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
$f$ is locally of finite presentation, or
$U \subset S$ 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). $\square$
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)