The Stacks Project


Tag: 06WM

This tag has label stacks-sheaves-lemma-lqc-colimits and it points to

The corresponding content:

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}

To cite this tag (see How to reference tags), use: 

\cite[\href{http://stacks.math.columbia.edu/tag/06WM}{Tag 06WM}]{stacks-project}

Comments (0)

There are no comments yet for this tag.

Add a comment on tag 06WM

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).




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 is tag 0321 you just have to write 0321. This captcha seems more appropriate than the usual illegible gibberish, right?