On the cohomological dimension of Artin-Schreier structures
An Artin-Schreier structure \((G,G',X)\) is a profinite group \(G\) acting on a Boolean space \(X\) with an open subgroup \(G'\) of index 2 acting freely on \(X\) such that the stabilizer \(G_x\) of every \(x\in X\) is of order 2 and the map \(X\to G\) sending \(x\) to the generator of \(G_x\) is continuous. The cohomology theory of Artin-Schreier structures was developed by the author [in J. Pure Appl. Algebra 69, 141-160 (1990; Zbl 0727.12010)]. In the present paper the author proceeds to study cohomologies of Artin-Schreier structures. The main result of the paper is Theorem B. If \(X\) is the set of involutions of \(G\) then \(cd_p(G,G',X)=cd_pG'\). As a consequence the author obtains: Theorem A. A pro-2-group \(G\) is a free pro-2-product of copies of \(\mathbb{Z}/2\mathbb{Z}\) with a free pro-2-group if and only if either \(G\) itself is free or \(G\) is a semidirect product of a free pro-2-group \(G'\) and a cyclic group of order 2 which coincides with its centralizer in \(G\). Note that the problem of a description of finite extensions of free groups in the category of pro-\(p\)-groups is open as well as closely related to the problem stated as follows: Problem (12.70 Kourovka Notebook). Let \(\alpha\) be an automorphism of a free pro-\(p\)-group of finite rank and \(\alpha^{p^n}=1\) for some natural \(n\). Is the subgroup of fixed points \(\text{Fix}_F(\alpha)\) finitely generated as pro-\(p\)-group?
scientific article; zbMATH DE number 495546