Lemma 37.22.7. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let

Then

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

$W$ is open in $X$,

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

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 $W' = (g')^{-1}(W)$.

## Comments (3)

Comment #2698 by Johan on

Comment #2736 by Takumi Murayama on

Comment #8292 by Matthieu Romagny on

There are also: