On a generalization of a result of Peskine and Szpiro

TL;DR

This paper extends a known vanishing result of local cohomology modules from positive characteristic to characteristic zero by linking Bass numbers of a specific local cohomology module to the vanishing of higher modules.

Contribution

It establishes that Bass numbers of $H^g_I(R)$ determine the vanishing of $H^i_I(R)$ for $i > g$ in characteristic zero, generalizing Peskine and Szpiro's result.

Findings

Bass numbers of $H^g_I(R)$ characterize vanishing of higher local cohomology modules.

The result bridges the gap between positive and zero characteristic cases.

Provides a criterion for vanishing based on Bass numbers.

Abstract

Let $(R,\mathfrak{m})$ be a regular local ring containing a field $K$. Let $I$ be a Cohen-Macaulay ideal of height $g$. If $\text{char } K = p > 0$ then by a result of Peskine and Szpiro the local cohomology modules $H^i_I(R)$ vanish for $i > g$. This result is not true if $\text{char } K = 0$. However we prove that the Bass numbers of the local cohomology module $H^g_I(R)$ completely determine whether $H^i_I(R)$ vanish for $i > g$.