The Stacks Project


Tag 0ATT

46.16. Compactifications

We interrupt the flow of the arguments for a little bit of geometry.

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 an 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 give a precise statement and a proof if we ever need this result).

Let $S$ be a quasi-compact and quasi-separated scheme. Let $X$ be a scheme over $S$. The category of compactifications of $X$ over $S$ is the category whose objects are open immersions $j : X \to \overline{X}$ over $S$ with $\overline{X} \to S$ proper and whose 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 $f \circ j' = j$.

Lemma 46.16.1. 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.

Proof. 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 closure $\overline{X}$ 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) and finishes the proof. $\square$

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 46.16.2. 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 $f : X \to Y$ be a morphism of schemes over $S$ with $Y$ separated and of finite type over $S$. 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 25.21.11) 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 46.16.3. 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 4.26.1) 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 46.16.4. 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 4.26.11) at the right multiplicative system of Lemma 46.16.3 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 4.26.16. 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 46.16.2 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 4.26.13 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 $(w, \overline{w})$. $\square$

    The code snippet corresponding to this tag is a part of the file duality.tex and is located in lines 3519–3734 (see updates for more information).

    \section{Compactifications}
    \label{section-compactify}
    
    \noindent
    We interrupt the flow of the arguments for a little bit of geometry.
    
    \medskip\noindent
    Let $S$ be a quasi-compact and quasi-separated scheme. We will say a
    scheme $X$ over $S$ {\it has a compactification over $S$}
    or {\it is compactifyable over $S$} if there exists
    an 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
    \cite{Lutkebohmert}, \cite{Conrad-Nagata}, \cite{Nagata-1},
    \cite{Nagata-2}, \cite{Nagata-3}, and \cite{Nagata-4}) that the converse is
    true as well (we will give a
    precise statement and a proof if we ever need this result).
    
    \medskip\noindent
    Let $S$ be a quasi-compact and quasi-separated scheme.
    Let $X$ be a scheme over $S$. The category
    of {\it compactifications of $X$ over $S$} is the category whose
    objects are open immersions $j : X \to \overline{X}$ over $S$ with
    $\overline{X} \to S$ proper and whose 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 $f \circ j' = j$.
    
    \begin{lemma}
    \label{lemma-compactifications-cofiltered}
    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.
    \end{lemma}
    
    \begin{proof}
    We have to check conditions (1), (2), (3) of
    Categories, Definition \ref{categories-definition-codirected}.
    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 closure $\overline{X}$
    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) and
    finishes the proof.
    \end{proof}
    
    \noindent
    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 $S$.
    
    \begin{lemma}
    \label{lemma-compactifyable}
    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 $Y$.
    \end{lemma}
    
    \begin{proof}
    Let $f : X \to Y$ be a morphism of schemes over $S$ with $Y$ separated
    and of finite type over $S$. 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 \ref{schemes-lemma-semi-diagonal})
    and hence $X \to \overline{X}'$ is an open immersion.
    \end{proof}
    
    \noindent
    Let $S$ be a quasi-compact and quasi-separated scheme.
    We define the {\it 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 $S$.
    
    \begin{lemma}
    \label{lemma-right-multiplicative-system}
    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 \ref{categories-definition-multiplicative-system})
    of arrows in the category of compactifications.
    \end{lemma}
    
    \begin{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.
    
    \medskip\noindent
    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.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-invert-right-multiplicative-system}
    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 \ref{categories-lemma-right-localization})
    at the right
    multiplicative system of Lemma \ref{lemma-right-multiplicative-system}
    to the category of compactifyable schemes over $S$.
    \end{lemma}
    
    \begin{proof}
    Denote $\mathcal{C}$ the category of compactifications and
    denote $Q : \mathcal{C} \to \mathcal{C}'$ the localization
    functor of Categories, Lemma
    \ref{categories-lemma-properties-right-localization}.
    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 \ref{lemma-compactifyable}
    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 \ref{categories-lemma-morphisms-right-localization}
    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 $(w, \overline{w})$.
    \end{proof}

    Comments (4)

    Comment #2064 by Hu Fei on June 13, 2016 a 2:55 pm UTC

    There are some typos before in the definition of the category of compactifications of $X$. It should be ...whose morphisms $(j' : X \to \overline{X}') \to (j : X \to \overline{X})$ are morphisms $f : \overline{X}' \to \overline{X}$ such that $f \circ j' = j$.'' Another thing: it seems thatcompactifyable scheme'' has not be exactly defined. I have to guess it means a scheme can be compactified. Although this is straightforward, but it would be better if give its definition after the definition of compactification. (I was checking duality theorem which involves the schemes of this kind).

    Comment #2065 by Hu Fei on June 13, 2016 a 2:59 pm UTC

    There are some typos in the definition of the category of compactifications of $X$. It should be ...whose morphisms $(j' : X \to \overline{X}') \to (j : X \to \overline{X})$ are morphisms $f : \overline{X}' \to \overline{X}$ such that $f \circ j' = j$.'' Another thing: it seems thatcompactifyable scheme'' has not be exactly defined. I have to guess it means a scheme can be compactified. Although this is straightforward, but it would be better if give its definition after the definition of compactification. (I was checking duality theorem which involves the schemes of this kind). Is it ``compactifiable''?

    Comment #2093 by Johan (site) on June 16, 2016 a 3:22 pm UTC

    Thanks, fixed here.

    Comment #2996 by Ko Aoki on November 14, 2017 a 10:36 am UTC

    The second paragraph says "If X has a compactification over S, then X→S is separated and of finite type," but it seems to be incorrect without a qc assumption for $X \to \overline{X}$.

    Add a comment on tag 0ATT

    Your email address will not be published. Required fields are marked.

    In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

    All contributions are licensed under the GNU Free Documentation License.




    In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?