Local cohomology of ideals and the $R_n$ condition of Serre

TL;DR

This paper investigates the local cohomology of ideals in regular rings of characteristic zero, establishing finiteness properties of associated primes under certain Serre's condition and regularity assumptions.

Contribution

It proves finiteness of associated primes of local cohomology modules when the quotient satisfies Serre's $R_i$ condition or is regular at non-maximal primes.

Findings

Finiteness of $Ass^{g+i+1}H^{g+1}_P(R)$ under $R_i$ condition.

Finiteness of associated primes of local cohomology modules when $R/P$ is regular at non-maximal primes.

Abstract

Let $R$ be a regular ring of dimension $d$ containing a field $K$ of characteristic zero. If $E$ is an $R$-module let $Ass^i E = \{ Q \in \ Ass E \mid \ height Q = i \}$. Let $P$ be a prime ideal in $R$ of height $g$. We show that if $R/P$ satisfies Serre's condition $R_i$ then $Ass^{g+i+1}H^{g+1}_P(R)$ is a finite set. As an application of our techniques we prove that if $P$ is a prime ideal in $R$ such that $(R/P)_\mathfrak{q}$ is regular for any non-maximal prime ideal $\mathfrak{q}$ then $H^i_P(R)$ has finitely many associate primes for all $i$.