The Stacks project

Lemma 37.22.4. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes. Assume that the fibres of $f$, $g$, and $g \circ f$ are locally Noetherian. Let $x \in X$ with images $y \in Y$ and $z \in Z$.

  1. If $f$ is Cohen-Macaulay at $x$ and $g$ is Cohen-Macaulay at $f(x)$, then $g \circ f$ is Cohen-Macaulay at $x$.

  2. If $f$ and $g$ are Cohen-Macaulay, then $g \circ f$ is Cohen-Macaulay.

  3. If $g \circ f$ is Cohen-Macaulay at $x$ and $f$ is flat at $x$, then $f$ is Cohen-Macaulay at $x$ and $g$ is Cohen-Macaulay at $f(x)$.

  4. If $g \circ f$ is Cohen-Macaulay and $f$ is flat, then $f$ is Cohen-Macaulay and $g$ is Cohen-Macaulay at every point in the image of $f$.

Proof. Consider the map of Noetherian local rings

\[ \mathcal{O}_{Y_ z, y} \to \mathcal{O}_{X_ z, x} \]

and observe that its fibre is

\[ \mathcal{O}_{X_ z, x}/\mathfrak m_{Y_ z, y}\mathcal{O}_{X_ z, x} = \mathcal{O}_{X_ y, x} \]

Thus the lemma this follows from Algebra, Lemma 10.163.3. $\square$

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