Closure operations in complete local rings of mixed characteristic

Abstract

The extended plus (epf) closure and the 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 the epf closure satisfies the usual colon-capturing property under mild conditions. In this paper, we extend their result and prove that the 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.