Lemma 76.26.4 (0E0X)—The Stacks project

Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.26: Cohen-Macaulay morphisms
Lemma 76.26.4 (cite)

Lemma 76.26.4. Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of algebraic spaces over $S$. Assume that the fibres of $f$, $g$, and $g \circ f$ are locally Noetherian. Let $x \in |X|$ with images $y \in |Y|$ and $z \in |Z|$.

If $f$ is Cohen-Macaulay at $x$ and $g$ is Cohen-Macaulay at $f(x)$, then $g \circ f$ is Cohen-Macaulay at $x$.
If $f$ and $g$ are Cohen-Macaulay, then $g \circ f$ is Cohen-Macaulay.
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 $f(x)$.
If $f \circ g$ 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. Working étale locally this follows from the corresponding result for schemes, see More on Morphisms, Lemma 37.22.4. Alternatively, we can use the equivalence of (a) and (b) in Lemma 76.26.3. Thus we consider the local homomorphism of Noetherian local rings

\[ \mathcal{O}_{Y, \overline{y}}/ \mathfrak m_{\overline{z}}\mathcal{O}_{Y, \overline{y}} \longrightarrow \mathcal{O}_{X, \overline{x}}/ \mathfrak m_{\overline{z}}\mathcal{O}_{X, \overline{x}} \]

whose fibre is

\[ \mathcal{O}_{X, \overline{x}}/ \mathfrak m_{\overline{y}}\mathcal{O}_{X, \overline{x}} \]

and we use Algebra, Lemma 10.163.3. $\square$

The Stacks project

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