Extended plus closure in complete local rings

Abstract

The (full) extended plus closure was developed as a replacement for tight closure in mixed characteristic rings. Here it is shown by adapting André'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)⊆(I:J)Sepf. 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 Briançon–Skoda theorem in mixed characteristic.