On the Frobenius closure of parameter ideals when the ring is F-injective on the punctured spectrum

TL;DR

This paper investigates the Frobenius closure of parameter ideals in certain local rings, establishing bounds on their length and describing their structure when contained in large powers of the maximal ideal.

Contribution

It proves an inequality relating the Frobenius closure length to local cohomology, and characterizes the Frobenius closure as a direct sum under specific conditions.

Findings

Bound on the length of Frobenius closure quotient in terms of local cohomology.

Structural isomorphism of Frobenius closure quotient to a direct sum of local cohomology modules.

Applicable to excellent generalized Cohen-Macaulay rings that are F-injective on the punctured spectrum.

Abstract

Let $(R,\frak m)$ be an excellent generalized Cohen-Macaulay local ring of dimension $d$ that is $F$-injective on the punctured spectrum. Let $\frak q$ be a standard parameter ideal of $R$. The aim of the paper is to prove that $$\ell_R({\frak q}^F/{\frak q})\leq \sum\limits_{i=0}^{d}\binom{d}{i}\ell_R(0^F_{H^i_{\frak m}(R)}).$$ Moreover, if $\frak q$ is contained in a large enough power of $\frak m$, we have $${\frak q}^F/{\frak q} \cong \bigoplus_{i=0}^d (0^F_{H^i_{\frak m}(R)})^{\binom{d}{i}}.$$