Normality of algebras over commutative rings and the Teichmüller class. I: The absolute case
Let \(S\) be a commutative ring with a unit, \(U(S)\) the group of invertible elements of \(S\), \(Q\) a (possibly, infinite) group and \(k _ {Q} : Q \to \mathrm{Aut}(S)\) an action of \(Q\) on \(S\) (not necessarily injective) by ring automorphisms. Suppose that \(A\) is a central \(S\)-algebra, denote by Inn\((A)\) the group of its inner automorphism, and let Out\((A) = \mathrm{Aut}(A)/\mathrm{Inn}(A)\) be the group of outer automorphisms of \(A\). By a normal\(Q\)-algebra, we mean a pair \((A, \sigma )\), where \(\sigma \) is a homomorphism of \(Q\) into Out\((A)\) that lifts the action \(k _ Q\) of \(Q\) on \(A\) in the sense that the composite of \(\sigma \) with the obvious map Out\((A) \to\mathrm{Aut}(S)\) coincides with \(k _ Q\). When \(S\) is a field, \(A\) is finite-dimensional and simple, and \(Q\) is a finite subgroup of Aut\((S)\), the property of being \(Q\)-normal means that each member of \(Q\) can be extended to an automorphism of \(A\); this is the same as \(Q\)-normality in the sense of \textit{S. Eilenberg} and \textit{S. MacLane} [Trans. Am. Math. Soc. 64, 1--20 (1948; Zbl 0031.34301)], since the restriction mapping Out\((A) \to \mathrm{Aut}(S)\) is injective, by the Skolem-Noether theorem. This special case has been studied by \textit{O. Teichmüller} [Deutsche Math. 5, 138--149 (1940; Zbl 0023.19805)]; following him, one can associate with any \(Q\)-normal central simple \(S\)-algebra a \(3\)-cocycle of \(Q\) with values in \(U(S)\), considering \(S\) to be a \(Q\)-module with respect to the structure coming from the action of \(Q\) on \(S\). The paper under review presents the first part of a research extending the results of Teichmüller, Eilenberg-MacLane, and others, to the general case of a commutative unitary ring \(S\) and an action \(k _ Q: Q \to \mathrm{Aut}(S)\) of \(Q\) on \(S\) by ring automorphisms (for the 2nd and 3rd part of this research, see [th author, J. Homotopy Relat. Struct. 13, No. 1, 71--125 (2018; Zbl 1439.13026); ibid. 13, No. 1, 127--142 (2018; Zbl 1428.13011)]). As a first step, the author associates with any \(Q\)-normal \(S\)-algebra \((A, \sigma )\) a crossed \(2\)-fold extension \(e _ {(A, \sigma )}\) (called the Teichmüller complex of \((A, \sigma )\)) starting at \(U(S)\) and ending at \(Q\); this complex, in turn, represents a class, the Teichmüller class, in the 3rd cohomology group \(H^3(Q, U(S))\) of \(Q\) with coefficients in \(U(S)\). Exploiting the description of the Teichmüller class in terms of the Teichmüller complex, he shows how the classical results related to the Teichmüller cocycle for finite-dimensional normal central simple algebras extend to general \(Q\)-normal algebras. Two of the main results of the paper give necessary and sufficient conditions for a \(Q\)-normal \(S\)-algebra \((A, \sigma )\) to have a zero Teichmüller class. When \(S/R\) is a Galois extension of commutative rings with Galois group \(Q\), where \(R = S^Q\), they show that a central \(S\)-algebra \(A\) with a \(Q\)-normal structure \(\sigma \) has a zero Teichmüller class if and only if \(A\) admits an embedding into a central \(R\)-algebra \(C\) so that (i) the centralizer of \(S\) in \(C\) equals \(A\), and (ii) each automorphism \(k _ Q(q)\) of \(S\), as \(q\) ranges over \(Q\), extends to an inner automorphism \(\alpha \) of \(C\) which (in view of (i)) maps \(A\) to itself in such a way that the class of \(\alpha \vert A\) in Out\((A)\) coincides with \(\sigma (q)\); moreover, if \(A\) is an Azumaya \(S\)-algebra, then \(C\) may be taken to be an Azumaya \(R\)-algebra. The results of the reviewed paper also concern the following major topics: normal algebras and their Teichmüller complexes; crossed products with normal algebras (two constructions and a proof of their equivalence); induced normal and equivariant structures; crossed Brauer group, generalized crossed Brauer group and Picard group; the equivariant Brauer group; the seven term exact sequence. The bibliography of the paper contains 47 items.
scientific article; zbMATH DE number 6860556
scientific article