The Stacks project

[IV Corollary 12.1.7(iii), EGA]

Lemma 37.22.7. 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. $W$ is 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 10.130.5. Part (4) follows either from Algebra, Lemma 10.130.7 or Varieties, Lemma 33.13.1. $\square$

Comments (3)

Comment #2698 by on

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

Comment #8292 by Matthieu Romagny on

In statement (3) : W is dense in every fibre of X→S

There are also:

  • 4 comment(s) on Section 37.22: Cohen-Macaulay morphisms

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