Existence of almost Cohen-Macaulay algebras implies the existence of big Cohen-Macaulay algebras

TL;DR

This paper demonstrates that the dagger closure satisfies the algebra axiom, leading to the existence of balanced big Cohen-Macaulay algebras over certain complete Noetherian local domains contained in almost Cohen-Macaulay domains.

Contribution

It extends the properties of dagger closure to satisfy the algebra axiom, establishing the existence of big Cohen-Macaulay algebras in new cases.

Findings

Dagger closure satisfies the algebra axiom.

Existence of balanced big Cohen-Macaulay algebras over certain domains.

Connection between almost Cohen-Macaulay domains and big Cohen-Macaulay algebras.

Abstract

In \cite{AB}, the dagger closure is extended over finitely generated modules over Noetherian local domain $(R,\fm)$ and it is proved to be a Dietz closure. In this short note we show that it also satisfies the `Algebra axiom' of \cite{R.G} and this leads to the following result of this paper: For a complete Noetherian local domain, if it is contained in an almost Cohen-Macaulay domain, then there exists a balanced big Cohen-Macaulay algebra over it.