Closure operations in complete local rings of mixed characteristic

TL;DR

This paper advances the understanding of closure operations in mixed characteristic rings by extending colon-capturing properties, introducing a new closure concept, and providing new proofs for the existence of big Cohen-Macaulay algebras.

Contribution

It introduces the weak epf closure, proves its colon-capturing properties, and offers new proofs for big Cohen-Macaulay algebra existence in mixed characteristic.

Findings

epf closure satisfies p-colon-capturing property

weak epf closure satisfies generalized colon-capturing

any module-finite extension is epf-phantom

Abstract

Extended plus (epf) closure and rank 1 (r1f) closure are two closure operations introduced by Raymond C. Heitmann for rings of mixed characteristic. Recently, he and Linquan Ma proved that epf closure satisfies the usual colon-capturing property under mild conditions. In this paper, we extend their result and prove that epf closure satisfies what we call the $p$-colon-capturing property. Based on that, we define a new closure notion called "weak epf closure", and prove that it satisfies the generalized colon-capturing property and some other colon-capturing properties. This gives a new proof of the existence of big Cohen-Macaulay algebras in the mixed characteristic case. We also show that any module-finite extension of a complete local domain is epf-phantom, which generalizes a result of Mel Hochster and Craig Huneke about "phantom extensions". Finally, we prove some related results in characteristic $p$.