Theorem 38.33.8. Let S be a quasi-compact and quasi-separated scheme. Let X \to S be a separated, finite type morphism. Then X has a compactification over S.
See [Lutkebohmert], [Conrad-Nagata], [Nagata-1], [Nagata-2], [Nagata-3], and [Nagata-4]
Proof. We first reduce to the Noetherian case. We strongly urge the reader to skip this paragraph. There exists a closed immersion X \to X' with X' \to S of finite presentation and separated. See Limits, Proposition 32.9.6. If we find a compactification of X' over S, then taking the scheme theoretic image of X in this will give a compactification of X over S. Thus we may assume X \to S is separated and of finite presentation. We may write S = \mathop{\mathrm{lim}}\nolimits S_ i as a directed limit of a system of Noetherian schemes with affine transition morphisms. See Limits, Proposition 32.5.4. We can choose an i and a morphism X_ i \to S_ i of finite presentation whose base change to S is X \to S, see Limits, Lemma 32.10.1. After increasing i we may assume X_ i \to S_ i is separated, see Limits, Lemma 32.8.6. If we can find a compactification of X_ i over S_ i, then the base change of this to S will be a compactification of X over S. This reduces us to the case discussed in the next paragraph.
Assume S is Noetherian. We can choose a finite affine open covering X = \bigcup _{i = 1, \ldots , n} U_ i such that U_1 \cap \ldots \cap U_ n is dense in X. This follows from Properties, Lemma 28.29.4 and the fact that X is quasi-compact with finitely many irreducible components. For each i we can choose an n_ i \geq 0 and an immersion U_ i \to \mathbf{A}^{n_ i}_ S by Morphisms, Lemma 29.39.2. Hence U_ i has a compactification over S for i = 1, \ldots , n by taking the scheme theoretic image in \mathbf{P}^{n_ i}_ S. Applying Lemma 38.33.7 (n - 1) times we conclude that the theorem is true. \square
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