Extended plus closure in complete local rings

2017/08/18 by Raymond C. Heitmann, Heitmann, Raymond, Linquan Ma +1 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1708.05761

openalex publication_date 2017/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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)⊆ (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 Briançon-Skoda theorem in mixed characteristic.

Cited by

• The Briançon-Skoda theorem for pseudo-rational and Du Bois singularities and uniformity in excellent rings

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