The Stacks project

Definition 76.26.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume the fibres of $f$ are locally Noetherian (Divisors on Spaces, Definition 71.4.2).

  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 equivalent conditions of Morphisms of Spaces, Lemma 67.22.5 hold for the property $\mathcal{P}$ described in Lemma 76.26.1.

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

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