A note on the vanishing of certain local cohomology modules

TL;DR

This paper explores conditions under which local cohomology modules vanish, revealing that certain classical formulas become trivial when using Matlis duals and spectral sequences, and introduces a new vanishing criterion.

Contribution

It presents a novel ring-theoretic vanishing criterion for local cohomology modules and analyzes the triviality of the formula involving cohomological dimension and module dimension.

Findings

The formula $ ext{cohdim}(m,M)= ext{dim} M$ becomes trivial with Matlis duals and spectral sequences.

A new vanishing criterion for local cohomology modules is established.

Insights into the behavior of local cohomology in equicharacteristic rings.

Abstract

For a finite module $M$ over a local, equicharacteristic ring $(R,m)$, we show that the well-known formula $\cohdim(m,M)=\dim M$ becomes trivial if ones uses Matlis duals of local cohomology modules together with spectral sequences. We also prove a new, ring-theoretic vanishing criterion for local cohomology modules.