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Tag 01LA

25.24. Functoriality for quasi-coherent modules

Let $X$ be a scheme. We denote $\mathit{QCoh}(\mathcal{O}_X)$ the category of quasi-coherent $\mathcal{O}_X$-modules as defined in Modules, Definition 17.10.1. We have seen in Section 25.7 that the category $\mathit{QCoh}(\mathcal{O}_X)$ has a lot of good properties when $X$ is affine. Since the property of being quasi-coherent is local on $X$, these properties are inherited by the category of quasi-coherent sheaves on any scheme $X$. We enumerate them here.

  1. A sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ is quasi-coherent if and only if the restriction of $\mathcal{F}$ to each affine open $U = \mathop{\mathrm{Spec}}(R)$ is of the form $\widetilde M$ for some $R$-module $M$.
  2. A sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ is quasi-coherent if and only if the restriction of $\mathcal{F}$ to each of the members of an affine open covering is quasi-coherent.
  3. Any direct sum of quasi-coherent sheaves is quasi-coherent.
  4. Any colimit of quasi-coherent sheaves is quasi-coherent.
  5. The kernel and cokernel of a morphism of quasi-coherent sheaves is quasi-coherent.
  6. Given a short exact sequence of $\mathcal{O}_X$-modules $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ if two out of three are quasi-coherent so is the third.
  7. Given a morphism of schemes $f : Y \to X$ the pullback of a quasi-coherent $\mathcal{O}_X$-module is a quasi-coherent $\mathcal{O}_Y$-module. See Modules, Lemma 17.10.4.
  8. Given two quasi-coherent $\mathcal{O}_X$-modules the tensor product is quasi-coherent, see Modules, Lemma 17.15.5.
  9. Given a quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ the tensor, symmetric and exterior algebras on $\mathcal{F}$ are quasi-coherent, see Modules, Lemma 17.19.6.
  10. Given two quasi-coherent $\mathcal{O}_X$-modules $\mathcal{F}$, $\mathcal{G}$ such that $\mathcal{F}$ is of finite presentation, then the internal hom $\mathop{\mathcal{H}\!\mathit{om}}\nolimits_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is quasi-coherent, see Modules, Lemma 17.20.5 and (5) above.

On the other hand, it is in general not the case that the pushforward of a quasi-coherent module is quasi-coherent. Here is a case where this does hold.

Lemma 25.24.1. Let $f : X \to S$ be a morphism of schemes. If $f$ is quasi-compact and quasi-separated then $f_*$ transforms quasi-coherent $\mathcal{O}_X$-modules into quasi-coherent $\mathcal{O}_S$-modules.

Proof. The question is local on $S$ and hence we may assume that $S$ is affine. Because $X$ is quasi-compact we may write $X = \bigcup_{i = 1}^n U_i$ with each $U_i$ open affine. Because $f$ is quasi-separated we may write $U_i \cap U_j = \bigcup_{k = 1}^{n_{ij}} U_{ijk}$ for some affine open $U_{ijk}$, see Lemma 25.21.7. Denote $f_i : U_i \to S$ and $f_{ijk} : U_{ijk} \to S$ the restrictions of $f$. For any open $V$ of $S$ and any sheaf $\mathcal{F}$ on $X$ we have \begin{eqnarray*} f_*\mathcal{F}(V) & = & \mathcal{F}(f^{-1}V) \\ & = & \mathop{\mathrm{Ker}}\left( \bigoplus\nolimits_i \mathcal{F}(f^{-1}V \cap U_i) \to \bigoplus\nolimits_{i, j, k} \mathcal{F}(f^{-1}V \cap U_{ijk})\right) \\ & = & \mathop{\mathrm{Ker}}\left( \bigoplus\nolimits_i f_{i, *}(\mathcal{F}|_{U_i})(V) \to \bigoplus\nolimits_{i, j, k} f_{ijk, *}(\mathcal{F}|_{U_{ijk}})\right)(V) \\ & = & \mathop{\mathrm{Ker}}\left( \bigoplus\nolimits_i f_{i, *}(\mathcal{F}|_{U_i}) \to \bigoplus\nolimits_{i, j, k} f_{ijk, *}(\mathcal{F}|_{U_{ijk}})\right)(V) \end{eqnarray*} In other words there is an exact sequence of sheaves $$ 0 \to f_*\mathcal{F} \to \bigoplus f_{i, *}\mathcal{F}_i \to \bigoplus f_{ijk, *}\mathcal{F}_{ijk} $$ where $\mathcal{F}_i, \mathcal{F}_{ijk}$ denotes the restriction of $\mathcal{F}$ to the corresponding open. If $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_X$-modules then $\mathcal{F}_i$, $\mathcal{F}_{ijk}$ is a quasi-coherent $\mathcal{O}_{U_i}$, $\mathcal{O}_{U_{ijk}}$-module. Hence by Lemma 25.7.3 we see that the second and third term of the exact sequence are quasi-coherent $\mathcal{O}_S$-modules. Thus we conclude that $f_*\mathcal{F}$ is a quasi-coherent $\mathcal{O}_S$-module. $\square$

Using this we can characterize (closed) immersions of schemes as follows.

