# The Stacks Project

## Tag: 06WI

This tag has label stacks-sheaves-lemma-characterize-quasi-coherent and it points to

The corresponding content:

Lemma 65.11.3. Let $p : \mathcal{X} \to (\textit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal{X}$-modules. Then $\mathcal{F}$ is quasi-coherent if and only if $x^*\mathcal{F}$ is a quasi-coherent sheaf on $(\textit{Sch}/U)_{fppf}$ for every object $x$ of $\mathcal{X}$ with $U = p(x)$.

Proof. By Lemma 65.11.2 the condition is necessary. Conversely, since $x^*\mathcal{F}$ is just the restriction to $\mathcal{X}_{fppf}/x$ we see that it is sufficient directly from the definition of a quasi-coherent sheaf (and the fact that the notion of being quasi-coherent is an intrinsic property of sheaves of modules, see Modules on Sites, Section 17.18). $\square$

\begin{lemma}
\label{lemma-characterize-quasi-coherent}
Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category
fibred in groupoids. Let $\mathcal{F}$
be a sheaf of $\mathcal{O}_\mathcal{X}$-modules. Then $\mathcal{F}$
is quasi-coherent if and only if $x^*\mathcal{F}$ is a quasi-coherent
sheaf on $(\Sch/U)_{fppf}$ for every object $x$ of
$\mathcal{X}$ with $U = p(x)$.
\end{lemma}

\begin{proof}
By
Lemma \ref{lemma-pullback-quasi-coherent}
the condition is necessary. Conversely, since $x^*\mathcal{F}$
is just the restriction to $\mathcal{X}_{fppf}/x$ we see that it
is sufficient directly from the definition of a quasi-coherent sheaf
(and the fact that the notion of being quasi-coherent is an intrinsic
property of sheaves of modules, see
Modules on Sites, Section \ref{sites-modules-section-intrinsic}).
\end{proof}


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