A variant of perfectoid Abhyankar's lemma and almost Cohen-Macaulay algebras

TL;DR

This paper extends perfectoid techniques to mixed characteristic, proving the existence of almost Cohen-Macaulay extensions with Frobenius surjectivity, advancing the understanding of algebraic structures in mixed characteristic settings.

Contribution

It introduces a mixed characteristic analogue of perfect closure results, establishing a new version of Abhyankar's lemma and Riemann's extension theorem for perfectoid rings.

Findings

Existence of integral extensions that are almost Cohen-Macaulay with Frobenius surjectivity.

Development of Witt-perfect decompletion of perfectoid rings.

Establishment of a mixed characteristic version of André's perfectoid Abhyankar's lemma.

Abstract

In this paper, we prove that a complete Noetherian local domain of mixed characteristic $p>0$ with perfect residue field has an integral extension that is an integrally closed, almost Cohen-Macaulay domain such that the Frobenius map is surjective modulo $p$. This result is seen as a mixed characteristic analogue of the fact that the perfect closure of a complete local domain in positive characteristic is almost Cohen-Macaulay. To this aim, we carry out a detailed study of decompletion of perfectoid rings and establish the Witt-perfect (decompleted) version of Andr\'e's perfectoid Abhyankar's lemma and Riemann's extension theorem.