Lemma 6.24.3. Let $f : X \to Y$ be a continuous map of topological spaces. Let $\mathcal{O}$ be a presheaf of rings on $Y$. Let $\mathcal{G}$ be a presheaf of $\mathcal{O}$-modules. Let $\mathcal{F}$ be a presheaf of $f_ p\mathcal{O}$-modules. Then

$\mathop{\mathrm{Mor}}\nolimits _{\textit{PMod}(f_ p\mathcal{O})}(f_ p\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PMod}(\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}).$

Here we use Lemmas 6.24.2 and 6.24.1, and we think of $f_*\mathcal{F}$ as an $\mathcal{O}$-module via the map $i_\mathcal {O} : \mathcal{O} \to f_*f_ p\mathcal{O}$ (defined first in the proof of Lemma 6.21.3).

Proof. Note that we have

$\mathop{\mathrm{Mor}}\nolimits _{\textit{PAb}(X)}(f_ p\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PAb}(Y)}(\mathcal{G}, f_*\mathcal{F}).$

according to Section 6.22. So what we have to prove is that under this correspondence, the subsets of module maps correspond. In addition, the correspondence is determined by the rule

$(\psi : f_ p\mathcal{G} \to \mathcal{F}) \longmapsto (f_*\psi \circ i_\mathcal {G} : \mathcal{G} \to f_* \mathcal{F})$

and in the other direction by the rule

$(\varphi : \mathcal{G} \to f_* \mathcal{F}) \longmapsto (c_\mathcal {F} \circ f_ p\varphi : f_ p\mathcal{G} \to \mathcal{F})$

where $i_\mathcal {G}$ and $c_\mathcal {F}$ are as in Section 6.22. Hence, using the functoriality of $f_*$ and $f_ p$ we see that it suffices to check that the maps $i_\mathcal {G} : \mathcal{G} \to f_* f_ p \mathcal{G}$ and $c_\mathcal {F} : f_ p f_* \mathcal{F} \to \mathcal{F}$ are compatible with module structures, which we leave to the reader. $\square$

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