Local cohomology under small perturbations

TL;DR

This paper investigates how local cohomology modules of a Noetherian local ring change under small perturbations of ideals, showing stability of certain properties like Cohen-Macaulayness, Buchsbaum, and Serre's conditions.

Contribution

It establishes that local cohomology modules remain isomorphic under small perturbations for generalized Cohen-Macaulay ideals, and proves stability of Buchsbaum and Serre's properties.

Findings

Local cohomology modules are invariant under small perturbations for generalized Cohen-Macaulay ideals.

Buchsbaum property is preserved under small perturbations.

Serre's property $(S_n)$ is maintained under certain conditions after perturbation.

Abstract

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $I$ an ideal of $R$. We study how local cohomology modules with support in $\mathfrak{m}$ change for small perturbations $J$ of $I$, that is, for ideals $J$ such that $I\equiv J\bmod \mathfrak{m}^N$ for large $N$, under the hypothesis that $I$ and $J$ share the same Hilbert function. As one of our main results, we show that if $R/I$ is generalized Cohen-Macaulay, then the local cohomology modules of $R/J$ are isomorphic to the corresponding local cohomology modules of $R/I$, except possibly the top one. In particular, this answers a question raised by Quy and V. D. Trung. Our approach also allows us to prove that if $R/I$ is Buchsbaum, then so is $R/J$. Finally, under some additional assumptions, we show that if $R/I$ satisfies Serre's property $(S_n)$, then so does $R/J$.