The Stacks project

Lemma 35.23.19. The property $\mathcal{P}(f) =$“$f$ is a closed immersion” is fpqc local on the base.

Proof. Let $f : X \to Y$ be a morphism of schemes. Let $\{ Y_ i \to Y\} $ be an fpqc covering. Assume that each $f_ i : Y_ i \times _ Y X \to Y_ i$ is a closed immersion. This implies that each $f_ i$ is affine, see Morphisms, Lemma 29.11.9. By Lemma 35.23.18 we conclude that $f$ is affine. It remains to show that $\mathcal{O}_ Y \to f_*\mathcal{O}_ X$ is surjective. For every $y \in Y$ there exists an $i$ and a point $y_ i \in Y_ i$ mapping to $y$. By Cohomology of Schemes, Lemma 30.5.2 the sheaf $f_{i, *}(\mathcal{O}_{Y_ i \times _ Y X})$ is the pullback of $f_*\mathcal{O}_ X$. By assumption it is a quotient of $\mathcal{O}_{Y_ i}$. Hence we see that

\[ \Big( \mathcal{O}_{Y, y} \longrightarrow (f_*\mathcal{O}_ X)_ y \Big) \otimes _{\mathcal{O}_{Y, y}} \mathcal{O}_{Y_ i, y_ i} \]

is surjective. Since $\mathcal{O}_{Y_ i, y_ i}$ is faithfully flat over $\mathcal{O}_{Y, y}$ this implies the surjectivity of $\mathcal{O}_{Y, y} \longrightarrow (f_*\mathcal{O}_ X)_ y$ as desired. $\square$

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