Extended plus closure in complete local rings

TL;DR

This paper demonstrates that extended plus closure in complete local rings with F-finite residue fields possesses the colon-capturing property, leading to significant algebraic consequences in mixed characteristic settings.

Contribution

It adapts perfectoid algebra techniques to prove colon-capturing for extended plus closure in complete local rings, confirming ideal closure and supporting key conjectures.

Findings

Extended plus closure has colon-capturing property.

All ideals in regular local rings are closed.

Supports the direct summand conjecture and Briançon-Skoda theorem in mixed characteristic.

Abstract

The (full) extended plus closure was developed as a replacement for tight closure in mixed characteristic rings. Here it is shown by adapting Andr\'{e}'s perfectoid algebra techniques that, for complete local rings that have F-finite residue fields, this closure has the colon-capturing property. In fact, more generally, if $R$ is a (possibly ramified) complete regular local ring of mixed characteristic that have F-finite residue fields, $I$ and $J$ are ideals of $R$, and the local domain $S$ is a finite $R$-module, then $(IS:J)\subseteq (I:J)S^{epf}$. A consequence is that all ideals in regular local rings are closed, a fact which implies the validity of the direct summand conjecture and the Brian\c{c}on-Skoda theorem in mixed characteristic.