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The Stacks project

Lemma 38.32.1. Let S be a quasi-compact and quasi-separated scheme. Let X be a compactifyable scheme over S.

  1. The category of compactifications of X over S is cofiltered.

  2. The full subcategory consisting of compactifications j : X \to \overline{X} such that j(X) is dense and scheme theoretically dense in \overline{X} is initial (Categories, Definition 4.17.3).

  3. If f : \overline{X}' \to \overline{X} is a morphism of compactifications of X such that j'(X) is dense in \overline{X}', then f^{-1}(j(X)) = j'(X).

Proof. To prove part (a) we have to check conditions (1), (2), (3) of Categories, Definition 4.20.1. Condition (1) holds exactly because we assumed that X is compactifyable. Let j_ i : X \to \overline{X}_ i, i = 1, 2 be two compactifications. Then we can consider the scheme theoretic image \overline{X} of (j_1, j_2) : X \to \overline{X}_1 \times _ S \overline{X}_2. This determines a third compactification j : X \to \overline{X} which dominates both j_ i:

\xymatrix{ (X, \overline{X}_1) & (X, \overline{X}) \ar[l] \ar[r] & (X, \overline{X}_2) }

Thus (2) holds. Let f_1, f_2 : \overline{X}_1 \to \overline{X}_2 be two morphisms between compactifications j_ i : X \to \overline{X}_ i, i = 1, 2. Let \overline{X} \subset \overline{X}_1 be the equalizer of f_1 and f_2. As \overline{X}_2 \to S is separated, we see that \overline{X} is a closed subscheme of \overline{X}_1 and hence proper over S. Moreover, we obtain an open immersion X \to \overline{X} because f_1|_ X = f_2|_ X = \text{id}_ X. The morphism (X \to \overline{X}) \to (j_1 : X \to \overline{X}_1) given by the closed immersion \overline{X} \to \overline{X}_1 equalizes f_1 and f_2 which proves condition (3).

Proof of (b). Let j : X \to \overline{X} be a compactification. If \overline{X}' denotes the scheme theoretic closure of X in \overline{X}, then X is dense and scheme theoretically dense in \overline{X}' by Morphisms, Lemma 29.7.7. This proves the first condition of Categories, Definition 4.17.3. Since we have already shown the category of compactifications of X is cofiltered, the second condition of Categories, Definition 4.17.3 follows from the first (we omit the solution to this categorical exercise).

Proof of (c). After replacing \overline{X}' with the scheme theoretic closure of j'(X) (which doesn't change the underlying topological space) this follows from Morphisms, Lemma 29.6.8. \square


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