Lemma 25.24.2. Let $f : X \to Y$ be a morphism of schemes. Suppose that

  1. $f$ induces a homeomorphism of $X$ with a closed subset of $Y$, and
  2. $f^\sharp : \mathcal{O}_Y \to f_*\mathcal{O}_X$ is surjective.

Then $f$ is a closed immersion of schemes.

Proof. Assume (1) and (2). By (1) the morphism $f$ is quasi-compact (see Topology, Lemma 5.12.3). Conditions (1) and (2) imply conditions (1) and (2) of Lemma 25.23.7. Hence $f : X \to Y$ is a monomorphism. In particular, $f$ is separated, see Lemma 25.23.3. Hence Lemma 25.24.1 above applies and we conclude that $f_*\mathcal{O}_X$ is a quasi-coherent $\mathcal{O}_Y$-module. Therefore the kernel of $\mathcal{O}_Y \to f_*\mathcal{O}_X$ is quasi-coherent by Lemma 25.7.8. Since a quasi-coherent sheaf is locally generated by sections (see Modules, Definition 17.10.1) this implies that $f$ is a closed immersion, see Definition 25.4.1. $\square$

We can use this lemma to prove the following lemma.

Lemma 25.24.3. A composition of immersions of schemes is an immersion, a composition of closed immersions of schemes is a closed immersion, and a composition of open immersions of schemes is an open immersion.

Proof. This is clear for the case of open immersions since an open subspace of an open subspace is also an open subspace.

Suppose $a : Z \to Y$ and $b : Y \to X$ are closed immersions of schemes. We will verify that $c = b \circ a$ is also a closed immersion. The assumption implies that $a$ and $b$ are homeomorphisms onto closed subsets, and hence also $c = b \circ a$ is a homeomorphism onto a closed subset. Moreover, the map $\mathcal{O}_X \to c_*\mathcal{O}_Z$ is surjective since it factors as the composition of the surjective maps $\mathcal{O}_X \to b_*\mathcal{O}_Y$ and $b_*\mathcal{O}_Y \to b_*a_*\mathcal{O}_Z$ (surjective as $b_*$ is exact, see Modules, Lemma 17.6.1). Hence by Lemma 25.24.2 above $c$ is a closed immersion.

Finally, we come to the case of immersions. Suppose $a : Z \to Y$ and $b : Y \to X$ are immersions of schemes. This means there exist open subschemes $V \subset Y$ and $U \subset X$ such that $a(Z) \subset V$, $b(Y) \subset U$ and $a : Z \to V$ and $b : Y \to U$ are closed immersions. Since the topology on $Y$ is induced from the topology on $U$ we can find an open $U' \subset U$ such that $V = b^{-1}(U')$. Then we see that $Z \to V = b^{-1}(U') \to U'$ is a composition of closed immersions and hence a closed immersion. This proves that $Z \to X$ is an immersion and we win. $\square$

    The code snippet corresponding to this tag is a part of the file schemes.tex and is located in lines 4524–4716 (see updates for more information).

    \section{Functoriality for quasi-coherent modules}
    \label{section-quasi-coherent}
    
