Lemma 67.31.6 (0CVU)—The Stacks project

Loading web-font TeX/Math/Italic

Lemma 67.31.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ . 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. Choose a surjective étale morphism $\varphi : V \to Y$ where $V$ is a scheme. Choose a surjective étale morphism $U \to V \times _ Y X$ where $U$ is a scheme. Then it suffices to prove the lemma for $U \to V$ and the quasi-coherent $\mathcal{O}_ V$ -module $\varphi ^*\mathcal{F}$ . Hence this lemma follows from the case of schemes, see Morphisms, Lemma 29.25.11. $\square$

Comments (0)

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).

Comments (0)

Post a comment