The Stacks project

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. $\square$