    \noindent
    Let $X$ be a scheme. We denote $\QCoh(\mathcal{O}_X)$ the
    category of quasi-coherent $\mathcal{O}_X$-modules as defined in
    Modules, Definition \ref{modules-definition-quasi-coherent}.
    We have seen in
    Section \ref{section-quasi-coherent-affine}
    that the category $\QCoh(\mathcal{O}_X)$ has a lot of good properties
    when $X$ is affine. Since the property of being quasi-coherent is
    local on $X$, these properties are inherited by the category
    of quasi-coherent sheaves on any scheme $X$. We enumerate them here.
    \begin{enumerate}
    \item A sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ is
    quasi-coherent if and only if the restriction of $\mathcal{F}$
    to each affine open $U = \Spec(R)$ is of the form
    $\widetilde M$ for some $R$-module $M$.
    \item A sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ is
    quasi-coherent if and only if the restriction of $\mathcal{F}$
    to each of the members of an affine open covering is quasi-coherent.
    \item Any direct sum of quasi-coherent sheaves is quasi-coherent.
    \item Any colimit of quasi-coherent sheaves is quasi-coherent.
    \item
    \label{item-kernel}
    The kernel and cokernel of a morphism of quasi-coherent sheaves
    is quasi-coherent.
    \item Given a short exact sequence of $\mathcal{O}_X$-modules
    $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$
    if two out of three are quasi-coherent so is the third.
    \item Given a morphism of schemes $f : Y \to X$ the pullback
    of a quasi-coherent $\mathcal{O}_X$-module is a quasi-coherent
    $\mathcal{O}_Y$-module. See
    Modules, Lemma \ref{modules-lemma-pullback-quasi-coherent}.
    \item Given two quasi-coherent $\mathcal{O}_X$-modules
    the tensor product is quasi-coherent, see
    Modules, Lemma \ref{modules-lemma-tensor-product-permanence}.
    \item Given a quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$
    the tensor, symmetric and exterior algebras on $\mathcal{F}$
    are quasi-coherent, see
    Modules, Lemma \ref{modules-lemma-whole-tensor-algebra-permanence}.
    \item Given two quasi-coherent $\mathcal{O}_X$-modules
    $\mathcal{F}$, $\mathcal{G}$ such that $\mathcal{F}$
    is of finite presentation, then the internal hom
    $\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$
    is quasi-coherent, see
    Modules, Lemma \ref{modules-lemma-internal-hom-locally-kernel-direct-sum}
    and (\ref{item-kernel}) above.
    \end{enumerate}
    On the other hand, it is in general not the case that the
    pushforward of a quasi-coherent module is quasi-coherent.
    Here is a case where this does hold.
    
    \begin{lemma}
    \label{lemma-push-forward-quasi-coherent}
    Let $f : X \to S$ be a morphism of schemes.
    If $f$ is quasi-compact and quasi-separated then
    $f_*$ transforms quasi-coherent $\mathcal{O}_X$-modules
    into quasi-coherent $\mathcal{O}_S$-modules.
    \end{lemma}
    
    \begin{proof}
    The question is local on $S$ and hence we may assume that
    $S$ is affine. Because $X$ is quasi-compact we may write
    $X = \bigcup_{i = 1}^n U_i$ with each $U_i$ open affine.
    Because $f$ is quasi-separated we may write
    $U_i \cap U_j = \bigcup_{k = 1}^{n_{ij}} U_{ijk}$ for some
    affine open $U_{ijk}$, see Lemma \ref{lemma-characterize-quasi-separated}.
    Denote $f_i : U_i \to S$ and $f_{ijk} : U_{ijk} \to S$ the
    restrictions of $f$. For any open $V$ of $S$ and any sheaf
    $\mathcal{F}$ on $X$ we have
    \begin{eqnarray*}
    f_*\mathcal{F}(V) & = & \mathcal{F}(f^{-1}V) \\
    & = &
    \Ker\left(
    \bigoplus\nolimits_i \mathcal{F}(f^{-1}V \cap U_i)
    \to
    \bigoplus\nolimits_{i, j, k} \mathcal{F}(f^{-1}V \cap U_{ijk})\right) \\
    & = &
    \Ker\left(
    \bigoplus\nolimits_i f_{i, *}(\mathcal{F}|_{U_i})(V)
    \to
    \bigoplus\nolimits_{i, j, k} f_{ijk, *}(\mathcal{F}|_{U_{ijk}})\right)(V) \\
    & = &
    \Ker\left(
    \bigoplus\nolimits_i f_{i, *}(\mathcal{F}|_{U_i})
    \to
    \bigoplus\nolimits_{i, j, k} f_{ijk, *}(\mathcal{F}|_{U_{ijk}})\right)(V)
    \end{eqnarray*}
    In other words there is an exact sequence of sheaves
    $$
    0 \to f_*\mathcal{F}
    \to \bigoplus f_{i, *}\mathcal{F}_i
    \to \bigoplus f_{ijk, *}\mathcal{F}_{ijk}
    $$
    where $\mathcal{F}_i, \mathcal{F}_{ijk}$ denotes the
    restriction of $\mathcal{F}$ to the corresponding open.
    If $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_X$-modules
    then $\mathcal{F}_i$, $\mathcal{F}_{ijk}$ is a quasi-coherent
    $\mathcal{O}_{U_i}$, $\mathcal{O}_{U_{ijk}}$-module.
    Hence by Lemma \ref{lemma-widetilde-pullback} we see that the second and
    third term of the exact sequence are quasi-coherent
    $\mathcal{O}_S$-modules. Thus we conclude that
    $f_*\mathcal{F}$ is a quasi-coherent $\mathcal{O}_S$-module.
    \end{proof}
    
