Abstract

In this series of lectures I will give an introduction to semidualizing modules over commutative noetherian rings. I will present aspects of the theory behind these objects and some of their applications within the study of rings and ring homomorphisms. I will begin by motivating the definition of the semidualizing property with a survey of some aspects of homological commutative algebra. I will then discuss methods for constructing examples of semidualizing modules and prove some of their basic properties. I will continue by introducing the Auslander and Bass classes associated to a semidualizing module and discussing the properties of these classes. It is these classes which give the semidualizing modules much of their power. The Bass classes give a way to endow the set of semidualizing modules over a fixed ring with the structure of an ordered set. An analysis of the ordering yields the first application which gives a lower bound for the growth of the Bass numbers of a local Cohen-Macaulay ring. Next I will discuss the Gorenstein dimension associated to a semidualizing module and, specifically, the G-dimension of a local ring homomorphism. This will allow me to give three more applications: (1) existence of the Bass series for a local homomorphism of finite G-dimension; (2) additional structure for quasi-deformations associated to modules of finite CI-dimension; and (3) a special case of the composition question for local ring homomorphisms of finite G-dimension.



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