# The Stacks Project

## Tag: 06WH

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

The corresponding content:

Lemma 64.11.2. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\textit{Sch}/S)_{fppf}$. The pullback functor $f^* = f^{-1} : \textit{Mod}(\mathcal{O}_\mathcal{Y}) \to \textit{Mod}(\mathcal{O}_\mathcal{X})$ preserves quasi-coherent sheaves.

Proof. This is a general fact, see Modules on Sites, Lemma 17.23.4. $\square$

\begin{lemma}
\label{lemma-pullback-quasi-coherent}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories
fibred in groupoids over $(\Sch/S)_{fppf}$.
The pullback functor
$f^* = f^{-1} : \textit{Mod}(\mathcal{O}_\mathcal{Y}) \to \textit{Mod}(\mathcal{O}_\mathcal{X})$
preserves quasi-coherent sheaves.
\end{lemma}

\begin{proof}
This is a general fact, see
Modules on Sites, Lemma \ref{sites-modules-lemma-local-pullback}.
\end{proof}


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