# The Stacks Project

## Tag: 06WN

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Lemma 64.11.8. Let $p : \mathcal{X} \to (\textit{Sch}/S)_{fppf}$ be a category fibred in groupoids.
1. The category $\textit{QCoh}(\mathcal{O}_\mathcal{X})$ has colimits and they agree with colimits in the category $\textit{Mod}(\mathcal{O}_\mathcal{X})$ as well as with colimits in the category $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$.
2. Given $\mathcal{F}, \mathcal{G}$ in $\textit{QCoh}(\mathcal{O}_\mathcal{X})$ the tensor product $\mathcal{F} \otimes_{\mathcal{O}_\mathcal{X}} \mathcal{G}$ in $\textit{Mod}(\mathcal{O}_\mathcal{X})$ is an object of $\textit{QCoh}(\mathcal{O}_\mathcal{X})$.
3. Given $\mathcal{F}, \mathcal{G}$ in $\textit{QCoh}(\mathcal{O}_\mathcal{X})$ with $\mathcal{F}$ locally of finite presentation on $\mathcal{X}_{fppf}$ the sheaf $\mathop{\mathcal{H}\!{\it om}}\nolimits_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})$ in $\textit{Mod}(\mathcal{O}_\mathcal{X})$ is an object of $\textit{QCoh}(\mathcal{O}_\mathcal{X})$.

Proof. Let $\mathcal{I} \to \textit{QCoh}(\mathcal{O}_\mathcal{X})$, $i \mapsto \mathcal{F}_i$ be a diagram. Consider the object $\mathcal{F} = \mathop{\rm colim}\nolimits_i \mathcal{F}_i$ of $\textit{Mod}(\mathcal{O}_\mathcal{X})$. For any object $x$ of $\mathcal{X}$ with $U = p(x)$ the pullback functor $x^*$ commutes with all colimits as it is a left adjoint. Hence $x^*\mathcal{F} = \mathop{\rm colim}\nolimits_i x^*\mathcal{F}_i$ in $\textit{Mod}((\textit{Sch}/U)_{fppf}, \mathcal{O})$. We conclude from Descent, Lemma 31.7.13 that $x^*\mathcal{F}$ is quasi-coherent, hence $\mathcal{F}$ is quasi-coherent, see Lemma 64.11.3. Thus we see that $\textit{QCoh}(\mathcal{O}_\mathcal{X})$ has colimits and they agree with colimits in the category $\textit{Mod}(\mathcal{O}_\mathcal{X})$. In particular the (fppf) sheaf $\mathcal{F}$ is also the colimit of the diagram in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal{X})$, hence $\mathcal{F}$ is also the colimit in $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$. This proves (1).

Parts (2) and (3) are proved in the same way. Details omitted. $\square$

\begin{lemma}
\label{lemma-qc-colimits}
Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category
fibred in groupoids.
\begin{enumerate}
\item The category $\textit{QCoh}(\mathcal{O}_\mathcal{X})$
has colimits and they agree with colimits in the category
$\textit{Mod}(\mathcal{O}_\mathcal{X})$ as well as with colimits
in the category $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$.
\item Given $\mathcal{F}, \mathcal{G}$ in
$\textit{QCoh}(\mathcal{O}_\mathcal{X})$
the tensor product $\mathcal{F} \otimes_{\mathcal{O}_\mathcal{X}} \mathcal{G}$
in $\textit{Mod}(\mathcal{O}_\mathcal{X})$
is an object of $\textit{QCoh}(\mathcal{O}_\mathcal{X})$.
\item Given $\mathcal{F}, \mathcal{G}$ in
$\textit{QCoh}(\mathcal{O}_\mathcal{X})$
with $\mathcal{F}$ locally of finite presentation on
$\mathcal{X}_{fppf}$ the sheaf
$\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})$
in $\textit{Mod}(\mathcal{O}_\mathcal{X})$
is an object of $\textit{QCoh}(\mathcal{O}_\mathcal{X})$.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $\mathcal{I} \to \textit{QCoh}(\mathcal{O}_\mathcal{X})$,
$i \mapsto \mathcal{F}_i$ be a diagram.
Consider the object $\mathcal{F} = \colim_i \mathcal{F}_i$ of
$\textit{Mod}(\mathcal{O}_\mathcal{X})$.
For any object $x$ of $\mathcal{X}$ with $U = p(x)$ the pullback functor
$x^*$ commutes with all colimits as it is a left adjoint. Hence
$x^*\mathcal{F} = \colim_i x^*\mathcal{F}_i$ in
$\textit{Mod}((\Sch/U)_{fppf}, \mathcal{O})$. We conclude from
Descent, Lemma \ref{descent-lemma-equivalence-quasi-coherent-limits}
that $x^*\mathcal{F}$ is quasi-coherent, hence $\mathcal{F}$
is quasi-coherent, see
Lemma \ref{lemma-characterize-quasi-coherent}.
Thus we see that $\textit{QCoh}(\mathcal{O}_\mathcal{X})$
has colimits and they agree with colimits in the category
$\textit{Mod}(\mathcal{O}_\mathcal{X})$. In particular the (fppf) sheaf
$\mathcal{F}$ is also the colimit of the diagram in
$\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal{X})$,
hence $\mathcal{F}$ is also the colimit in
$\textit{LQCoh}(\mathcal{O}_\mathcal{X})$. This proves (1).

\medskip\noindent
Parts (2) and (3) are proved in the same way.
Details omitted.
\end{proof}


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