The Stacks project

Lemma 35.20.20. The property $\mathcal{P}(f) =$“$f$ is quasi-affine” is fpqc local on the base.

Proof. Let $f : X \to Y$ be a morphism of schemes. Let $\{ g_ i : Y_ i \to Y\} $ be an fpqc covering. Assume that each $f_ i : Y_ i \times _ Y X \to Y_ i$ is quasi-affine. This implies that each $f_ i$ is quasi-compact and separated. By Lemmas 35.20.1 and 35.20.6 this implies that $f$ is quasi-compact and separated. Consider the sheaf of $\mathcal{O}_ Y$-algebras $\mathcal{A} = f_*\mathcal{O}_ X$. By Schemes, Lemma 26.24.1 it is a quasi-coherent $\mathcal{O}_ Y$-algebra. Consider the canonical morphism

\[ j : X \longrightarrow \underline{\mathop{\mathrm{Spec}}}_ Y(\mathcal{A}) \]

see Constructions, Lemma 27.4.7. By flat base change (see for example Cohomology of Schemes, Lemma 30.5.2) we have $g_ i^*f_*\mathcal{O}_ X = f_{i, *}\mathcal{O}_{X'}$ where $g_ i : Y_ i \to Y$ are the given flat maps. Hence the base change $j_ i$ of $j$ by $g_ i$ is the canonical morphism of Constructions, Lemma 27.4.7 for the morphism $f_ i$. By assumption and Morphisms, Lemma 29.13.3 all of these morphisms $j_ i$ are quasi-compact open immersions. Hence, by Lemmas 35.20.1 and 35.20.16 we see that $j$ is a quasi-compact open immersion. Hence by Morphisms, Lemma 29.13.3 again we conclude that $f$ is quasi-affine. $\square$


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