# The Stacks Project

## Tag 07B1

Lemma 94.12.3. Let $\mathcal{X}$ be an algebraic stack. Let $\mathcal{M}_\mathcal{X}$ be the category of locally quasi-coherent $\mathcal{O}_\mathcal{X}$-modules with the flat base change property.

1. With $g$ as in Lemma 94.11.2 for the lisse-étale site we have
1. the functors $g^{-1}$ and $g_!$ define mutually inverse functors $$\xymatrix{ \mathit{QCoh}(\mathcal{O}_\mathcal{X}) \ar@<1ex>[r]^-{g^{-1}} & \mathit{QCoh}(\mathcal{X}_{lisse,{\acute{e}tale}}, \mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}) \ar@<1ex>[l]^-{g_!} }$$
2. if $\mathcal{F}$ is in $\mathcal{M}_\mathcal{X}$ then $g^{-1}\mathcal{F}$ is in $\mathit{QCoh}(\mathcal{X}_{lisse,{\acute{e}tale}}, \mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$ and
3. $Q(\mathcal{F}) = g_!g^{-1}\mathcal{F}$ where $Q$ is as in Lemma 94.9.1.
2. With $g$ as in Lemma 94.11.2 for the flat-fppf site we have
1. the functors $g^{-1}$ and $g_!$ define mutually inverse functors $$\xymatrix{ \mathit{QCoh}(\mathcal{O}_\mathcal{X}) \ar@<1ex>[r]^-{g^{-1}} & \mathit{QCoh}(\mathcal{X}_{flat,fppf}, \mathcal{O}_{\mathcal{X}_{flat,fppf}}) \ar@<1ex>[l]^-{g_!} }$$
2. if $\mathcal{F}$ is in $\mathcal{M}_\mathcal{X}$ then $g^{-1}\mathcal{F}$ is in $\mathit{QCoh}(\mathcal{X}_{flat,fppf}, \mathcal{O}_{\mathcal{X}_{flat,fppf}})$ and
3. $Q(\mathcal{F}) = g_!g^{-1}\mathcal{F}$ where $Q$ is as in Lemma 94.9.1.

Proof. Pullback by any morphism of ringed topoi preserves categories of quasi-coherent modules, see Modules on Sites, Lemma 18.23.4. Hence $g^{-1}$ preserves the categories of quasi-coherent modules. The same is true for $g_!$ by Lemma 94.12.2. We know that $\mathcal{H} \to g^{-1}g_!\mathcal{H}$ is an isomorphism by Lemma 94.11.2. Conversely, if $\mathcal{F}$ is in $\mathit{QCoh}(\mathcal{O}_\mathcal{X})$ then the map $g_!g^{-1}\mathcal{F} \to \mathcal{F}$ is a map of quasi-coherent modules on $\mathcal{X}$ whose restriction to any scheme smooth over $\mathcal{X}$ is an isomorphism. Then the discussion in Sheaves on Stacks, Sections 87.13 and 87.14 (comparing with quasi-coherent modules on presentations) shows it is an isomorphism. This proves (1)(a) and (2)(a).

Let $\mathcal{F}$ be an object of $\mathcal{M}_\mathcal{X}$. By Lemma 94.9.2 the kernel and cokernel of the map $Q(\mathcal{F}) \to \mathcal{F}$ are parasitic. Hence by Lemma 94.11.5 and since $g^* = g^{-1}$ is exact, we conclude $g^*Q(\mathcal{F}) \to g^*\mathcal{F}$ is an isomorphism. Thus $g^*\mathcal{F}$ is quasi-coherent. This proves (1)(b) and (2)(b). Finally, (1)(c) and (2)(c) follow because $g_!g^*Q(\mathcal{F}) \to Q(\mathcal{F})$ is an isomorphism by our arguments above. $\square$

The code snippet corresponding to this tag is a part of the file stacks-cohomology.tex and is located in lines 2271–2314 (see updates for more information).

