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

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 045U. Beware of the difference between the letter 'O' and the digit '0'.