Weak functoriality of Cohen-Macaulay algebras

TL;DR

This paper establishes the weak functoriality of big Cohen-Macaulay algebras in mixed characteristic, advancing the understanding of homological conjectures in commutative algebra using perfectoid and tilting techniques.

Contribution

It proves the weak functoriality of Cohen-Macaulay algebras in mixed characteristic, extending known results from equal characteristic cases using perfectoid methods.

Findings

Weak functoriality of Cohen-Macaulay algebras in mixed characteristic established.

Compatible homomorphisms exist between Cohen-Macaulay algebras over related local domains.

The proof combines perfectoid techniques with a tilting argument.

Abstract

We prove the weak functoriality of (big) Cohen-Macaulay algebras, which controls the whole skein of "homological conjectures" in commutative algebra [H1][HH2]. Namely, for any local homomorphism $ R\to R'$ of complete local domains, there exists a compatible homomorphism between some Cohen-Macaulay $R$-algebra and some Cohen-Macaulay $R'$-algebra. When $R$ contains a field, this is already known [[3.9]{HH2}]. When $R$ is of mixed characteristic, our strategy of proof is reminiscent of G. Dietz's refined treatment [D] of weak functoriality of Cohen-Macaulay algebras in characteristic $p$; in fact, developing a "tilting argument" due to K. Shimomoto, we combine the perfectoid techniques of [A1][A2] with Dietz's result.