On the structure of endomorphisms of projective modules

Abstract

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.