Characterizing finite length local cohomology in terms of bounds on Koszul cohomology

TL;DR

This paper characterizes when local cohomology modules have finite length by examining the asymptotic behavior of Koszul cohomology on parameter ideals, linking Cohen-Macaulay properties to bounds on Koszul cohomology.

Contribution

It provides a new characterization of modules with finite length local cohomology in terms of bounds on Koszul cohomology, connecting asymptotic properties to Cohen-Macaulay conditions.

Findings

Finite length local cohomology characterized by Koszul cohomology bounds.

Asymptotic Cohen-Macaulayness equivalent to Cohen-Macaulay on punctured spectrum.

Boundedness of Koszul cohomology implies Cohen-Macaulay properties.

Abstract

Let $(R,m, \kappa)$ be a local ring. We give a characterization of $R$-modules $M$ whose local cohomology is finite length up to some index in terms of asymptotic vanishing of Koszul cohomology on parameter ideals up to the same index. In particular, we show that a quasi-unmixed module $M$ is asymptotically Cohen-Macaulay if and only if $M$ is Cohen-Macaulay on the punctured spectrum if and only if $\sup\{\ell(H^i(f_1, \ldots, f_d;M))\mid \sqrt{f_1, \ldots, f_d} = m \mbox{, } i< d\}<\infty$.