce a canonical homomorphism \(H^*(G(E),\mathbb{F}_p)\to\lim_{{\mathcal A}_*(G)}H^*(C_G(E),\mathbb{F}_p)\). The main result of the paper is the following Theorem. Let \(G\) be a profinite group and assume that \(H^*(G,\mathbb{F}_p)\) is a finitely generated \(\mathbb{F}_p\)-algebra. Then the canonical map \( H^*(G(E),\mathbb{F}_p)\to\lim_{{\mathcal A}_*(G)}H^*(C_G(E),\mathbb{F}_p)\) has finite kernel and cokernel. As a corollary the author deduces that if all elementary \(p\)-subgroups of \(G\) are cyclic, then the restriction maps induce a homomorphism \(H^*(G(E),\mathbb{F}_p)\to\prod_EH^*(C_G(E),\mathbb{F}_p)^{N_G(E)}\) with finite kernel and cokernel, where the direct product is taken over conjugacy classes of cyclic subgroups of order \(p\) and \(N_G(E)\) is the normalizer of \(E\) in \(G\).
complexes of elementary Abelian \(p\)-subgroups
scientific article; zbMATH DE number 1425085
scientific article