Theorem 81.14.9. Let S be a scheme. Let B be a quasi-compact and quasi-separated algebraic space over S. Let X \to B be a separated, finite type morphism. Then X has a compactification over B.
[CLO]
Proof. We first reduce to the Noetherian case. We strongly urge the reader to skip this paragraph. First, we may replace S by \mathop{\mathrm{Spec}}(\mathbf{Z}). See Spaces, Section 65.16 and Properties of Spaces, Definition 66.3.1. There exists a closed immersion X \to X' with X' \to B of finite presentation and separated. See Limits of Spaces, Proposition 70.11.7. If we find a compactification of X' over B, then taking the scheme theoretic closure of X in this will give a compactification of X over B. Thus we may assume X \to B is separated and of finite presentation. We may write B = \mathop{\mathrm{lim}}\nolimits B_ i as a directed limit of a system of Noetherian algebraic spaces of finite type over \mathop{\mathrm{Spec}}(\mathbf{Z}) with affine transition morphisms. See Limits of Spaces, Proposition 70.8.1. We can choose an i and a morphism X_ i \to B_ i of finite presentation whose base change to B is X \to B, see Limits of Spaces, Lemma 70.7.1. After increasing i we may assume X_ i \to B_ i is separated, see Limits of Spaces, Lemma 70.6.9. If we can find a compactification of X_ i over B_ i, then the base change of this to B will be a compactification of X over B. This reduces us to the case discussed in the next paragraph.
Assume B is of finite type over \mathbf{Z} in addition to being quasi-compact and quasi-separated. Let U \to X be an étale morphism of algebraic spaces such that U has a compactification Y over \mathop{\mathrm{Spec}}(\mathbf{Z}). The morphism
is separated and quasi-finite by Morphisms of Spaces, Lemma 67.27.10 (the displayed morphism factors into an immersion hence is a monomorphism). Hence by Zariski's main theorem (More on Morphisms of Spaces, Lemma 76.34.3) there is an open immersion of U into an algebraic space Y' finite over B \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} Y. Then Y' \to B is proper as the composition Y' \to B \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} Y \to B of two proper morphisms (use Morphisms of Spaces, Lemmas 67.45.9, 67.40.4, and 67.40.3). We conclude that U has a compactification over B.
There is a dense open subspace U \subset X which is a scheme. (Properties of Spaces, Proposition 66.13.3). In fact, we may choose U to be an affine scheme (Properties, Lemmas 28.5.7 and 28.29.1). Thus U has a compactification over \mathop{\mathrm{Spec}}(\mathbf{Z}); this is easily shown directly but also follows from the theorem for schemes, see More on Flatness, Theorem 38.33.8. By the previous paragraph U has a compactification over B. By Noetherian induction we can find a maximal dense open subspace U \subset X which has a compactification over B. We will show that the assumption that U \not= X leads to a contradiction. Namely, by Lemma 81.14.8 we can find a strictly larger open U \subset W \subset X and a distinguished square (U \subset W, f : V \to W) with V affine and U \times _ W V dense image in U. Since V is affine, as before it has a compactification over B. Hence Lemma 81.14.7 applies to show that W has a compactification over B which is the desired contradiction. \square
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