## 38.32 Compactifications

Let $S$ be a quasi-compact and quasi-separated scheme. We will say a scheme $X$ over $S$ *has a compactification over $S$* or *is compactifyable over $S$* if there exists a quasi-compact open immersion $X \to \overline{X}$ into a scheme $\overline{X}$ proper over $S$. If $X$ has a compactification over $S$, then $X \to S$ is separated and of finite type. It is a theorem of Nagata, see [Lutkebohmert], [Conrad-Nagata], [Nagata-1], [Nagata-2], [Nagata-3], and [Nagata-4], that the converse is true as well. We will prove this theorem in the next section, see Theorem 38.33.8.

Let $S$ be a quasi-compact and quasi-separated scheme. Let $X \to S$ be a separated finite type morphism of schemes. The category of *compactifications of $X$ over $S$* is the category defined as follows:

Objects are open immersions $j : X \to \overline{X}$ over $S$ with $\overline{X} \to S$ proper.

Morphisms $(j' : X \to \overline{X}') \to (j : X \to \overline{X})$ are morphisms $f : \overline{X}' \to \overline{X}$ of schemes over $S$ such that $f \circ j' = j$.

If $j : X \to \overline{X}$ is a compactification, then $j$ is a quasi-compact open immersion, see Schemes, Remark 26.21.18.

**Warning.** We do *not* assume compactifications $j : X \to \overline{X}$ to have dense image. Consequently, if $f : \overline{X}' \to \overline{X}$ is a morphism of compactifications, it may not be the case that $f^{-1}(j(X)) = j'(X)$.

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

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

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).

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.
$\square$

We can also consider the category of all compactifications (for varying $X$). It turns out that this category, localized at the set of morphisms which induce an isomorphism on the interior is equivalent to the category of compactifyable schemes over $S$.

Lemma 38.32.2. Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \to Y$ be a morphism of schemes over $S$ with $Y$ separated and of finite type over $S$ and $X$ compactifyable over $S$. Then $X$ has a compactification over $Y$.

**Proof.**
Let $j : X \to \overline{X}$ be a compactification of $X$ over $S$. Then we let $\overline{X}'$ be the scheme theoretic image of $(j, f) : X \to \overline{X} \times _ S Y$. The morphism $\overline{X}' \to Y$ is proper because $\overline{X} \times _ S Y \to Y$ is proper as a base change of $\overline{X} \to S$. On the other hand, since $Y$ is separated over $S$, the morphism $(1, f) : X \to X \times _ S Y$ is a closed immersion (Schemes, Lemma 26.21.10) and hence $X \to \overline{X}'$ is an open immersion.
$\square$

Let $S$ be a quasi-compact and quasi-separated scheme. We define the *category of compactifications* to be the category whose objects are pairs $(X, \overline{X})$ where $\overline{X}$ is a scheme proper over $S$ and $X \subset \overline{X}$ is a quasi-compact open and whose morphisms are commutative diagrams

\[ \xymatrix{ X \ar[d] \ar[r]_ f & Y \ar[d] \\ \overline{X} \ar[r]^{\overline{f}} & \overline{Y} } \]

of morphisms of schemes over $S$.

Lemma 38.32.3. Let $S$ be a quasi-compact and quasi-separated scheme. The collection of morphisms $(u, \overline{u}) : (X', \overline{X}') \to (X, \overline{X})$ such that $u$ is an isomorphism forms a right multiplicative system (Categories, Definition 4.27.1) of arrows in the category of compactifications.

