Lemma 29.25.11 (0CVT)—The Stacks project

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Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.25: Flat morphisms
Lemma 29.25.11 (cite)

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Lemma 29.25.11. Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$ -module. Assume $f$ locally finite presentation, $\mathcal{F}$ of finite type, $X = \text{Supp}(\mathcal{F})$ , and $\mathcal{F}$ flat over $Y$ . Then $f$ is universally open.

Proof. By Lemmas 29.25.7, 29.21.4, and 29.5.3 the assumptions are preserved under base change. By Lemma 29.23.2 it suffices to show that generalizations lift along $f$ . This follows from Algebra, Lemma 10.41.12. $\square$

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