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

1. $W = \bigcup _{d \geq 0} U_ d$ is dense in every fibre of $f$, and

2. $U_ d \to S$ 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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