Lemma 37.22.9 (054T)—The Stacks project

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Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.22: Cohen-Macaulay morphisms
Lemma 37.22.9 (cite)

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Lemma 37.22.9. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. For $d \geq 0$ there exist opens $U_ d \subset X$ with the following properties

$W = \bigcup _{d \geq 0} U_ d$ is dense in every fibre of $f$, and
$U_ d \to S$ is of relative dimension $d$ (see Morphisms, Definition 29.29.1).

Proof. This follows by combining Lemma 37.22.7 with Morphisms, Lemma 29.29.4. $\square$

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