Weak functoriality of Cohen-Macaulay algebras

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 → R ′ R→ R’ of complete local domains, there exists a compatible homomorphism between some Cohen-Macaulay R R -algebra and some Cohen-Macaulay R ′ R’ -algebra. When R R contains a field, this is already known. When R 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 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.