\begin{lemma}
\label{lemma-quasi-coherent}
Let $\mathcal{X}$ be an algebraic stack. Let $\mathcal{M}_\mathcal{X}$
be the category of locally quasi-coherent $\mathcal{O}_\mathcal{X}$-modules
with the flat base change property.
\begin{enumerate}
\item With $g$ as in Lemma \ref{lemma-lisse-etale}
for the lisse-\'etale site we have
\begin{enumerate}
\item the functors $g^{-1}$ and $g_!$ define mutually inverse functors
$$\xymatrix{ \QCoh(\mathcal{O}_\mathcal{X}) \ar@<1ex>[r]^-{g^{-1}} & \QCoh(\mathcal{X}_{lisse,\etale}, \mathcal{O}_{\mathcal{X}_{lisse,\etale}}) \ar@<1ex>[l]^-{g_!} }$$
\item if $\mathcal{F}$ is in $\mathcal{M}_\mathcal{X}$
then $g^{-1}\mathcal{F}$ is in
$\QCoh(\mathcal{X}_{lisse,\etale}, \mathcal{O}_{\mathcal{X}_{lisse,\etale}})$ and
\item $Q(\mathcal{F}) = g_!g^{-1}\mathcal{F}$ where $Q$ is as in
\end{enumerate}
\item With $g$ as in Lemma \ref{lemma-lisse-etale}
for the flat-fppf site we have
\begin{enumerate}
\item the functors $g^{-1}$ and $g_!$ define mutually inverse functors
$$\xymatrix{ \QCoh(\mathcal{O}_\mathcal{X}) \ar@<1ex>[r]^-{g^{-1}} & \QCoh(\mathcal{X}_{flat,fppf}, \mathcal{O}_{\mathcal{X}_{flat,fppf}}) \ar@<1ex>[l]^-{g_!} }$$
\item if $\mathcal{F}$ is in $\mathcal{M}_\mathcal{X}$
then $g^{-1}\mathcal{F}$ is in
$\QCoh(\mathcal{X}_{flat,fppf}, \mathcal{O}_{\mathcal{X}_{flat,fppf}})$
and
\item $Q(\mathcal{F}) = g_!g^{-1}\mathcal{F}$ where $Q$ is as in
\end{enumerate}
\end{enumerate}
\end{lemma}

\begin{proof}
Pullback by any morphism of ringed topoi preserves categories of quasi-coherent
modules, see
Modules on Sites, Lemma \ref{sites-modules-lemma-local-pullback}.
Hence $g^{-1}$ preserves the categories of quasi-coherent modules.
The same is true for $g_!$ by
Lemma \ref{lemma-shriek-quasi-coherent}.
We know that $\mathcal{H} \to g^{-1}g_!\mathcal{H}$ is an isomorphism by
Lemma \ref{lemma-lisse-etale}.
Conversely, if $\mathcal{F}$ is in $\QCoh(\mathcal{O}_\mathcal{X})$
then the map $g_!g^{-1}\mathcal{F} \to \mathcal{F}$ is a map of quasi-coherent
modules on $\mathcal{X}$ whose restriction to any scheme smooth over
$\mathcal{X}$ is an isomorphism. Then the discussion in
Sheaves on Stacks, Sections
\ref{stacks-sheaves-section-quasi-coherent-presentation} and
\ref{stacks-sheaves-section-quasi-coherent-algebraic-stacks}
(comparing with quasi-coherent modules on presentations)
shows it is an isomorphism. This proves (1)(a) and (2)(a).

\medskip\noindent
Let $\mathcal{F}$ be an object of $\mathcal{M}_\mathcal{X}$. By
the kernel and cokernel of the map
$Q(\mathcal{F}) \to \mathcal{F}$ are parasitic. Hence by
Lemma \ref{lemma-parasitic-in-terms-flat-fppf}
and since $g^* = g^{-1}$ is exact, we conclude
$g^*Q(\mathcal{F}) \to g^*\mathcal{F}$ is an isomorphism. Thus
$g^*\mathcal{F}$ is quasi-coherent. This proves (1)(b) and (2)(b).
Finally, (1)(c) and (2)(c) follow because
$g_!g^*Q(\mathcal{F}) \to Q(\mathcal{F})$ is an isomorphism by
our arguments above.
\end{proof}

Comment #3192 by anonymous on February 11, 2018 a 8:48 am UTC

Concerning the first two sentences of the proof: The functor $g^{-1}$ has source $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_{\mathcal{X}})$ and not $\textit{Mod}(\mathcal{O}_\mathcal{X})$. What is the relation between quasi-coherent $\mathcal{O}_\mathcal{X}$-modules and quasi-coherent $\mathcal{O}_{\mathcal{X}_{{\acute{e}tale}}}$-modules?

Comment #3196 by Johan (site) on February 11, 2018 a 3:20 pm UTC

Please see comment #3195

There are also 2 comments on Section 94.12: Cohomology of Algebraic Stacks.

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