Finiteness properties of local cohomology for F-pure local rings

TL;DR

This paper proves that in F-pure local rings, local cohomology modules have finitely many Frobenius compatible submodules, answering an open question and establishing finiteness properties under certain conditions.

Contribution

It establishes finiteness properties of local cohomology modules in F-pure local rings, including the finiteness of Frobenius compatible submodules and finite length in specific categories.

Findings

Frobenius compatible submodules are finite in F-pure local rings.

Local cohomology modules have finite length under certain conditions.

Finiteness properties are preserved under localization.

Abstract

In this paper, we show that for an $F$-pure local ring $(R,\m)$, all local cohomology modules $H_{\m}^i(R)$ have finitely many Frobenius compatible submodules. This answers positively an open question raised by F.Enescu and M.Hochster. We also prove that if $(R,\m)$ is excellent and is $F$-pure on the punctured spectrum, then all local cohomology modules have finite length in the category of $R$-modules with Frobenius action. Finally, we show that the property that all $H_{\m}^(R)$ have finitely many Frobenius compatible submodules passes to localizations.