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.

中文翻译：

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Cohen-Macaulay 代数的弱函子性

我们证明了（大的）Cohen-Macaulay 代数的弱函子性，它控制着交换代数 [H1][HH2] 中“同调猜想”的整个系列。即，对于完全局部域的任何局部同态$R\toR'$，在一些Cohen-Macaulay $R$-代数和一些Cohen-Macaulay $R'$-代数之间存在兼容同态。当 $R$ 包含一个字段时，这是已知的 [[3.9]{HH2}]。当 $R$ 具有混合特征时，我们的证明策略让人想起 G. Dietz 对 Cohen-Macaulay 代数在特征 $p$ 中的弱泛函性的精细处理 [D]；事实上，由于 K. Shimomoto，我们开发了一个“倾斜论证”，我们将 [A1][A2] 的完美技术与迪茨的结果结合起来。