Lemma 6.24.4. 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. Let \mathcal{G} be a presheaf of f_*\mathcal{O}-modules. Then
\mathop{\mathrm{Mor}}\nolimits _{\textit{PMod}(\mathcal{O})}( \mathcal{O} \otimes _{p, f_ pf_*\mathcal{O}} f_ p\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PMod}(f_*\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}).
Here we use Lemmas 6.24.2 and 6.24.1, and we use the map c_\mathcal {O} : f_ pf_*\mathcal{O} \to \mathcal{O} in the definition of the tensor product.
Proof.
This follows from the equalities
\begin{eqnarray*} \mathop{\mathrm{Mor}}\nolimits _{\textit{PMod}(\mathcal{O})}( \mathcal{O} \otimes _{p, f_ pf_*\mathcal{O}} f_ p\mathcal{G}, \mathcal{F}) & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{PMod}(f_ pf_*\mathcal{O})}( f_ p\mathcal{G}, \mathcal{F}_{f_ pf_*\mathcal{O}}) \\ & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{PMod}(f_*\mathcal{O})}(\mathcal{G}, f_*(\mathcal{F}_{f_ pf_*\mathcal{O}})) \\ & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{PMod}(f_*\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}). \end{eqnarray*}
The first equality is Lemma 6.6.2. The second equality is Lemma 6.24.3. The third equality is given by the equality f_*(\mathcal{F}_{f_ pf_*\mathcal{O}}) = f_*\mathcal{F} of abelian sheaves which is f_*\mathcal{O}-linear. Namely, \text{id}_{f_*\mathcal{O}} corresponds to c_\mathcal {O} under the adjunction described in the proof of Lemma 6.21.3 and thus \text{id}_{f_*\mathcal{O}} = f_*c_\mathcal {O} \circ i_{f_*\mathcal{O}}.
\square
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