Lemma 6.24.1. Let f : X \to Y be a continuous map of topological spaces. Let \mathcal{O} be a presheaf of rings on X. Let \mathcal{F} be a presheaf of \mathcal{O}-modules. There is a natural map of underlying presheaves of sets
f_*\mathcal{O} \times f_*\mathcal{F} \longrightarrow f_*\mathcal{F}
which turns f_*\mathcal{F} into a presheaf of f_*\mathcal{O}-modules. This construction is functorial in \mathcal{F}.
Proof.
Let V \subset Y is open. We define the map of the lemma to be the map
f_*\mathcal{O}(V) \times f_*\mathcal{F}(V) = \mathcal{O}(f^{-1}V) \times \mathcal{F}(f^{-1}V) \to \mathcal{F}(f^{-1}V) = f_*\mathcal{F}(V).
Here the arrow in the middle is the multiplication map on X. We leave it to the reader to see this is compatible with restriction mappings and defines a structure of f_*\mathcal{O}-module on f_*\mathcal{F}.
\square
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