The Stacks project

Proof. By Lemma 87.17.1 we may choose an affine formal algebraic space $Y$ and a representable surjective étale morphism $Y \to X$. Write $Y = \mathop{\mathrm{colim}}\nolimits Y_\lambda$ as in Definition 87.9.1.

Pick $\lambda \in \Lambda$. Then $Y_\lambda \times _ X Y$ is a scheme by Lemma 87.9.11. The reduction (Lemma 87.12.1) of $Y_\lambda \times _ X Y$ is equal to the reduction of $Y_{red} \times _{X_{red}} Y_{red}$ which is quasi-compact as $X$ is quasi-separated and $Y_{red}$ is affine. Therefore $Y_\lambda \times _ X Y$ is a quasi-compact scheme. Hence there exists a $\mu \geq \lambda$ such that $\text{pr}_2 : Y_\lambda \times _ X Y \to Y$ factors through $Y_\mu$, see Lemma 87.9.4. Let $Z_\lambda$ be the scheme theoretic image of the morphism $\text{pr}_2 : Y_\lambda \times _ X Y \to Y_\mu$. This is independent of the choice of $\mu$ and we can and will think of $Z_\lambda \subset Y$ as the scheme theoretic image of the morphism $\text{pr}_2 : Y_\lambda \times _ X Y \to Y$. Observe that $Z_\lambda$ is also equal to the scheme theoretic image of the morphism $\text{pr}_1 : Y \times _ X Y_\lambda \to Y$ since this is isomorphic to the morphism used to define $Z_\lambda$. We claim that $Z_\lambda \times _ X Y = Y \times _ X Z_\lambda$ as subfunctors of $Y \times _ X Y$. Namely, since $Y \to X$ is étale we see that $Z_\lambda \times _ X Y$ is the scheme theoretic image of the morphism

by Morphisms of Spaces, Lemma 67.16.3. By the same token, $Y \times _ X Z_\lambda$ is the scheme theoretic image of the morphism

The claim follows. Then $R_\lambda = Z_\lambda \times _ X Y = Y \times _ X Z_\lambda$ together with the morphism $R_\lambda \to Z_\lambda \times _ S Z_\lambda$ defines an étale equivalence relation. In this way we obtain an algebraic space $X_\lambda = Z_\lambda /R_\lambda$. By construction the diagram

is cartesian (because $X$ is the coequalizer of the two projections $R = Y \times _ X Y \to Y$, because $Z_\lambda \subset Y$ is $R$-invariant, and because $R_\lambda$ is the restriction of $R$ to $Z_\lambda$). Hence $X_\lambda \to X$ is representable and a closed immersion, see Spaces, Lemma 65.11.5. On the other hand, since $Y_\lambda \subset Z_\lambda$ we see that $(X_\lambda )_{red} = X_{red}$, in other words, $X_\lambda \to X$ is a thickening. Finally, we claim that

We have $Y \times _ X X_\lambda = Z_\lambda \supset Y_\lambda$. Every morphism $T \to X$ where $T$ is a scheme over $S$ lifts étale locally to a morphism into $Y$ which lifts étale locally into a morphism into some $Y_\lambda$. Hence $T \to X$ lifts étale locally on $T$ to a morphism into $X_\lambda$. This finishes the proof. $\square$