# The Stacks Project

## Tag 008X

Lemma 6.24.6. Let $f : X \to Y$ be a continuous map of topological spaces. Let $\mathcal{O}$ be a sheaf of rings on $Y$. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}$-modules. There is a natural map of underlying presheaves of sets $$f^{-1}\mathcal{O} \times f^{-1}\mathcal{G} \longrightarrow f^{-1}\mathcal{G}$$ which turns $f^{-1}\mathcal{G}$ into a sheaf of $f^{-1}\mathcal{O}$-modules.

Proof. Recall that $f^{-1}$ is defined as the composition of the functor $f_p$ and sheafification. Thus the lemma is a combination of Lemma 6.24.2 and Lemma 6.20.1. $\square$

The code snippet corresponding to this tag is a part of the file sheaves.tex and is located in lines 2828–2841 (see updates for more information).

\begin{lemma}
\label{lemma-pullback-module}
Let $f : X \to Y$ be a continuous map of topological spaces.
Let $\mathcal{O}$ be a sheaf of rings on $Y$. Let
$\mathcal{G}$ be a sheaf of $\mathcal{O}$-modules.
There is a natural map of underlying presheaves of sets
$$f^{-1}\mathcal{O} \times f^{-1}\mathcal{G} \longrightarrow f^{-1}\mathcal{G}$$
which turns $f^{-1}\mathcal{G}$ into a
sheaf of $f^{-1}\mathcal{O}$-modules.
\end{lemma}

\begin{proof}
Recall that $f^{-1}$ is defined as the composition of the
functor $f_p$ and sheafification. Thus the lemma
is a combination of Lemma \ref{lemma-pullback-presheaf-module}
and Lemma \ref{lemma-sheafification-presheaf-modules}.
\end{proof}

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