## Tag `07AZ`

Chapter 94: Cohomology of Algebraic Stacks > Section 94.12: Quasi-coherent modules, II

Lemma 94.12.1. Let $\mathcal{X}$ be an algebraic stack.

- Let $f_j : \mathcal{X}_j \to \mathcal{X}$ be a family of smooth morphisms of algebraic stacks with $|\mathcal{X}| =\bigcup |f_j|(|\mathcal{X}_j|)$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal{X}$-modules on $\mathcal{X}_{\acute{e}tale}$. If each $f_j^{-1}\mathcal{F}$ is quasi-coherent, then so is $\mathcal{F}$.
- Let $f_j : \mathcal{X}_j \to \mathcal{X}$ be a family of flat and locally finitely presented morphisms of algebraic stacks with $|\mathcal{X}| =\bigcup |f_j|(|\mathcal{X}_j|)$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal{X}$-modules on $\mathcal{X}_{fppf}$. If each $f_j^{-1}\mathcal{F}$ is quasi-coherent, then so is $\mathcal{F}$.

Proof.Proof of (1). We may replace each of the algebraic stacks $\mathcal{X}_j$ by a scheme $U_j$ (using that any algebraic stack has a smooth covering by a scheme and that compositions of smooth morphisms are smooth, see Morphisms of Stacks, Lemma 92.32.2). The pullback of $\mathcal{F}$ to $(\mathit{Sch}/U_j)_{\acute{e}tale}$ is still locally quasi-coherent, see Sheaves on Stacks, Lemma 87.11.2. Then $f = \coprod f_j : U = \coprod U_j \to \mathcal{X}$ is a smooth surjective morphism. Let $x : V \to \mathcal{X}$ be an object of $\mathcal{X}$. By Sheaves on Stacks, Lemma 87.18.10 there exists an étale covering $\{x_i \to x\}_{i \in I}$ such that each $x_i$ lifts to an object $u_i$ of $(\mathit{Sch}/U)_{\acute{e}tale}$. This just means that $x_i$ lives over a scheme $V_i$, that $\{V_i \to V\}$ is an étale covering, and that $x_i$ comes from a morphism $u_i : V_i \to U$. Then $x_i^*\mathcal{F} = u_i^*f^*\mathcal{F}$ is quasi-coherent. This implies that $x^*\mathcal{F}$ on $(\mathit{Sch}/V)_{\acute{e}tale}$ is quasi-coherent, for example by Modules on Sites, Lemma 18.23.3. By Sheaves on Stacks, Lemma 87.11.3 we see that $\mathcal{F}$ is quasi-coherent.Proof of (2). This is proved using exactly the same argument, which we fully write out here. We may replace each of the algebraic stacks $\mathcal{X}_j$ by a scheme $U_j$ (using that any algebraic stack has a smooth covering by a scheme and that flat and locally finite presented morphisms are preserved by composition, see Morphisms of Stacks, Lemmas 92.24.2 and 92.26.2). The pullback of $\mathcal{F}$ to $(\mathit{Sch}/U_j)_{\acute{e}tale}$ is still locally quasi-coherent, see Sheaves on Stacks, Lemma 87.11.2. Then $f = \coprod f_j : U = \coprod U_j \to \mathcal{X}$ is a surjective, flat, and locally finitely presented morphism. Let $x : V \to \mathcal{X}$ be an object of $\mathcal{X}$. By Sheaves on Stacks, Lemma 87.18.10 there exists an fppf covering $\{x_i \to x\}_{i \in I}$ such that each $x_i$ lifts to an object $u_i$ of $(\mathit{Sch}/U)_{\acute{e}tale}$. This just means that $x_i$ lives over a scheme $V_i$, that $\{V_i \to V\}$ is an fppf covering, and that $x_i$ comes from a morphism $u_i : V_i \to U$. Then $x_i^*\mathcal{F} = u_i^*f^*\mathcal{F}$ is quasi-coherent. This implies that $x^*\mathcal{F}$ on $(\mathit{Sch}/V)_{\acute{e}tale}$ is quasi-coherent, for example by Modules on Sites, Lemma 18.23.3. By Sheaves on Stacks, Lemma 87.11.3 we see that $\mathcal{F}$ is quasi-coherent. $\square$

