The study of multiplicities has been of major importance in commutative algebra since the beginnings of the subject. It developed from the notion of the multiplicity of a root of a polynomial and from counting multiplicities of intersections in algebraic geometry, and it has influenced the development of the subject in many ways. On the other hand, the notion of Chern classes was originally developed in topology, and it is comparatively recently that it has been used in algebraic geometry and even more recently that it has made its way into commutative algebra. It is the aim of this book to present the theories of multiplicities and of Chern classes in an algebraic setting and to describe their relations with each other and with other topics in the field.

There are two somewhat different notions of multiplicities that will be discussed at length in this book. They both originate from algebraic geometry but from somewhat different sources. The first notion of multiplicity that we consider comes from the multiplicity of a variety at a point. This is best illustrated by an example. Suppose we have a curve in a plane that crosses itself, perhaps more than once, at a point. It then makes sense to take the multiplicity of the curve at the point to be the number of components locally at the point.