## 68.18 A weak version of Chow's lemma

In this section we quickly prove the following lemma in order to help us prove the basic results on cohomology of coherent modules on proper algebraic spaces.

Lemma 68.18.1. Let $A$ be a ring. Let $X$ be an algebraic space over $\mathop{\mathrm{Spec}}(A)$ whose structure morphism $X \to \mathop{\mathrm{Spec}}(A)$ is separated of finite type. Then there exists a proper surjective morphism $X' \to X$ where $X'$ is a scheme which is H-quasi-projective over $\mathop{\mathrm{Spec}}(A)$.

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 67.5.1). 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 67.8.1. Let

$V \subset W \times _ X W \times _ X \ldots \times _ X W$

($d$ 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 66.39.10 and 66.38.9. 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

$Z \subset Y \times _ A Y \times _ A \ldots \times _ A Y$

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

$X' = \bigcup g_ i^{-1}(W) \subset Z$

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 66.17.8. 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 66.42.5 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

$\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & V \ar[d] \\ \mathop{\mathrm{Spec}}(R) \ar[r] & X }$

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 66.42.4. 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

$\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & Z \ar[d]_{g_ i}\\ \mathop{\mathrm{Spec}}(R) \ar[r]^ w & Y }$

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. $\square$

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