    \noindent
    Using this we can characterize (closed) immersions of schemes
    as follows.
    
    \begin{lemma}
    \label{lemma-characterize-closed-immersions}
    Let $f : X \to Y$ be a morphism of schemes.
    Suppose that
    \begin{enumerate}
    \item $f$ induces a homeomorphism of $X$ with a
    closed subset of $Y$, and
    \item $f^\sharp : \mathcal{O}_Y \to f_*\mathcal{O}_X$
    is surjective.
    \end{enumerate}
    Then $f$ is a closed immersion of schemes.
    \end{lemma}
    
    \begin{proof}
    Assume (1) and (2). By (1) the morphism $f$ is quasi-compact
    (see Topology, Lemma \ref{topology-lemma-closed-in-quasi-compact}).
    Conditions (1) and (2) imply conditions (1) and (2) of
    Lemma \ref{lemma-injective-points-surjective-stalks}.
    Hence $f : X \to Y$ is a monomorphism. In particular, $f$
    is separated, see Lemma \ref{lemma-monomorphism-separated}.
    Hence Lemma \ref{lemma-push-forward-quasi-coherent}
    above applies and we conclude that $f_*\mathcal{O}_X$ is a quasi-coherent
    $\mathcal{O}_Y$-module. Therefore the kernel of
    $\mathcal{O}_Y \to f_*\mathcal{O}_X$ is quasi-coherent
    by Lemma \ref{lemma-extension-quasi-coherent}. Since a quasi-coherent
    sheaf is locally generated by sections (see
    Modules, Definition \ref{modules-definition-quasi-coherent})
    this implies that $f$ is a closed immersion, see
    Definition \ref{definition-closed-immersion-locally-ringed-spaces}.
    \end{proof}
    
    \noindent
    We can use this lemma to prove the following lemma.
    
    \begin{lemma}
    \label{lemma-composition-immersion}
    A composition of immersions of schemes is an immersion,
    a composition of closed immersions of schemes is a closed immersion, and
    a composition of open immersions of schemes is an open immersion.
    \end{lemma}
    
    \begin{proof}
    This is clear for the case of open immersions since an open subspace of
    an open subspace is also an open subspace.
    
    \medskip\noindent
    Suppose $a : Z \to Y$ and $b : Y \to X$ are closed immersions of schemes.
    We will verify that $c = b \circ a$ is also a closed immersion.
    The assumption implies that $a$ and $b$ are homeomorphisms onto closed subsets,
    and hence also $c = b \circ a$ is a homeomorphism onto a closed subset.
    Moreover, the map $\mathcal{O}_X \to c_*\mathcal{O}_Z$ is surjective
    since it factors as the composition of the surjective maps
    $\mathcal{O}_X \to b_*\mathcal{O}_Y$ and
    $b_*\mathcal{O}_Y \to b_*a_*\mathcal{O}_Z$ (surjective as $b_*$ is exact,
    see Modules, Lemma \ref{modules-lemma-i-star-exact}).
    Hence by Lemma \ref{lemma-characterize-closed-immersions}
    above $c$ is a closed immersion.
    
    \medskip\noindent
    Finally, we come to the case of immersions.
    Suppose $a : Z \to Y$ and $b : Y \to X$ are immersions of schemes.
    This means there exist open subschemes
    $V \subset Y$ and $U \subset X$ such that
    $a(Z) \subset V$, $b(Y) \subset U$ and $a : Z \to V$
    and $b : Y \to U$ are closed immersions.
    Since the topology on $Y$ is induced from the topology on
    $U$ we can find an open $U' \subset U$ such that
    $V = b^{-1}(U')$. Then we see that
    $Z \to V = b^{-1}(U') \to U'$ is a composition of
    closed immersions and hence a closed immersion.
    This proves that $Z \to X$ is an immersion and we win.
    \end{proof}
    
    
    
    
    
    
    
    
    
    
    
    \input{chapters}

    Comments (2)

    Comment #1719 by Keenan Kidwell on December 3, 2015 a 11:56 pm UTC

    In the second sentence of the text block following the list of properties of quasi-coherent modules, the word "it" should be removed.

    Comment #1758 by Johan (site) on December 15, 2015 a 7:06 pm UTC

    Thanks, fixed here.

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