On the structure of endomorphisms of projective modules
type: Item
instance of: scholarly article
title: On the structure of endomorphisms of projective modules
zbMATH Open document ID: 1386.13027
DOI: 10.4171/LEM/60-1/2-7
author: Daniel Ferrand
author: Dan Laksov
published in: L'Enseignement Mathématique. 2e Série
publication date: 2014-12-19T00:00:00Z
review text: Summary: Taking as a model the completed theory of vector space endomorphisms, the present text aims at extending this theory to endomorphisms of finitely generated projective modules over a general commutative ring; now analogous results often require totally different methods of proof. The first important result is a structure theorem for such modules when the characteristic polynomial of the endomorphism is separable. The second topic deals with the minimal polynomial, whose mere existence is shown to require additional hypotheses, even over a domain. In the third topic we extend the classical notion of 'cyclic modules' as the modules which are invertible over the ring of polynomials modulo the characteristic polynomial. { } Regarding the diagonalization of endomorphisms, we show that a classical criterion of being diagonalizable over some extension of the base field can be transferred nearly verbatim to rings, provided that diagonalization is expected only after some faithfully flat base change. Many results that hold over a field, like the fact that commuting diagonalizable endomorphisms are simultaneously diagonalizable, hold over arbitrary rings, with this extended meaning of diagonalization. The Jordan-Chevalley-Dunford decomposition, shown as a particular case of the lifting property of étale algebras, also holds over rings.{ } Finally, in several reasonable situations, the eigenspace associated with any root of the characteristic polynomial is shown to be given a more concrete description as the image of a map. In these situations the classical theory generalizes to rings.
Mathematics Subject Classification ID: 13C10
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 14A05
Mathematics Subject Classification ID: 14B25
Mathematics Subject Classification ID: 15-01
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A21
zbMATH DE Number: 6381842
zbMATH Keywords: projective modules
zbMATH Keywords: characteristic polynomials
zbMATH Keywords: eigenspaces
zbMATH Keywords: étale algebras
zbMATH Keywords: Jordan decomposition
MaRDI profile type: Publication
cites work: Q3963102
cites work: Q3843687
cites work: Endomorphisms of finitely generated projective modules over a commutative ring
cites work: Q5580320
cites work: Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas (Quatrième partie). Rédigé avec la colloboration de J. Dieudonné
cites work: Two "Generic" Proofs of the Spectral Mapping Theorem
cites work: SPLITTING ALGEBRAS, SYMMETRIC FUNCTIONS AND GALOIS THEORY
description: scientific article; zbMATH DE number 6381842