The Stacks project

Lemma 37.66.7. The property of being ind-quasi-affine is fpqc local on the base.

Proof. The stability of ind-quasi-affineness under base change supplied by Lemma 37.66.6 gives one direction. For the other, let $f : X \to Y$ be a morphism of schemes and let $\{ g_ i : Y_ i \to Y\} $ be an fpqc covering such that the base change $f_ i : X_ i \to Y_ i$ is ind-quasi-affine for all $i$. We need to show $f$ is ind-quasi-affine.

By Lemma 37.66.2, we may work Zariski locally on $Y$, so we assume that $Y$ is affine. Then we use stability under base change ensured by Lemma 37.66.6 to refine the cover and assume that it is given by a single affine, faithfully flat morphism $g : Y' \to Y$. For any quasi-compact open $U \subset X$, its $Y'$-base change $U \times _ Y Y' \subset X \times _ Y Y'$ is also quasi-compact. It remains to observe that, by Descent, Lemma 35.23.20, the map $U \to Y$ is quasi-affine if and only if so is $U \times _ Y Y' \to Y'$. $\square$

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