# Tight closure of parameter ideals in local rings $F$-rational on the punctured spectrum

## TL;DR

This paper investigates the behavior of tight closure of parameter ideals in certain local rings, showing independence of length from the choice of ideal under specific $F$-singularity conditions.

## Contribution

It demonstrates that in $F$-injective local rings that are $F$-rational on the punctured spectrum, the length of the tight closure difference is independent of the parameter ideal chosen.

## Key findings

- The length $ ext{length}_R(rak{q}^*/rak{q})$ is invariant under different parameter ideals.
- The result applies to excellent equidimensional local rings of characteristic $p>0$.
- Provides insight into the structure of tight closure in rings with specific $F$-singularities.

## Abstract

Let $(R, \mathfrak{m}, k)$ be an excellent equidimensional local ring of characteristic $p>0$. The aim of this paper is to show that $\ell_R(\mathfrak{q}^*/\mathfrak{q})$ does not depend on the choice of parameter ideal $\mathfrak{q}$ provided $R$ is an $F$-injective local ring that is $F$-rational on the punctured spectrum.