Cycles and Commutative Algebra

The aim of this article is to describe several applications of the theory of cycles to problems in Commutative Algebra. The main topic is the use of the theory of local Chern characters defined in the Chow group of a ring to answer some questions on modules of finite homological dimension and to clarify others. In the first section, we describe the origins of these problems in Intersection Theory, and in particular Serre’s definition of intersection multiplicities and the conjectures which arose from this definition. We next give an overview of the main facts on projective dimension and related areas of Commutative Algebra. We then present an outline of the main properties of the Chow group and local Chern characters, together with their relation to modules of finite projective dimension. Finally, in the last two sections we describe applications of the theory and present the current state of research in this area. This paper is based on talks given at the conference on cycles in Morelia, Mexico in June 2003.