Lemma 29.28.6 (02G0)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.28: Morphisms and dimensions of fibres
Lemma 29.28.6 (cite)

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Lemma 29.28.6. Let $f : X \to S$ be a morphism of schemes. Let $n \geq 0$ . Assume $f$ is locally of finite presentation. The open

$U_ n = \{ x \in X \mid \dim _ x X_{f(x)} \leq n\}$

of Lemma 29.28.4 is retrocompact in $X$ . (See Topology, Definition 5.12.1.)

Proof. The topological space $X$ has a basis for its topology consisting of affine opens $U \subset X$ such that the induced morphism $f|_ U : U \to S$ factors through an affine open $V \subset S$ . Hence it is enough to show that $U \cap U_ n$ is quasi-compact for such a $U$ . Note that $U_ n \cap U$ is the same as the open $\{ x \in U \mid \dim _ x U_{f(x)} \leq n\}$ . This reduces us to the case where $X$ and $S$ are affine. In this case the lemma follows from Algebra, Lemma 10.125.8 (and Lemma 29.21.2). $\square$

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