Almost Cohen-Macaulay algebras in mixed characteristic via Fontaine rings

TL;DR

This paper proves that complete local domains of mixed characteristic possess weakly almost Cohen-Macaulay algebras using Fontaine rings and Witt vectors, advancing understanding in mixed characteristic algebra and related conjectures.

Contribution

It introduces a method to construct weakly almost Cohen-Macaulay algebras in mixed characteristic via Fontaine rings, linking to the Monomial Conjecture.

Findings

Existence of weakly almost Cohen-Macaulay algebras in mixed characteristic

Construction over absolute integral closure using Fontaine rings and Witt vectors

Connection established with the Monomial Conjecture

Abstract

In the present paper, it is proved that any complete local domain of mixed characteristic has a weakly almost Cohen-Macaulay algebra in the sense that some system of parameters is a weakly almost regular sequence, which is a notion defined via a valuation. The central idea of this result originates from the main statement obtained by Heitmann to prove the Monomial Conjecture in dimension 3. In fact, A weakly almost Cohen-Macaulay algebra is constructed over the absolute integral closure of a complete local domain by applying the methods of Fontaine rings and Witt vectors. A connection of the main theorem with the Monomial Conjecture is also discussed.