Intersection Multiplicities

Abstract

As in Chap. 6, consider a fibre square

$$ \begin{array}{*{20}{c}} W \to V \\ \downarrow {}{ \downarrow {}^f} \\ X{\mathop \to \limits_i }Y \end{array} $$

with i a regular imbedding of codimension d, V a k-dimensional variety. If Z is an irreducible component of W of dimension k — d, the intersection multiplicity i (Z, X·V; Y) is defined to be the coefficient of Z in the intersection class X·V ∈ A _k-d (W). The intersection multiplicity is a positive integer, satisfying

$$ i\left({Z,X \cdot V;Y} \right) \leqq length\left({{\upsilon _{z,w}}} \right) $$

Examples show that this inequality may be strict; equality holds, however, if ${\mathcal{O}_{z,v}}$ is a Cohen-Macaulay ring.

On the other hand, the criterion of multiplicity one asserts that i (Z, X·V;Y) is one precisely when ${\mathcal{O}_{z,v}}$ is a regular local ring with maximal ideal generated by the ideal of X in Y.

The standard properties of intersection multiplicities, worked out in the examples, follow from the basic properties of the general intersection product which were proved in Chapter 6.