Normality of algebras over commutative rings and the Teichmüller class. II: Crossed pairs and the relative case
type: Item
instance of: scholarly article
title: Normality of algebras over commutative rings and the Teichmüller class. II: Crossed pairs and the relative case
zbMATH Open document ID: 1439.13026
author: J. Huebschmann
published in: Journal of Homotopy and Related Structures
publication date: 2018-04-16T00:00:00Z
review text: The paper under review contains the second part of a research presented in three papers (whose sections share a common numeration), which is devoted to the study of the generalized Teichmüller cocycle map associated with normal algebras over commutative unitary rings (for the other two parts, see [the author, J. Homotopy Relat. Struct. 13, No. 1, 1--70 (2018; Zbl 1439.13025); ibid. 13, No. 1, 127--142 (2018; Zbl 1428.13011)]). Let \(S\) be a commutative ring with a unit, \(U(S)\) the group of invertible elements of \(S\), \(Q\) a group acting on \(S\) by ring automorphisms via a homomorphism \(k_ {Q} : Q \to\mathrm{Aut}(S)\), and let \(R\) denote the subring of those elements of \(S\) which are fixed under \(Q\). Recall that a normal \(Q\)-algebra means a pair \((A, \sigma )\), where \(A\) is a central \(S\)-algebra and \(\sigma \) is a homomorphism of \(Q\) into the group Out\((A)\) of outer automorphisms of \(A\) that lifts on \(A\) the action of \(Q\) on \(S\) so that the composite of \(\sigma \) with the obvious map Out\((A) \to\mathrm{Aut}(S)\) coincides with \(k_ Q\). The action of \(Q\) on \(S\) induces on \(U(S)\) a structure of a\(Q\)-module, which allows one to associate with \((A, \sigma )\) its Teichmüller complex, a certain crossed \(2\)-fold extension \(e(A, \sigma )\) starting at \(U(S)\) and ending at \(Q\). This crossed \(2\)-fold extension represents a the Teichmüller class of \((A, \sigma )\) in the third cohomology group \(H ^ 3(Q, U(S))\). In the paper under review, the author introduces the concept of a \(Q\)-normal Galois extension of commutative rings, and associates to such an extension \(T \vert S\) a structure extension \(e_ {(T \vert S)}: N \rightarrowtail G \twoheadrightarrow Q\) of \(Q\) by the Galois group \(N =\mathrm{Aut}(T \vert S)\) and an action of \(G\) on \(T\) by ring automorphisms. Using this, he develops the relative theory of the generalized Teichmüller cocycle map. One of the basic constructions of the theory is the \(Q\)-n…
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: 6860557
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: s40062-017-0174-2
OpenAlex ID: W3168342231
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DOI: 10.1007/S40062-017-0174-2
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description: scientific article; zbMATH DE number 6860557