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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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