The Grade Conjecture and Asymptotic Intersection Multiplicity

TL;DR

This paper investigates the asymptotic intersection multiplicity of modules over local rings in characteristic p, establishing conditions linked to the Grade Conjecture and providing new insights into the relationship between module properties and intersection theory.

Contribution

It proves a criterion for positivity of asymptotic intersection multiplicity and advances the understanding of the Grade Conjecture in commutative algebra.

Findings

Existence of parameter systems where intersection multiplicity positivity characterizes Ext modules.

New conditions relating the Grade Conjecture to asymptotic intersection multiplicity.

Results supporting the validity of the Grade Conjecture for modules with finite projective dimension.

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 $\underline{x} = (x_1, \ldots, x_r)$ 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 $\dim \Ext^{d-r}(M, A) = r$, 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 $\pd M < \infty$.