Torsion in the Picard group and extension of scalars
type: Item
instance of: scholarly article
title: Torsion in the Picard group and extension of scalars
zbMATH Open document ID: 0655.13005
DOI: 10.1016/0022-4049(88)90092-8
author: Robert Guralnick
author: Hyman Bass
published in: Journal of Pure and Applied Algebra
publication date: 1988-01-01T00:00:00Z
review text: Let Pic(R) be the group of rank one projective modules over a commutative ring R. The torsion elements I of Pic(R) are characterized by the following equivalent properties: (1) \(I\otimes_ RS\simeq S\) (as S-modules) for some commutative extension S of R, which is a finite rank projective R-module. (2) nI\(\simeq nR\) for some \(n>0.\) Such a result is used, in particular, to prove that, for M, N two finitely generated R-modules, with \(End_ R(M)\simeq R\), the following facts are equivalent: (1) nM\(\simeq nN\) for some \(n>0.\) (2) There exist a commutative extension S of R, which is a finite rank projective R-module, such that: \(M\otimes_ RS\simeq N\otimes_ RS.\) Another application is made to investigate the automorphisms of Azumaya algebras.
Mathematics Subject Classification ID: 13C10
Mathematics Subject Classification ID: 13D15
Mathematics Subject Classification ID: 16H05
zbMATH DE Number: 4069031
zbMATH Keywords: Pic
zbMATH Keywords: torsion elements
zbMATH Keywords: automorphisms of Azumaya algebras
Wikidata QID: Q105596764
reviewed by: Carla Massaza
MaRDI profile type: Publication
full work available at URL: 0022-4049(88)90092-8
OpenAlex ID: W2038748952
cites work: Q5562621
cites work: Projective Modules with Free Multiples and Powers
cites work: Separable algebras over commutative rings
cites work: Algebras over zero-dimensional rings
cites work: The genus of a module. II: Roiter's theorem, power cancellation and extension of scalars
cites work: Algèbres d'Azumaya et modules projectifs
description: scientific article; zbMATH DE number 4069031