A VARIANT OF PERFECTOID ABHYANKAR'S LEMMA AND ALMOST COHEN-MACAULAY ALGEBRAS

In this article, 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 analog 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 Andre's perfectoid Abhyankar's lemma and Riemann's extension theorem.

Automated summary

In this article, it is shown 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 serves as a mixed characteristic analogy to the fact that the perfect closure of a complete local domain in positive characteristic is almost Cohen-Macaulay. In order to achieve this, a detailed study of decompletion of perfectoid rings is carried out, and the Witt-perfect (decompleted) version of Andre's perfectoid Abhyankar's lemma and Riemann's extension theorem is established.