F-thresholds, tight closure, integral closure, and multiplicity bounds

TL;DR

This paper explores the use of F-thresholds to detect ideal containments and proposes a conjecture relating F-thresholds to multiplicities, proving it in specific cases within positive characteristic algebra.

Contribution

It introduces a conjecture linking F-thresholds with multiplicities for zero-dimensional ideals and proves it for monomial and homogeneous parameter ideals in Cohen-Macaulay rings.

Findings

F-thresholds can detect integral and tight closures of parameter ideals.

The conjecture relating F-thresholds to multiplicities is proven for monomial ideals.

The conjecture is also proven for homogeneous parameter ideals in Cohen-Macaulay graded rings.

Abstract

The F-threshold $c^J(\a)$ of an ideal $\a$ with respect to the ideal $J$ is a positive characteristic invariant obtained by comparing the powers of $\a$ with the Frobenius powers of $J$. We show that under mild assumptions, we can detect the containment in the integral closure or the tight closure of a parameter ideal using F-thresholds. We formulate a conjecture bounding $c^J(\a)$ in terms of the multiplicities $e(\a)$ and $e(J)$, when $\a$ and $J$ are zero-dimensional ideals, and $J$ is generated by a system of parameters. We prove the conjecture when $J$ is a monomial ideal in a polynomial ring, and also when $\a$ and $J$ are generated by homogeneous systems of parameters in a Cohen-Macaulay graded $k$-algebra.