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.
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 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.
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