Noncommutative descent and non-abelian cohomology
type: Item
instance of: scholarly article
title: Noncommutative descent and non-abelian cohomology
zbMATH Open document ID: 0884.18015
DOI: 10.1023/A:1007734431772
author: Philippe Nuss
published in: \(K\)-Theory
publication date: 1997-10-23T00:00:00Z
review text: This paper gives an elegant presentation of descent for modules over extensions of non-commutative rings. Let \(\phi:R\to S\) be a \(k\)-algebra homomorphism, \(k\) a commutative ring. Descent theory answers two questions: I. Given a right \(S\)-module \(M\), decide if there is a right \(R\)-module \(N\) so that \(M\cong N\otimes S\) (\(\otimes=\otimes_R\)). II. Given an \(R\)-module \(N_0\), classify the twisted forms of \(N_0\), that is, right \(R\)-modules \(N\) so that \(N_0\otimes S\cong N\otimes S\). Question I is answered by trying to attach to \(M\) a descent datum: that is, finding an \(S\)-linear map \(f:M\to M\otimes S\) which is a section of the multiplication map \(M\otimes S\to M\) and satisfies a certain cocycle conditition. If \(\phi:R\to S\) makes \(S\) into a faithfully flat left \(R\)-module, then \textit{M. Cipolla} [Rend. Circ. Math. Palermo, II. Ser. 25, 43-46 (1976; Zbl 0391.16022)] proved that the functor \(N\mapsto N\otimes S\) yields an equivalence from the category of right \(R\)-modules to the category of right \(S\)-modules with descent data. The author elicidates this result by defining two other categories of right \(S\)-modules. To do so, the author identifies \(S\otimes S\) with \(S\oplus\Omega(S/R)\) where \(\Omega(S/R)\), the kernel of the multiplication map \(S\otimes S\to S\), is called the module of differential 1-forms of \(S\) over \(R\): if \(\Omega(S/R)\) is viewed as an ideal of square 0, then the resulting multiplication makes \(S\otimes S\) into a \(k\)-algebra. An \(R\)-linear map \(\Delta\) from \(M\) to \(M\otimes_S\Omega(S/R)\) is called a connection; \(\Delta\) is flat if the induced ``curvature'' map \(\Delta^2\) from \(M\) to \(M\otimes_S\Omega(S/R)\otimes_S\Omega(S/R)\) is zero. The collection of right \(S\)-modules with flat connections forms a category. The author then defines two different \(S\otimes S\)-module structures on \(M\otimes S\), subscripted \(_1\) and \(_2\). A twist is an \(S\otimes S\)-module isom…
Mathematics Subject Classification ID: 18G50
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16D10
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 46L87
zbMATH DE Number: 1059651
zbMATH Keywords: descent
zbMATH Keywords: descent data
zbMATH Keywords: Hopf Galois extension
zbMATH Keywords: Amitsur cohomology
zbMATH Keywords: connection
zbMATH Keywords: curvature
zbMATH Keywords: symmetry
zbMATH Keywords: Yang-Baxter equation
zbMATH Keywords: braid equation
zbMATH Keywords: Hermitian module
zbMATH Keywords: twisted form
reviewed by: Lindsay N. Childs
MaRDI profile type: Publication
full work available at URL: a:1007734431772
OpenAlex ID: W1980248199
description: scientific article; zbMATH DE number 1059651