On the associated primes of local cohomology

TL;DR

This paper proves finiteness of associated primes of local cohomology modules in certain rings of positive characteristic, extending previous results and providing criteria related to singularities for finiteness in all characteristics.

Contribution

It offers a new, concise proof for finiteness of associated primes in rings with finite $F$-representation type or finite singular locus, and establishes criteria linking singularities to associated prime finiteness.

Findings

Finiteness of associated primes for local cohomology in rings with finite $F$-representation type.

Extension of previous results by Takagi-Takahashi.

Criteria relating singularities to finiteness of associated primes in all characteristics.

Abstract

Let $R$ be a commutative Noetherian ring of prime characteristic $p$. In this paper we give a short proof using filter regular sequences that the set of associated prime ideals of $H^t_I(R)$ is finite for any ideal $I$ and for any $t \ge 0$ when $R$ has finite $F$-representation type or finite singular locus. This extends a previous result by Takagi-Takahashi and gives affirmative answers for a problem of Huneke in many new classes of rings in positive characteristic. We also give a criterion about the singularities of $R$ (in any characteristic) to guarantee that the set of associated primes of $H^2_I(R)$ is always finite.