The code snippet corresponding to this tag is a part of the file `stacks-cohomology.tex` and is located in lines 2136–2153 (see updates for more information).

```
\begin{lemma}
\label{lemma-check-qc-on-etale-covering}
Let $\mathcal{X}$ be an algebraic stack.
\begin{enumerate}
\item Let $f_j : \mathcal{X}_j \to \mathcal{X}$ be a family of smooth
morphisms of algebraic stacks with
$|\mathcal{X}| =\bigcup |f_j|(|\mathcal{X}_j|)$.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal{X}$-modules
on $\mathcal{X}_\etale$. If each $f_j^{-1}\mathcal{F}$
is quasi-coherent, then so is $\mathcal{F}$.
\item Let $f_j : \mathcal{X}_j \to \mathcal{X}$ be a family of flat and
locally finitely presented morphisms of algebraic stacks with
$|\mathcal{X}| =\bigcup |f_j|(|\mathcal{X}_j|)$.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal{X}$-modules
on $\mathcal{X}_{fppf}$. If each $f_j^{-1}\mathcal{F}$
is quasi-coherent, then so is $\mathcal{F}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Proof of (1). We may replace each of the algebraic stacks $\mathcal{X}_j$
by a scheme $U_j$ (using that any algebraic stack has a smooth covering by
a scheme and that compositions of smooth morphisms are smooth, see
Morphisms of Stacks, Lemma \ref{stacks-morphisms-lemma-composition-smooth}).
The pullback of $\mathcal{F}$ to $(\Sch/U_j)_\etale$ is still
locally quasi-coherent, see
Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-pullback-quasi-coherent}.
Then $f = \coprod f_j : U = \coprod U_j \to \mathcal{X}$ is a smooth surjective
morphism. Let $x : V \to \mathcal{X}$ be an object of $\mathcal{X}$. By
Sheaves on Stacks, Lemma
\ref{stacks-sheaves-lemma-surjective-flat-locally-finite-presentation}
there exists an \'etale covering $\{x_i \to x\}_{i \in I}$
such that each $x_i$ lifts to an object $u_i$ of $(\Sch/U)_\etale$.
This just means that $x_i$ lives over a scheme $V_i$, that
$\{V_i \to V\}$ is an \'etale covering, and that $x_i$ comes from
a morphism $u_i : V_i \to U$. Then
$x_i^*\mathcal{F} = u_i^*f^*\mathcal{F}$ is quasi-coherent.
This implies that $x^*\mathcal{F}$ on $(\Sch/V)_\etale$
is quasi-coherent, for example by
Modules on Sites, Lemma \ref{sites-modules-lemma-local-final-object}.
By Sheaves on Stacks, Lemma
\ref{stacks-sheaves-lemma-characterize-quasi-coherent}
we see that $\mathcal{F}$ is quasi-coherent.
\medskip\noindent
Proof of (2). This is proved using exactly the same argument, which we fully
write out here. We may replace each of the algebraic stacks $\mathcal{X}_j$
by a scheme $U_j$ (using that any algebraic stack has a smooth covering by
a scheme and that flat and locally finite presented morphisms are preserved
by composition, see Morphisms of Stacks, Lemmas
\ref{stacks-morphisms-lemma-composition-flat} and
\ref{stacks-morphisms-lemma-composition-finite-presentation}).
The pullback of $\mathcal{F}$ to $(\Sch/U_j)_\etale$ is still
locally quasi-coherent, see
Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-pullback-quasi-coherent}.
Then $f = \coprod f_j : U = \coprod U_j \to \mathcal{X}$ is a surjective,
flat, and locally finitely presented morphism. Let
$x : V \to \mathcal{X}$ be an object of $\mathcal{X}$. By
Sheaves on Stacks, Lemma
\ref{stacks-sheaves-lemma-surjective-flat-locally-finite-presentation}
there exists an fppf covering $\{x_i \to x\}_{i \in I}$
such that each $x_i$ lifts to an object $u_i$ of $(\Sch/U)_\etale$.
This just means that $x_i$ lives over a scheme $V_i$, that
$\{V_i \to V\}$ is an fppf covering, and that $x_i$ comes from
a morphism $u_i : V_i \to U$. Then
$x_i^*\mathcal{F} = u_i^*f^*\mathcal{F}$ is quasi-coherent.
This implies that $x^*\mathcal{F}$ on $(\Sch/V)_\etale$
is quasi-coherent, for example by
Modules on Sites, Lemma \ref{sites-modules-lemma-local-final-object}.
By Sheaves on Stacks, Lemma
\ref{stacks-sheaves-lemma-characterize-quasi-coherent}
we see that $\mathcal{F}$ is quasi-coherent.
\end{proof}
```

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