11 - Local Chern Characters

Summary

This chapter is devoted to developing the theory of local Chern characters of complexes. The main aim is to prove the necessary results for the applications to intersection multiplicities in Chapter 13. However, we also point out relations between local Chern characters and multiplicities of ideals and describe connections with other topics discussed in this book.

We first treat matrices with support at the maximal ideal of a local ring, where it makes sense to define Chern classes of an individual matrix and where these classes can be interpreted as numbers rather than classes in a Chow group (Roberts). In the second section we show that this definition generalizes that of the multiplicity of an m-primary ideal.

However, for the important applications of the theory it is necessary to give a more general definition of the Chern character of a complex of locally free sheaves with given support, which we do in the following section. Essentially, this is simply the alternating sum of Chern characters of the maps in the complex, but it is more convenient to carry out the construction for all the maps at once.

The remainder of this chapter and most of the following one are devoted to proving the important properties of these invariants that will be used in the applications.