Multiplicity for ideals of maximal analytic spread and intersection theory

Abstract

We define a multiplicity y ( I , A) for an ideal I in a local ring (A, m) that is not necessarily m-primary, b u t whose analytic spread is equal to the dimension of A . If the ideal I is m-primary, p ( I , A ) is the usual Samuel m ultiplicity. In the case w hen / and A arise from an in a projective space, it(1, A) coincides with the number of Stiickrad and Vogel for a K-rational component (maybe embedded) of (see [SV 1 ]). O ur multiplicity can be computed by using certain filter-regular sequences of elements in /, which generate a minimal reduction of I in the sense of Northcott and Rees [NR] and do always exist if A/m is infinite. We will call such sequences super-reductions a n d w e p rove tha t /2(1, A ) i s th e length o f a n ideal that is constructed by an intersection algorithm (local counterpart of the StiickradVogel algorithm) from a super-reduction. In order to prove this we m ust also consider algorithms in th e associated graded ring o f A with respect t o / and study their relationship to the algorithms in A . H ence our investigations about various algorithms reflect van Gastel's result that the Stiickrad-Vogel cycle is invariant under the deformation to the normal cone (see [vG]). O ur approach to num bers by local algebra leads to a better understanding o f th e contribution o f non-isolated com ponents to theory (see [V ], c h . 3 ) o f algebraic and , w e hope , a lso o f complex analytic varieties. F urtherm ore , the algorithmic aspect of Stiickrad and Vogel's approach, transferred to the local case with precise meaning o f generic, is now more useful for concrete calculations. The paper is divided into three p a r ts . In th e first section we fix notation, which will be used throughout the paper, we define a multiplicity for ideals of