an:07103854
Zbl 1426.55017
Barthel, Tobias; Castellana, Nat??lia; Heard, Drew; Valenzuela, Gabriel
Stratification and duality for homotopical groups
EN
Adv. Math. 354, Article ID 106733, 61 p. (2019).
00438217
2019
j
55R35 20J05 13D45 55P42
\(p\)-local compact groups; support theory; \(F\)-isomorphism theorem; stratification; costratification; Gorenstein duality
Let \(G\) be a finite group. Some of the most celebrated results in group cohomology are the \(F\)-isomorphism theorem of Quillen and the study of the spectrum of the cohomology ring of \(G\) with coefficients in a field \(k\) whose characteristic divides the order of \(G\). There have been generalizations of these results to other instances like \(p\)-compact groups and \(p\)-local finite groups. The main objective of this paper is to generalize these (and many related) results to \(p\)-local compact groups. Much of the work is devoted to establishing the technical tools to achieve these results. We provide some examples. A \(p\)-local compact group \(\mathcal{G}=(S,\mathcal{F})\) satisfies Choinard's theorem if induction and coinduction along the morphism induced by restriction \[ C^{*}(B\mathcal{G},k)\to \prod_{\mathcal{E}(\mathcal{G})} C^{*}(BE,k), \] is conservative, where \(\mathcal{E}(\mathcal{G})\) denotes a set of representatives of \(\mathcal{F}\)-conjugacy classes of elementary abelian subgroups of \(S\) and \(C^{*}(B(-),k)\) is the spectrum of \(k\)-valued cochains on \(B(-)\). The authors prove that connected \(p\)-local groups satisfy Choinard's theorem. They also prove the generalization of Quillen's \(F\)-isomorphism theorem to these groups:
Theorem. Let \(\mathcal{G}=(S,\mathcal{F})\) be a \(p\)-local compact group, the the restriction to elementary abelian subgroups of \(S\) induces an \(F\)-isomorphism \[ H^{*}(B\mathcal{G},\mathbf{F}_p)\to \lim_{\mathcal{F}^{e}} H^{*}(BE,\mathbb{F}_p), \] where \(\mathbb{F}^{e}\) denotes the full subcategory of \(\mathbb{F}\) consisting elementary abelian \(p\)-groups.
They also prove that \(p\)-local compact groups satisfy Quillen's stratification.
Daniel Juan Pineda (Michoacan)