Normality of algebras over commutative rings and the Teichmüller class. III: Examples
type: Item
instance of: scholarly article
title: Normality of algebras over commutative rings and the Teichmüller class. III: Examples
zbMATH Open document ID: 1428.13011
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 concluding 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. It describes various non-trivial examples that illustrate the approach to the ``Teichmüller cocycle map'' developed in the former two parts (see [the author, J. Homotopy Relat. Struct. 13, No. 1, 71--125 (2018; Zbl 1439.13026); ibid. 13, No. 1, 1--70 (2018; Zbl 1439.13025)]) in terms of crossed \(2\)-fold extensions. First, the author recalls the classical situation for a number field \(K\) with a finite group \(Q\) of operators. In this case \(K/K _ 0\) is a Galois extension with a Galois group \(\mathcal{G}(K/K _ 0) = Q\), where \(K _ 0\) is the fixed field of \(Q\); moreover, the Teichmüller \(3\) class is nontrivial if and only if the global degree of \(K/K _ 0\) is different from the least common multiple of its the local degrees. The author shows how this extends to the setting \(S/R\), where \(S\) and \(R\) are the rings of algebraic integers in \(K\) and \(K _ 0\), respectively. The second class of examples is based on Grothendieck's identification of the Brauer group Br\((S)\) of the algebra \(S\) of continuous complex-valued functions of a finite CW-complex \(X\) with the torsion part of the third integral cohomology group \(H ^ 3(X)\) (see Theorem~1.6, due to [\textit{A. Grothendieck}, in: Dix Exposés Cohomologie Schémas, Advanced Studies Pure Math. 3, 46--66 (1968; Zbl 0193.21503)]). The third class of examples arises by interpreting various group \(3\)-cocycles constructed in the theory of \(C ^{\ast }\)-algebras as variants of the Teichmüller \(3\)-cocycle.
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: 6860558
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
zbMATH Keywords: number fields
zbMATH Keywords: rings of integers
zbMATH Keywords: Brauer group of a topological space
zbMATH Keywords: metacyclic group
zbMATH Keywords: \({\mathrm {C}}^*\)-algebra
MaRDI profile type: Publication
full work available at URL: s40062-017-0175-1
OpenAlex ID: W2738717290
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DOI: 10.1007/S40062-017-0175-1
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