Almost Cohen–Macaulay algebras in mixed characteristic via Fontaine rings

Abstract

In the present paper, it is proved that any complete local domain of mixed characteristic has a weakly almost Cohen–Macaulay algebra $B$ in the sense that a system of parameters is a weakly almost regular sequence in $B$, which is a notion defined via a valuation. In fact, the central idea of this result originates from the main statement obtained by Heitmann to prove the Monomial Conjecture in dimension 3. 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.