Properties Of Modular Automorphisms

The concept of multiplicity of an module is an important concept,which plays an important role in commutative algebra.It can be used to classfy Noetherian local rings into various interesting types.Although it is slightly different from multiplicity of an module,the multiplicity of an endomorphism of a finitely generated module over a Noetherian local ring still has many properties similar to multiplicity of an module,such as additivity,nonnegativity and so on.The main purpose of the paper is to make a further study on its properties and obtain the following main results:(1)A computing formula for the multiplicity of an endomorphism of a finitely generated module over a Noetherian local domain;(2)Under a condition that the complex of endomorphisms is not necessarily exact,it is proved that the multiplicity of automorphisms is still additive;(3)The multiplicity of M of d-1 dimension can be expressed by multiplicity related to the Fitting ideal of M,if the d dimensional ring(R,m)is a normal local domain or the finitely generated R-module M has finite free resolution.