Cohomology theory of Artin-Schreier structures
Artin-Schreier structures have been defined in [\textit{D. Haran} and \textit{M. Jarden}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 12, 449--489 (1985; Zbl 0595.12013)] to solve the problem of characterization of the absolute Galois groups of pseudo real closed fields. Let \(G\) be a profinite group with a continuous left action on a Boolean space \(X\) and \(G'\) be an open subgroup of \(G\), of index \(1\) or \(2\) such that (1) \(G_x=\{\sigma \in G\mid \sigma x=x\}\) is of order \(2\) and \(G_x\cap G'=1\), for every \(x\in X\), (2) the map \(d: X\to G\), where \(d(x)\) is the generator of \(G_x\), is continuous. \(\mathbb G=<G,G',X>\) is called an Artin-Schreier structure. Let \(L\) be a Galois extension of a field \(K\) and \(\sqrt{-1}\in L\). Denote by \(X(L/K)\) the space of maximal ordered subfields of \(L\) containing \(K\). Then \(\mathbb G(L/K)=<\mathrm{Gal}(L/K), \mathrm{Gal}(L/K(\sqrt{-1})),X(L/K)>\) is an Artin-Schreier structure. If \(\tilde K\) is the separable closure of \(K\) then \(\mathbb G(K)=\mathbb G(\tilde K/K)\) is called the absolute Artin-Schreier structure of \(K\). The author generalizes the Galois group cohomology theory to the cohomology theory of Artin-Schreier structures. Let \(K\) be a field. Then the group \(\mathbb G(K)\) is real projective if and only if the cohomological dimension of the Artin-Schreier structure \(\mathbb G(K)\) is at most \(1\). The \(p\)-cohomological dimension of \(\mathbb G(K)\) is \(0, 1\) or \(2\) for every prime \(p\), and there is an algebraic criterion to compute it.
scientific article; zbMATH DE number 4200342
scientific article