# The Stacks Project

## Tag: 06WL

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

The corresponding content:

Lemma 65.11.6. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\textit{Sch}/S)_{fppf}$. The pullback functor $f^* = f^{-1} : \textit{Mod}(\mathcal{Y}_{\acute{e}tale}, \mathcal{O}_\mathcal{Y}) \to \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal{X})$ preserves locally quasi-coherent sheaves.

Proof. Let $\mathcal{G}$ be locally quasi-coherent on $\mathcal{Y}$. Choose an object $x$ of $\mathcal{X}$ lying over the scheme $U$. The restriction $x^*f^*\mathcal{G}|_{U_{\acute{e}tale}}$ equals $(f \circ x)^*\mathcal{G}|_{U_{\acute{e}tale}}$ hence is a quasi-coherent sheaf by assumption on $\mathcal{G}$. $\square$

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

\begin{proof}
Let $\mathcal{G}$ be locally quasi-coherent on $\mathcal{Y}$.
Choose an object $x$ of $\mathcal{X}$ lying over the scheme $U$.
The restriction $x^*f^*\mathcal{G}|_{U_\etale}$ equals
$(f \circ x)^*\mathcal{G}|_{U_\etale}$
hence is a quasi-coherent sheaf by assumption on $\mathcal{G}$.
\end{proof}


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