Extended plus closure in complete local rings

The (full) extended plus closure was developed as a replacement for tight closure in mixed characteristic rings. Here it is shown by adapting Andre'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 has an F-finite residue field, I and J are ideals of R, and the local domain S is a finite R-module, then (IS : J) subset of (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 Briancon-Skoda theorem in mixed characteristic. (C) 2018 Elsevier Inc. All rights reserved.

Automated summary

This article introduces the concept of the full extended plus closure and proves that complete local rings with F-finite residue fields have the properties of this closure, further extending to regular local rings in mixed characteristic environments.