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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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