Tight closure with respect to a multiplicatively closed subset of an $F$-pure local ring

TL;DR

This paper explores the structure of certain radical ideals in $F$-pure local rings related to tight closure and test ideals, revealing their interrelations and properties, especially in complete rings.

Contribution

It characterizes ideals in the set ${ m extbf{ extit I}}$ as $S'$-test ideals for some multiplicative set $S'$, and shows their closure properties under test ideals in complete rings.

Findings

Ideals in ${ m extbf{ extit I}}$ correspond to $S'$-test ideals.

In complete rings, ${ m extbf{ extit I}}$ is closed under taking test ideals.

$F$-purity of $R/C$ is preserved for $C$ in ${ m extbf{ extit I}}$.

Abstract

Let $R$ be a (commutative Noetherian) local ring of prime characteristic that is $F$-pure. This paper studies a certain finite set ${\mathcal I}$ of radical ideals of $R$ that is naturally defined by the injective envelope of the simple $R$-module. This set ${\mathcal I}$ contains $0$ and $R$, and is closed under taking primary components. For a multiplicatively closed subset $S$ of $R$, the concept of tight closure with respect to $S$, or $S$-tight closure, is discussed, together with associated concepts of $S$-test element and $S$-test ideal. It is shown that an ideal of $R$ belongs to ${\mathcal I}$ if and only if it is the $S'$-test ideal of $R$ for some multiplicatively closed subset $S'$ of $R$. When $R$ is complete, ${\mathcal I}$ is also `closed under taking test ideals', in the following sense: for each proper ideal $C$ in ${\mathcal I}$, it turns out that $R/C$ is again $F$-pure, and if $J$ and $K$ are the unique ideals of $R$ that contain $C$ and are such that $J/C$ is the (tight closure) test ideal of $R/C$ and $K/C$ is the big test ideal of $R/C$, then both $J$ and $K$ belong to ${\mathcal I}$. The paper ends with several examples.