# The local cohomology of a parameter ideal with respect to an arbitrary ideal

## TL;DR

This paper investigates the associated primes of local cohomology modules related to parameter ideals, providing counterexamples, establishing finiteness conditions, and generalizing key isomorphisms, with implications for regular rings.

## Contribution

It offers a negative answer to a longstanding question, establishes a new criterion linking associated primes of different local cohomology modules, and generalizes an important isomorphism in the field.

## Key findings

- Existence of modules with infinitely many associated primes
- Finiteness of associated primes depends on hypotheses on R/J
- Regularity of R/J in prime characteristic ensures finiteness

## Abstract

Let $R$ be a regular ring, let $J$ be an ideal generated by a regular sequence of codimension at least $2$, and let $I$ be an ideal containing $J$. We give an example of a module $H^3_I(J)$ with infinitely many associated primes, answering a question of Hochster and N\'u\~nez-Betancourt in the negative. In fact, for $i\leq 4$, we show that under suitable hypotheses on $R/J$, $\text{Ass}\,H^{i}_I(J)$ is finite if and only if $\text{Ass}\,H^{i-1}_I(R/J)$ is finite. Our proof of this statement involves a novel generalization of an isomorphism of Hellus, which may be of some independent interest. The finiteness comparison between $\text{Ass}\, H^i_I(J)$ and $\text{Ass}\, H^{i-1}_I(R/J)$ tends to improve as our hypotheses on $R/J$ become more restrictive. To illustrate the extreme end of this phenomenon, at least in the prime characteristic $p>0$ setting, we show that if $R/J$ is regular, then $\text{Ass}\, H^i_I(J)$ is finite for all $i\geq 0$.