WEAK FUNCTORIALITY OF COHEN-MACAULAY ALGEBRAS

International audienceWe prove the weak functoriality of (big) Cohen-Macaulay algebras, which controls the whole skein of "homological conjectures" in commutative algebra [10][13]; namely, for any local homomorphism R → 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 [13, 3.9]. When R is of mixed characteristic , our strategy of proof is reminiscent of G. Dietz's refined treatment [7] of weak functoriality of Cohen-Macaulay algebras in characteristic p; in fact, developing a "tilt-ing argument" due to K. Shimomoto, we combine the perfectoid techniques of [1][2] with Dietz's result