The Grade Conjecture and Asymptotic Intersection Multiplicity

Abstract

Given a finitely generated module $M$ over a local ring $A$ of characteristic $p$ with $\pd M < \infty$, we study the asymptotic intersection multiplicity ${\chi }_{\infty }(M,A/\underline{x})$, where $\underline{x}=(x_1,\dots ,x_r)$ is a system of parameters for $M$. We show that there exists a system of parameters such that ${\chi }_{\infty }$ is positive if and only if $\dim \Ext^{d-r}(M, A) = r$, where $d=\dim A$ and $r=\dim M$. We use this to prove several results relating to the Grade Conjecture, which states that $\grade M + \dim M = \dim A$ for any module $M$ with $\pd M < \infty$.