This tag has label schemes-section-quasi-coherent, it is called Functoriality for quasi-coherent modules in the Stacks project and it points to
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22.24. Functoriality for quasi-coherent modules
Let $X$ be a scheme. We denote $\textit{QCoh}(\mathcal{O}_X)$ or $\textit{QCoh}(X)$ the category of quasi-coherent $\mathcal{O}_X$-modules as defined in Modules, Definition 16.10.1. We have seen in Section 22.7 that the category $\textit{QCoh}(\mathcal{O}_X)$ 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.
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 it this does hold.
- 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 $U = \mathop{\rm Spec}(R)$ is of the form $\widetilde M$ for some $R$-module $M$.
- 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.
- Any direct sum of quasi-coherent sheaves is quasi-coherent.
- Any colimit of quasi-coherent sheaves is quasi-coherent.
- The kernel and cokernel of a morphism of quasi-coherent sheaves is quasi-coherent.
- 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.
- 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 16.10.4.
- Given two quasi-coherent $\mathcal{O}_X$-modules the tensor product is quasi-coherent, see Modules, Lemma 16.15.5.
- 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 16.18.6.
- Given two quasi-coherent $\mathcal{O}_X$-modules $\mathcal{F}$, $\mathcal{G}$ such that $\mathcal{F}$ is of finite presentation, then the internal hom $\mathop{\mathcal{H}\!{\it om}}\nolimits_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is quasi-coherent, see Modules, Lemma 16.19.4 and (5) above.
Lemma 22.24.1. 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 22.21.7. 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) \\ & = & \text{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) \\ & = & \text{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) \\ & = & \text{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 a short 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 22.7.3 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 22.24.2. Let $f : X \to Y$ be a morphism of schemes. Suppose that
Then $f$ is a closed immersion of schemes.
- $f$ induces a homeomorphism of $X$ with a closed subset of $Y$, and
- $f^\sharp : \mathcal{O}_Y \to f_*\mathcal{O}_X$ is surjective.
Proof. Assume (1) and (2). By (1) the morphism $f$ is quasi-compact (see Topology, Lemma 5.9.3). Conditions (1) and (2) imply conditions (1) and (2) of Lemma 22.23.6. Hence $f : X \to Y$ is a monomorphism. In particular, $f$ is separated, see Lemma 22.23.3. Hence Lemma 22.24.1 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 22.7.8. Since a quasi-coherent sheaf is locally generated by sections (see Modules, Definition 16.10.1) this implies that $f$ is a closed immersion, see Definition 22.4.1. $\square$
We can use this lemma to prove the following lemma.
Lemma 22.24.3. 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 16.6.1). Hence by Lemma 22.24.2 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. $\square$
\section{Functoriality for quasi-coherent modules}
\label{section-quasi-coherent}
\noindent
Let $X$ be a scheme. We denote $\textit{QCoh}(\mathcal{O}_X)$ or
$\textit{QCoh}(X)$ 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 $\textit{QCoh}(\mathcal{O}_X)$ 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 it 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) \\
& = &
\text{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) \\
& = &
\text{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) \\
& = &
\text{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 a short 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}
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