Normality of algebras over commutative rings and the Teichmüller class. I: The absolute case
type: Item
instance of: scholarly article
title: Normality of algebras over commutative rings and the Teichmüller class. I: The absolute case
zbMATH Open document ID: 1439.13025
author: J. Huebschmann
published in: Journal of Homotopy and Related Structures
publication date: 2018-04-16T00:00:00Z
review text: 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.…
reviewed by: Ivan D. Chipchakov
Mathematics Subject Classification ID: 13B05
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16K50
Mathematics Subject Classification ID: 16S35
Mathematics Subject Classification ID: 20J06
zbMATH DE Number: 6860556
zbMATH Keywords: Teichmüller cocycle
zbMATH Keywords: crossed module
zbMATH Keywords: crossed pair
zbMATH Keywords: normal algebra
zbMATH Keywords: crossed product
zbMATH Keywords: Deuring embedding problem
zbMATH Keywords: Group cohomology
zbMATH Keywords: Galois theory of commutative rings
zbMATH Keywords: Azumaya algebra
zbMATH Keywords: Brauer group
zbMATH Keywords: Galois cohomology
zbMATH Keywords: non-commutative Galois theory
zbMATH Keywords: non-abelian cohomology
MaRDI profile type: Publication
full work available at URL: s40062-017-0173-3
OpenAlex ID: W2999097549
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DOI: 10.1007/S40062-017-0173-3
description: scientific article; zbMATH DE number 6860556