# The Stacks Project

## Tag: 06WM

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Lemma 64.11.7. Let $p : \mathcal{X} \to (\textit{Sch}/S)_{fppf}$ be a category fibred in groupoids.
1. The category $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$ has colimits and they agree with colimits in the category $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal{X})$.
2. The category $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$ is abelian with kernels and cokernels computed in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal{X})$, in other words the inclusion functor is exact.
3. Given a short exact sequence $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ of $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal{X})$ if two out of three are locally quasi-coherent so is the third.
4. Given $\mathcal{F}, \mathcal{G}$ in $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$ the tensor product $\mathcal{F} \otimes_{\mathcal{O}_\mathcal{X}} \mathcal{G}$ in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal{X})$ is an object of $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$.
5. Given $\mathcal{F}, \mathcal{G}$ in $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$ with $\mathcal{F}$ locally of finite presentation on $\mathcal{X}_{\acute{e}tale}$ the sheaf $\mathop{\mathcal{H}\!{\it om}}\nolimits_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})$ in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal{X})$ is an object of $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$.

Proof. Each of these statements follows from the corresponding statement of Descent, Lemma 31.7.13. For example, suppose that $\mathcal{I} \to \textit{LQCoh}(\mathcal{O}_\mathcal{X})$, $i \mapsto \mathcal{F}_i$ is a diagram. Consider the object $\mathcal{F} = \mathop{\rm colim}\nolimits_i \mathcal{F}_i$ of $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal{X})$. 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 $x^*\mathcal{F} = \mathop{\rm colim}\nolimits_i x^*\mathcal{F}_i$. Similarly we have $x^*\mathcal{F}|_{U_{\acute{e}tale}} = \mathop{\rm colim}\nolimits_i x^*\mathcal{F}_i|_{U_{\acute{e}tale}}$. Now by assumption each $x^*\mathcal{F}_i|_{U_{\acute{e}tale}}$ is quasi-coherent, hence the colimit is quasi-coherent by the aforementioned Descent, Lemma 31.7.13. This proves (1).

It follows from (1) that cokernels exist in $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$ and agree with the cokernels computed in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal{X})$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$ and let $\mathcal{K} = \text{Ker}(\varphi)$ computed in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal{X})$. If we can show that $\mathcal{K}$ is a locally quasi-coherent module, then the proof of (2) is complete. To see this, note that kernels are computed in the category of presheaves (no sheafification necessary). Hence $\mathcal{K}|_{U_{\acute{e}tale}}$ is the kernel of the map $\mathcal{F}|_{U_{\acute{e}tale}} \to \mathcal{G}|_{U_{\acute{e}tale}}$, i.e., is the kernel of a map of quasi-coherent sheaves on $U_{\acute{e}tale}$ whence quasi-coherent by Descent, Lemma 31.7.13. This proves (2).

Parts (3), (4), and (5) follow in exactly the same way. Details omitted. $\square$

\begin{lemma}
\label{lemma-lqc-colimits}
Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in
groupoids.
\begin{enumerate}
\item The category $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$
has colimits and they agree with colimits in the category
$\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal{X})$.
\item The category $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$
is abelian with kernels and cokernels computed in
$\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal{X})$,
in other words the inclusion functor is exact.
\item Given a short exact sequence
$0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ of
$\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal{X})$
if two out of three are locally quasi-coherent so is the third.
\item Given $\mathcal{F}, \mathcal{G}$ in
$\textit{LQCoh}(\mathcal{O}_\mathcal{X})$
the tensor product $\mathcal{F} \otimes_{\mathcal{O}_\mathcal{X}} \mathcal{G}$
in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal{X})$
is an object of $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$.
\item Given $\mathcal{F}, \mathcal{G}$ in
$\textit{LQCoh}(\mathcal{O}_\mathcal{X})$
with $\mathcal{F}$ locally of finite presentation on
$\mathcal{X}_{\acute{e}tale}$ the sheaf
$\SheafHom_{\mathcal{O}_\mathcal{X}}(\mathcal{F}, \mathcal{G})$
in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal{X})$
is an object of $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$.
\end{enumerate}
\end{lemma}

\begin{proof}
Each of these statements follows from the corresponding statement of
Descent, Lemma \ref{descent-lemma-equivalence-quasi-coherent-limits}.
For example, suppose that
$\mathcal{I} \to \textit{LQCoh}(\mathcal{O}_\mathcal{X})$,
$i \mapsto \mathcal{F}_i$ is a diagram.
Consider the object $\mathcal{F} = \colim_i \mathcal{F}_i$ of
$\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal{X})$.
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
$x^*\mathcal{F} = \colim_i x^*\mathcal{F}_i$. Similarly we have
$x^*\mathcal{F}|_{U_{\acute{e}tale}} = \colim_i x^*\mathcal{F}_i|_{U_{\acute{e}tale}}$.
Now by assumption each $x^*\mathcal{F}_i|_{U_{\acute{e}tale}}$
is quasi-coherent, hence the colimit is quasi-coherent by the
aforementioned
Descent, Lemma \ref{descent-lemma-equivalence-quasi-coherent-limits}.
This proves (1).

\medskip\noindent
It follows from (1) that cokernels exist in
$\textit{LQCoh}(\mathcal{O}_\mathcal{X})$ and agree with the cokernels computed
in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal{X})$.
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of
$\textit{LQCoh}(\mathcal{O}_\mathcal{X})$ and let
$\mathcal{K} = \text{Ker}(\varphi)$ computed in
$\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal{X})$.
If we can show that $\mathcal{K}$ is a locally quasi-coherent module,
then the proof of (2) is complete. To see this, note that kernels
are computed in the category of presheaves (no sheafification necessary).
Hence $\mathcal{K}|_{U_{\acute{e}tale}}$ is the kernel of the map
$\mathcal{F}|_{U_{\acute{e}tale}} \to \mathcal{G}|_{U_{\acute{e}tale}}$,
i.e., is the kernel of a map of quasi-coherent sheaves on $U_{\acute{e}tale}$
whence quasi-coherent by
Descent, Lemma \ref{descent-lemma-equivalence-quasi-coherent-limits}.
This proves (2).

\medskip\noindent
Parts (3), (4), and (5) follow in exactly the same way. Details omitted.
\end{proof}


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