Associated primes of local cohomology modules and of Frobenius powers

TL;DR

This paper constructs specific hypersurfaces with local cohomology modules having infinitely many associated primes, addressing a key question in tight closure theory and Frobenius powers.

Contribution

It introduces new examples of hypersurfaces with infinitely many associated primes in local cohomology, impacting the understanding of Frobenius powers and singularities.

Findings

Existence of hypersurfaces with infinitely many associated primes

Resolution of a question in tight closure theory

Connections between local cohomology and Frobenius powers

Abstract

We construct normal hypersurfaces whose local cohomology modules have infinitely many associated primes. These include unique factorization domains of characteristic zero with rational singularities, as well as F-regular unique factorization domains of positive characteristic. As a consequence, we answer a question on the associated primes of Frobenius powers of ideals, which arose from the localization problem in tight closure theory.