Modules of finite projective dimension with negative intersection multiplicities

Abstract

Let x~ be the image of X~ in R. The main object of this paper is to study modules of finite length and finite projective dimension over R. One consequence of our study will be counterexamples to several conjectures concerning the behavior of intersection multiplicities for modules of finite projective dimension. Let / denote length. If S is a Noetherian ring, M, N are finitely generated S-modules, l(M| ) is finite, and d=pdsM<~, we may define, following Serre IS] d Zs(M, N)= z(M, N) = ~ ( 1)'/(TorS(M, N)).