[IV Corollary 12.1.7(iii), EGA]

Lemma 37.21.7. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let

$W = \{ x \in X \mid f\text{ is Cohen-Macaulay at }x\}$

Then

1. $W = \{ x \in X \mid \mathcal{O}_{X_{f(x)}, x}\text{ is Cohen-Macaulay}\}$,

2. $W$ is open in $X$,

3. $W$ dense in every fibre of $X \to S$,

4. the formation of $W$ commutes with arbitrary base change of $f$: For any morphism $g : S' \to S$, consider the base change $f' : X' \to S'$ of $f$ and the projection $g' : X' \to X$. Then the corresponding set $W'$ for the morphism $f'$ is equal to $W' = (g')^{-1}(W)$.

Proof. As $f$ is flat with locally Noetherian fibres the equality in (1) holds by definition. Parts (2) and (3) follow from Algebra, Lemma 10.130.6. Part (4) follows either from Algebra, Lemma 10.130.8 or Varieties, Lemma 33.13.1. $\square$

Comment #2698 by on

A reference is EGA IV_3, Corollary 12.1.7 (iii).

Comment #7372 by Torsten Wedhorn on

A rather trivial observation: The flatness hypothesis on $f$ seems to be superfluous except for (1) because of openness of flatness. So I think, one could slightly strengthen the result by supposing only that $f$ is locally of finite presentation and add in (1) the condition that $f$ is flat in $x$ in the description of $W$.

Comment #7375 by Torsten Wedhorn on

A rather trivial observation: The flatness hypothesis on $f$ seems to be superfluous except for (1) because of openness of flatness. So I think, one could slightly strengthen the result by supposing only that $f$ is locally of finite presentation and add in (1) the condition that $f$ is flat in $x$ in the description of $W$.

Sorry, I just realized that the flatness is of course also needed for (3). So please forget my comment.

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