On the Teichmüller cocycle map
type: Item
instance of: scholarly article
title: On the Teichmüller cocycle map
zbMATH Open document ID: 0694.13004
DOI: 10.1016/0021-8693(89)90143-9
author: K.-H. Ulbrich
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: Let G be a finite group acting as automorphisms of a commutative ring S, and let R be the subring of elements fixed by G. L. N. Childs defined a group \(H^ 3_{0,1}(G)\) and investigated a sequence of groups: \[ 1\to Br(S/R)\to Br(R)\to H^ 0(G,Br(S))\to H^ 3_{0,1}(G), \] where Br(S) denotes the Brauer group of Azumaya S-algebras and Br(S/R) is the kernel of the group homomorphism of Br(R) into Br(S) induced by extension of the ring of scalars [see \textit{L. N. Childs}, J. Algebra 23, 1-17 (1972; Zbl 0243.16003)]. If S is a Galois extension of R, then the sequence is exact, \(H^ 0(G,Br(S))\) is isomorphic to the Amitsur cohomology group \(H^ 0(S/R,Br)\), and \(H^ 3_{0,1}(G)\) is isomorphic to the group \(H^ 3(J)\) defined by \textit{O. E. Villamayor} and \textit{D. Zelinsky} [J. Pure Appl. Algebra 10, 19-55 (1977; Zbl 0383.13002)]. The author gives another description of \(H^ 3_{0,1}(G)\) and an explicit definition of a homomorphism t of \(H^ 0(G,Br(S))\) into \(H^ 3_{0,1}(G)\). When S is a Galois extension of R, the above sequence is exact for this \(\hom omorphism\quad t;\) and the author conjectures that t coincides with the homomorphism of \(H^ 0(S/R,Br)\) into \(H^ 3(J)\) described by Villamayor and Zelinsky.
Mathematics Subject Classification ID: 14F22
Mathematics Subject Classification ID: 13B05
Mathematics Subject Classification ID: 20J06
zbMATH DE Number: 4136084
zbMATH Keywords: Teichmüller cocycle map
zbMATH Keywords: Brauer group
zbMATH Keywords: Galois extension
reviewed by: H. F. Kreimer
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90143-9
OpenAlex ID: W2085376219
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description: scientific article; zbMATH DE number 4136084