The Stacks project

Lemma 76.26.6. 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. Let $Y' \to Y$ be locally of finite type. Let $f' : X' \to Y'$ be the base change of $f$. Let $x' \in |X'|$ be a point with image $x \in |X|$.

  1. If $f$ is Cohen-Macaulay at $x$, then $f' : X' \to Y'$ is Cohen-Macaulay at $x'$.

  2. If $f$ is flat at $x$ and $f'$ is Cohen-Macaulay at $x'$, then $f$ is Cohen-Macaulay at $x$.

  3. If $Y' \to Y$ is flat at $f'(x')$ and $f'$ is Cohen-Macaulay at $x'$, then $f$ is Cohen-Macaulay at $x$.

Proof. Denote $y \in |Y|$ and $y' \in |Y'|$ the image of $x'$. Choose a surjective étale morphism $V \to Y$ where $V$ is a scheme. Choose a surjective étale morphism $U \to X \times _ Y V$ where $U$ is a scheme. Choose a surjectiev étale morphism $V' \to Y' \times _ Y V$ where $V'$ is a scheme. Then $U' = U \times _ V V'$ is a scheme which comes equipped with a surjective étale morphism $U' \to X'$. Choose $u' \in U'$ mapping to $x'$. Denote $u \in U$ the image of $u'$. Then the lemma follows from the lemma for $U \to V$ and its base change $U' \to V'$ and the points $u'$ and $u$ (this follows from the definitions). Thus the lemma follows from the case of schemes, see More on Morphisms, Lemma 37.22.6. $\square$

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