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Tag 008X

Chapter 6: Sheaves on Spaces > Section 6.24: Continuous maps and sheaves of modules

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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