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

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 ${\ell }_R({\mathfrak{q}}^{\ast }/\mathfrak{q})$ 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.

The title is changed. Final version, to appear in J. Pure Appl. Algebra