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

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 algebra such that the Frobenius map is surjective modulo p. This is seen as a mixed characteristic analogue of the classical result of Hochster-Huneke for the absolute integral closure in characteristic p > 0. To achieve this goal, we establish the Witt-perfect version of André's perfectoid Abhyankar's lemma and Riemann's extension theorem. The advantage of Witt-perfet rings is that they are not assumed to be p-adically complete.