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arXiv · 2010.12564

Closure operations in complete local rings of mixed characteristic

Zhan Jiang
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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 $p$.

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Zhan Jiang. 2020-10-23. Closure operations in complete local rings of mixed characteristic. https://doi.org/10.1016/j.jalgebra.2021.03.038

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