Definition 37.20.1. Let $f : X \to Y$ be a morphism of schemes. Assume that all the fibres $X_ y$ are locally Noetherian schemes.

1. Let $x \in X$, and $y = f(x)$. We say that $f$ is Cohen-Macaulay at $x$ if $f$ is flat at $x$, and the local ring of the scheme $X_ y$ at $x$ is Cohen-Macaulay.

2. We say $f$ is a Cohen-Macaulay morphism if $f$ is Cohen-Macaulay at every point of $X$.

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