Torsion in the Picard group and extension of scalars
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.
scientific article; zbMATH DE number 4069031