The Stacks project

Lemma 73.11.20. The property $\mathcal{P}(f) =$“$f$ is quasi-affine” is fpqc local on the base.

Proof. We will use Lemma 73.10.4 to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 66.21.3. Let $Z' \to Z$ be a surjective flat morphism of affine schemes over $S$. Let $f : X \to Z$ be a morphism of algebraic spaces, and assume that the base change $f' : Z' \times _ Z X \to Z'$ is quasi-affine. Let $X'$ be a scheme representing $Z' \times _ Z X$. We obtain a canonical isomorphism

\[ \varphi : X' \times _ Z Z' \longrightarrow Z' \times _ Z X' \]

since both schemes represent the algebraic space $Z' \times _ Z Z' \times _ Z X$. This is a descent datum for $X'/Z'/Z$, see Descent, Definition 35.34.1 (verification omitted, compare with Descent, Lemma 35.39.1). Since $X' \to Z'$ is quasi-affine this descent datum is effective, see Descent, Lemma 35.38.1. Thus there exists a scheme $Y \to Z$ over $Z$ and an isomorphism $\psi : Z' \times _ Z Y \to X'$ compatible with descent data. Of course $Y \to Z$ is quasi-affine (by construction or by Descent, Lemma 35.23.20). Note that $\mathcal{Y} = \{ Z' \times _ Z Y \to Y\} $ is a fpqc covering, and interpreting $\psi $ as an element of $X(Z' \times _ Z Y)$ we see that $\psi \in \check{H}^0(\mathcal{Y}, X)$. By the sheaf condition for $X$ (see Properties of Spaces, Proposition 65.17.1) we obtain a morphism $Y \to X$. By construction the base change of this to $Z'$ is an isomorphism, hence an isomorphism by Lemma 73.11.15. This proves that $X$ is representable by a quasi-affine scheme and we win. $\square$


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