The grade conjecture and asymptotic intersection multiplicity

Abstract:

Given a finitely generated module $M$ over a local ring $A$ of characteristic $p$ with $\operatorname {pd} M < \infty$, we study the asymptotic intersection multiplicity $\chi _\infty (M, A/\underline {x})$, where $\underline {x} = (x_1, \ldots , x_r)$ 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 $\dim \operatorname {Ext}^{d-r}(M, A) = r$, where $d = \dim A$ and $r = \dim M$. We use this to prove several results relating to the grade conjecture, which states that $\operatorname {grade} M + \dim M = \dim A$ for any module $M$ with $\operatorname {pd} M < \infty$.