This chapter is devoted to several applications of the theory of local Chern characters to local algebra. In the first section we show how to extend the definition of intersection multiplicities in the nonregular case. The second section consists of two examples of negative intersection multiplicities, and in the third we discuss their implications to the theory of local Chern characters. In the fourth section we prove the Peskine-Szpiro intersection theorem in mixed characteristic.

Intersection Multiplicities

As mentioned in the introduction, one of the motivations behind much of the theory discussed in this book is the problem of defining intersection multiplicities of two subschemes that meet at a point. While there is still not a complete solution to this question, the use of local Chern characters makes it possible to extend the definitions to more general situations. Essentially, it allows a definition of intersection multiplicities for two cycles in the Chow group when both cycles are associated to the homology of a complex of finite length of locally free sheaves. We limit ourselves here to the case of bounded complexes of free modules over a local ring.

Let F• be a bounded free complex over a local integral domain A of dimension d, and suppose that the support of F• has dimension at most k.