The Stacks project

Lemma 38.11.3. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite type. Let $U \subset S$ be a dense open such that $X_ U \to U$ has relative dimension $\leq e$, see Morphisms, Definition 29.29.1. If also either

  1. $f$ is locally of finite presentation, or

  2. $U \subset S$ is retrocompact,

then $f$ has relative dimension $\leq e$.

Proof. Proof in case (1). Let $W \subset X$ be the open subscheme constructed and studied in More on Morphisms, Lemmas 37.22.7 and 37.22.9. Note that every generic point of every fibre is contained in $W$, hence it suffices to prove the result for $W$. Since $W = \bigcup _{d \geq 0} U_ d$, it suffices to prove that $U_ d = \emptyset $ for $d > e$. Since $f$ is flat and locally of finite presentation it is open hence $f(U_ d)$ is open (Morphisms, Lemma 29.25.10). Thus if $U_ d$ is not empty, then $f(U_ d) \cap U \not= \emptyset $ as desired.

Proof in case (2). We may replace $S$ by its reduction. Then $U$ is scheme theoretically dense. Hence $f$ is locally of finite presentation by Lemma 38.11.2. In this way we reduce to case (1). $\square$

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