Weak functoriality of Cohen-Macaulay algebras

Abstract:

We prove the weak functoriality of (big) Cohen-Macaulay algebras, which controls the whole skein of “homological conjectures” in commutative algebra; 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. When $R$ is of mixed characteristic, our strategy of proof is reminiscent of G. Dietz’s refined treatment 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 the author’s earlier work with Dietz’s result.