**Proof.**
Assume $U_ i \to X_ i \subset X$ are as in (3). To prove (4) choose for each $i$ a finite affine open covering $U_ i = U_{i1} \cup \ldots \cup U_{in_ i}$ such that each $U_{ij}$ maps into an affine open of $S$. The compositions $U_{ij} \to U_ i \to X_ i$ are étale and quasi-compact (see Spaces, Lemma 64.5.4). Let $X_{ij} \subset X_ i$ be the open subspace corresponding to the image of $|U_{ij}| \to |X_ i|$, see Lemma 65.4.10. Note that $U_{ij} \to X_{ij}$ is quasi-compact as $X_{ij} \subset X_ i$ is a monomorphism and as $U_{ij} \to X$ is quasi-compact. Then $X = \bigcup X_{ij}$ is a covering as in (4). The implication (4) $\Rightarrow $ (3) is immediate.

Assume (4). To show that $X$ is Zariski locally quasi-separated over $S$ it suffices to show that $X_ i$ is quasi-separated over $S$. Hence we may assume there exists an affine scheme $U$ mapping into an affine open of $S$ and a quasi-compact surjective étale morphism $U \to X$. Consider the fibre product square

\[ \xymatrix{ U \times _ X U \ar[r] \ar[d] & U \times _ S U \ar[d] \\ X \ar[r]^-{\Delta _{X/S}} & X \times _ S X } \]

The right vertical arrow is surjective étale (see Spaces, Lemma 64.5.7) and $U \times _ S U$ is affine (as $U$ maps into an affine open of $S$, see Schemes, Section 26.17), and $U \times _ X U$ is quasi-compact because the projection $U \times _ X U \to U$ is quasi-compact as a base change of $U \to X$. It follows from Spaces, Lemma 64.11.4 that $\Delta _{X/S}$ is quasi-compact as desired.

Assume (1). To prove (3) there is an immediate reduction to the case where $X$ is quasi-separated over $S$. By Lemma 65.6.2 we can find a Zariski open covering $X = \bigcup X_ i$ such that each $X_ i$ maps into an affine open of $S$, and such that there exist affine schemes $U_ i$ and surjective étale morphisms $U_ i \to X_ i$. Since $U_ i \to S$ maps into an affine open of $S$ we see that $U_ i \times _ S U_ i$ is affine, see Schemes, Section 26.17. As $X$ is quasi-separated over $S$, the morphisms

\[ R_ i = U_ i \times _{X_ i} U_ i = U_ i \times _ X U_ i \longrightarrow U_ i \times _ S U_ i \]

as base changes of $\Delta _{X/S}$ are quasi-compact. Hence we conclude that $R_ i$ is a quasi-compact scheme. This in turn implies that each projection $R_ i \to U_ i$ is quasi-compact. Hence, applying Spaces, Lemma 64.11.4 to the covering $U_ i \to X_ i$ and the morphism $U_ i \to X_ i$ we conclude that the morphisms $U_ i \to X_ i$ are quasi-compact as desired.

At this point we see that (1), (3), and (4) are equivalent. Since (3) does not refer to the base scheme we conclude that these are also equivalent with (2).
$\square$

## Comments (0)