Lemma 37.66.2. For a morphism of schemes $f : X \to Y$, the following are equivalent:

1. $f$ is ind-quasi-affine,

2. for every affine open subscheme $V \subset Y$ and every quasi-compact open subscheme $U \subset f^{-1}(V)$, the induced morphism $U \to V$ is quasi-affine.

3. for some cover $\{ V_ j \} _{j \in J}$ of $Y$ by quasi-compact and quasi-separated open subschemes $V_ j \subset Y$, every $j \in J$, and every quasi-compact open subscheme $U \subset f^{-1}(V_ j)$, the induced morphism $U \to V_ j$ is quasi-affine.

4. for every quasi-compact and quasi-separated open subscheme $V \subset Y$ and every quasi-compact open subscheme $U \subset f^{-1}(V)$, the induced morphism $U \to V$ is quasi-affine.

In particular, the property of being an ind-quasi-affine morphism is Zariski local on the base.

Proof. The equivalence (1) $\Leftrightarrow$ (2) follows from the definitions and Morphisms, Lemma 29.13.3. For (2) $\Rightarrow$ (4), let $U$ and $V$ be as in (4). By Schemes, Lemma 26.21.14, the induced morphism $U \to V$ is quasi-compact. Thus, for every affine open $V' \subset V$, the fiber product $V' \times _ V U$ is quasi-compact, so, by (2), the induced map $V' \times _ V U \to V'$ is quasi-affine. Thus, $U \to V$ is also quasi-affine by Morphisms, Lemma 29.13.3. This argument also gives (3) $\Rightarrow$ (4): indeed, keeping the same notation, those affine opens $V' \subset V$ that lie in one of the $V_ j$ cover $V$, so one needs to argue that the quasi-compact map $V' \times _ V U \to V'$ is quasi-affine. However, by (3), the composition $V' \times _ V U \to V' \to V_ j$ is quasi-affine and, by Schemes, Lemma 26.21.13, the map $V' \to V_ j$ is quasi-separated. Thus, $V' \times _ V U \to V'$ is quasi-affine by Morphisms, Lemma 29.13.8. The final implications (4) $\Rightarrow$ (2) and (4) $\Rightarrow$ (3) are evident. $\square$

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