The Stacks project

Proof. Let $W$ be an affine scheme and let $f : W \to X$ be a surjective étale morphism. There exists an integer $d$ such that all geometric fibres of f have $\leq d$ points (because $X$ is a separated algebraic hence reasonable, see Decent Spaces, Lemma 68.5.1). Picking $d$ minimal we get a nonempty open $U \subset X$ such that $f^{-1}(U) \to U$ is finite étale of degree $d$, see Decent Spaces, Lemma 68.8.1. Let

($d$ factors in the fibre product) be the complement of all the diagonals. Because $W \to X$ is separated the diagonal $W \to W \times _ X W$ is a closed immersion. Since $W \to X$ is étale the diagonal $W \to W \times _ X W$ is an open immersion, see Morphisms of Spaces, Lemmas 67.39.10 and 67.38.9. Hence the diagonals are open and closed subschemes of the quasi-compact scheme $W \times _ X \ldots \times _ X W$. In particular we conclude $V$ is a quasi-compact scheme. Choose an open immersion $W \subset Y$ with $Y$ H-projective over $A$ (this is possible as $W$ is affine and of finite type over $A$; for example we can use Morphisms, Lemmas 29.39.2 and 29.43.11). Let

be the scheme theoretic image of the composition $V \to W \times _ X \ldots \times _ X W \to Y \times _ A \ldots \times _ A Y$. Observe that this morphism is quasi-compact since $V$ is quasi-compact and $Y \times _ A \ldots \times _ A Y$ is separated. Note that $V \to Z$ is an open immersion as $V \to Y \times _ A \ldots \times _ A Y$ is an immersion, see Morphisms, Lemma 29.7.7. The projection morphisms give $d$ morphisms $g_ i : Z \to Y$. These morphisms $g_ i$ are projective as $Y$ is projective over $A$, see material in Morphisms, Section 29.43. We set

There is a morphism $X' \to X$ whose restriction to $g_ i^{-1}(W)$ is the composition $g_ i^{-1}(W) \to W \to X$. Namely, these morphisms agree over $V$ hence agree over $g_ i^{-1}(W) \cap g_ j^{-1}(W)$ by Morphisms of Spaces, Lemma 67.17.8. Claim: the morphism $X' \to X$ is proper.

If the claim holds, then the lemma follows by induction on $d$. Namely, by construction $X'$ is H-quasi-projective over $\mathop{\mathrm{Spec}}(A)$. The image of $X' \to X$ contains the open $U$ as $V$ surjects onto $U$. Denote $T$ the reduced induced algebraic space structure on $X \setminus U$. Then $T \times _ X W$ is a closed subscheme of $W$, hence affine. Moreover, the morphism $T \times _ X W \to T$ is étale and every geometric fibre has $< d$ points. By induction hypothesis there exists a proper surjective morphism $T' \to T$ where $T'$ is a scheme H-quasi-projective over $\mathop{\mathrm{Spec}}(A)$. Since $T$ is a closed subspace of $X$ we see that $T' \to X$ is a proper morphism. Thus the lemma follows by taking the proper surjective morphism $X' \amalg T' \to X$.

Proof of the claim. By construction the morphism $X' \to X$ is separated and of finite type. We will check conditions (1) – (4) of Morphisms of Spaces, Lemma 67.42.5 for the morphisms $V \to X'$ and $X' \to X$. Conditions (1) and (2) we have seen above. Condition (3) holds as $X' \to X$ is separated (as a morphism whose source is a separated algebraic space). Thus it suffices to check liftability to $X'$ for diagrams

where $R$ is a valuation ring with fraction field $K$. Note that the top horizontal map is given by $d$ pairwise distinct $K$-valued points $w_1, \ldots , w_ d$ of $W$. In fact, this is a complete set of inverse images of the point $x \in X(K)$ coming from the diagram. Since $W \to X$ is surjective, we can, after possibly replacing $R$ by an extension of valuation rings, lift the morphism $\mathop{\mathrm{Spec}}(R) \to X$ to a morphism $w : \mathop{\mathrm{Spec}}(R) \to W$, see Morphisms of Spaces, Lemma 67.42.4. Since $w_1, \ldots , w_ d$ is a complete collection of inverse images of $x$ we see that $w|_{\mathop{\mathrm{Spec}}(K)}$ is equal to one of them, say $w_ i$. Thus we see that we get a commutative diagram

By the valuative criterion of properness for the projective morphism $g_ i$ we can lift $w$ to $z : \mathop{\mathrm{Spec}}(R) \to Z$, see Morphisms, Lemma 29.43.5 and Schemes, Proposition 26.20.6. The image of $z$ is in $g_ i^{-1}(W) \subset X'$ and the proof is complete. $\square$