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Tag: 06WK

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Lemma 65.11.5. Let $p : \mathcal{X} \to (\textit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_\mathcal{X}$-modules. Then $\mathcal{F}$ is quasi-coherent if and only if the following two conditions hold
1. $\mathcal{F}$ is locally quasi-coherent, and
2. for any morphism $\varphi : x \to y$ of $\mathcal{X}$ lying over $f : U \to V$ the comparison map $c_\varphi : f_{small}^*\mathcal{F}|_{V_{\acute{e}tale}} \to \mathcal{F}|_{U_{\acute{e}tale}}$ of (65.9.4.1) is an isomorphism.

Proof. Assume $\mathcal{F}$ is quasi-coherent. Then $\mathcal{F}$ is a sheaf for the fppf topology, hence a sheaf for the étale topology. Moreover, any pullback of $\mathcal{F}$ to a ringed topos is quasi-coherent, hence the restrictions $x^*\mathcal{F}|_{U_{\acute{e}tale}}$ are quasi-coherent. This proves $\mathcal{F}$ is locally quasi-coherent. Let $y$ be an object of $\mathcal{X}$ with $V = p(y)$. We have seen that $\mathcal{X}/y = (\textit{Sch}/V)_{fppf}$. By Descent, Proposition 31.7.11 it follows that $y^*\mathcal{F}$ is the quasi-coherent module associated to a (usual) quasi-coherent module $\mathcal{F}_V$ on the scheme $V$. Hence certainly the comparison maps (65.9.4.1) are isomorphisms.

Conversely, suppose that $\mathcal{F}$ satisfies (1) and (2). Let $y$ be an object of $\mathcal{X}$ with $V = p(y)$. Denote $\mathcal{F}_V$ the quasi-coherent module on the scheme $V$ corresponding to the restriction $y^*\mathcal{F}|_{V_{\acute{e}tale}}$ which is quasi-coherent by assumption (1), see Descent, Proposition 31.7.11. Condition (2) now signifies that the restrictions $x^*\mathcal{F}|_{U_{\acute{e}tale}}$ for $x$ over $y$ are each isomorphic to the (étale sheaf associated to the) pullback of $\mathcal{F}_V$ via the corresponding morphism of schemes $U \to V$. Hence $y^*\mathcal{F}$ is the sheaf on $(\textit{Sch}/V)_{fppf}$ associated to $\mathcal{F}_V$. Hence it is quasi-coherent (by Descent, Proposition 31.7.11 again) and we see that $\mathcal{F}$ is quasi-coherent on $\mathcal{X}$ by Lemma 65.11.3. $\square$

\begin{lemma}
\label{lemma-quasi-coherent}
Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category
fibred in groupoids. Let $\mathcal{F}$
be a presheaf of $\mathcal{O}_\mathcal{X}$-modules. Then $\mathcal{F}$
is quasi-coherent if and only if the following two conditions hold
\begin{enumerate}
\item $\mathcal{F}$ is locally quasi-coherent, and
\item for any morphism $\varphi : x \to y$ of $\mathcal{X}$ lying over
$f : U \to V$ the comparison map
$c_\varphi : f_{small}^*\mathcal{F}|_{V_\etale} \to \mathcal{F}|_{U_\etale}$ of
(\ref{equation-comparison-modules}) is an isomorphism.
\end{enumerate}
\end{lemma}

\begin{proof}
Assume $\mathcal{F}$ is quasi-coherent. Then $\mathcal{F}$ is a sheaf
for the fppf topology, hence a sheaf for the \'etale topology. Moreover,
any pullback of $\mathcal{F}$ to a ringed topos is quasi-coherent, hence
the restrictions $x^*\mathcal{F}|_{U_\etale}$ are quasi-coherent.
This proves $\mathcal{F}$ is locally quasi-coherent.
Let $y$ be an object of $\mathcal{X}$ with $V = p(y)$.
We have seen that $\mathcal{X}/y = (\Sch/V)_{fppf}$. By
Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent}
it follows that $y^*\mathcal{F}$ is the quasi-coherent module
associated to a (usual) quasi-coherent module $\mathcal{F}_V$ on
the scheme $V$. Hence certainly the comparison maps
(\ref{equation-comparison-modules}) are isomorphisms.

\medskip\noindent
Conversely, suppose that $\mathcal{F}$ satisfies (1) and (2).
Let $y$ be an object of $\mathcal{X}$ with $V = p(y)$. Denote
$\mathcal{F}_V$ the quasi-coherent module on
the scheme $V$ corresponding to the restriction
$y^*\mathcal{F}|_{V_\etale}$ which is quasi-coherent by
assumption (1), see
Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent}.
Condition (2) now signifies that the restrictions
$x^*\mathcal{F}|_{U_\etale}$ for $x$ over $y$ are each
isomorphic to the (\'etale sheaf associated to the) pullback of $\mathcal{F}_V$
via the corresponding morphism of schemes $U \to V$.
Hence $y^*\mathcal{F}$ is the sheaf on $(\Sch/V)_{fppf}$
associated to $\mathcal{F}_V$. Hence it is quasi-coherent (by
Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent}
again) and we see that $\mathcal{F}$ is quasi-coherent on $\mathcal{X}$ by
Lemma \ref{lemma-characterize-quasi-coherent}.
\end{proof}


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