The Stacks project

Proof. Let $W$ be an affine scheme and let $\varphi : W \to X$ be a surjective étale morphism, see Properties of Spaces, Lemma 66.6.3. If (1) holds, then $\varphi ^{-1}(U) \to W$ is quasi-compact, hence $\varphi ^{-1}(U)$ is quasi-compact, hence $U$ is quasi-compact (as $|\varphi ^{-1}(U)| \to |U|$ is surjective). If (2) holds, then $\varphi ^{-1}(U)$ is quasi-compact because $\varphi $ is quasi-compact since $X$ is quasi-separated (Morphisms of Spaces, Lemma 67.8.10). Hence $\varphi ^{-1}(U) \to W$ is a quasi-compact morphism of schemes by Properties, Lemma 28.24.1. It follows that $U \to X$ is quasi-compact by Morphisms of Spaces, Lemma 67.8.8. Thus (1) and (2) are equivalent.

Assume (1) and (2). By Properties of Spaces, Lemma 66.12.3 there exists a unique quasi-coherent sheaf of ideals $\mathcal{J}$ cutting out the reduced induced closed subspace structure on $|X| \setminus |U|$. Note that $\mathcal{J}|_ U = \mathcal{O}_ U$ which is an $\mathcal{O}_ U$-modules of finite type. As $U$ is quasi-compact it follows from Lemma 70.9.2 that there exists a quasi-coherent subsheaf $\mathcal{I} \subset \mathcal{J}$ which is of finite type and has the property that $\mathcal{I}|_ U = \mathcal{J}|_ U$. Then $|X| \setminus |U| = |V(\mathcal{I})|$ and we obtain (3). Conversely, if $\mathcal{I}$ is as in (3), then $\varphi ^{-1}(U) \subset W$ is a quasi-compact open by the lemma for schemes (Properties, Lemma 28.24.1) applied to $\varphi ^{-1}\mathcal{I}$ on $W$. Thus (2) holds. $\square$