**Proof.**
Axiom RMS1 is trivial to verify. Let us check RMS2 holds. Suppose given a diagram

\[ \xymatrix{ & (X', \overline{X}') \ar[d]_{(u, \overline{u})} \\ (Y, \overline{Y}) \ar[r]^{(f, \overline{f})} & (X, \overline{X}) } \]

with $u : X' \to X$ an isomorphism. Then we let $Y' = Y \times _ X X'$ with the projection map $v : Y' \to Y$ (an isomorphism). We also set $\overline{Y}' = \overline{Y} \times _{\overline{X}} \overline{X}'$ with the projection map $\overline{v} : \overline{Y}' \to \overline{Y}$ It is clear that $Y' \to \overline{Y}'$ is an open immersion. The diagram

\[ \xymatrix{ (Y', \overline{Y}') \ar[r]_{(g, \overline{g})} \ar[d]_{(v, \overline{v})} & (X', \overline{X}') \ar[d]_{(u, \overline{u})} \\ (Y, \overline{Y}) \ar[r]^{(f, \overline{f})} & (X, \overline{X}) } \]

shows that axiom RMS2 holds.

Let us check RMS3 holds. Suppose given a pair of morphisms $(f, \overline{f}), (g, \overline{g}) : (X, \overline{X}) \to (Y, \overline{Y})$ of compactifications and a morphism $(v, \overline{v}) : (Y, \overline{Y}) \to (Y', \overline{Y}')$ such that $v$ is an isomorphism and such that $(v, \overline{v}) \circ (f, \overline{f}) = (v, \overline{v}) \circ (g, \overline{g})$. Then $f = g$. Hence if we let $\overline{X}' \subset \overline{X}$ be the equalizer of $\overline{f}$ and $\overline{g}$, then $(u, \overline{u}) : (X, \overline{X}') \to (X, \overline{X})$ will be a morphism of the category of compactifications such that $(f, \overline{f}) \circ (u, \overline{u}) = (g, \overline{g}) \circ (u, \overline{u})$ as desired.
$\square$

Lemma 38.32.4. Let $S$ be a quasi-compact and quasi-separated scheme. The functor $(X, \overline{X}) \mapsto X$ defines an equivalence from the category of compactifications localized (Categories, Lemma 4.27.11) at the right multiplicative system of Lemma 38.32.3 to the category of compactifyable schemes over $S$.

**Proof.**
Denote $\mathcal{C}$ the category of compactifications and denote $Q : \mathcal{C} \to \mathcal{C}'$ the localization functor of Categories, Lemma 4.27.16. Denote $\mathcal{D}$ the category of compactifyable schemes over $S$. It is clear from the lemma just cited and our choice of multiplicative system that we obtain a functor $\mathcal{C}' \to \mathcal{D}$. This functor is clearly essentially surjective. If $f : X \to Y$ is a morphism of compactifyable schemes, then we choose an open immersion $Y \to \overline{Y}$ into a scheme proper over $S$, and then we choose an embedding $X \to \overline{X}$ into a scheme $\overline{X}$ proper over $\overline{Y}$ (possible by Lemma 38.32.2 applied to $X \to \overline{Y}$). This gives a morphism $(X, \overline{X}) \to (Y, \overline{Y})$ of compactifications which produces our given morphism $X \to Y$. Finally, suppose given a pair of morphisms in the localized category with the same source and target: say

\[ a = ((f, \overline{f}) : (X', \overline{X}') \to (Y, \overline{Y}), (u, \overline{u}) : (X', \overline{X}') \to (X, \overline{X})) \]

and

\[ b = ((g, \overline{g}) : (X'', \overline{X}'') \to (Y, \overline{Y}), (v, \overline{v}) : (X'', \overline{X}'') \to (X, \overline{X})) \]

which produce the same morphism $X \to Y$ over $S$, in other words $f \circ u^{-1} = g \circ v^{-1}$. By Categories, Lemma 4.27.13 we may assume that $(X', \overline{X}') = (X'', \overline{X}'')$ and $(u, \overline{u}) = (v, \overline{v})$. In this case we can consider the equalizer $\overline{X}''' \subset \overline{X}'$ of $\overline{f}$ and $\overline{g}$. The morphism $(w, \overline{w}) : (X', \overline{X}''') \to (X', \overline{X}')$ is in the multiplicative subset and we see that $a = b$ in the localized category by precomposing with $(w, \overline{w})$.
$\square$

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