Lemma 6.24.7 (008Y)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 6: Sheaves on Spaces
Section 6.24: Continuous maps and sheaves of modules
Lemma 6.24.7 (cite)

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Lemma 6.24.7. 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. Let $\mathcal{F}$ be a sheaf of $f^{-1}\mathcal{O}$ -modules. Then

$\mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(f^{-1}\mathcal{O})}(f^{-1}\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}).$

Here we use Lemmas 6.24.6 and 6.24.5, and we think of $f_*\mathcal{F}$ as an $\mathcal{O}$ -module by restriction via $\mathcal{O} \to f_*f^{-1}\mathcal{O}$ .

Proof. Argue by the equalities

$\begin{eqnarray*} \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(f^{-1}\mathcal{O})}(f^{-1}\mathcal{G}, \mathcal{F}) & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(f_ p\mathcal{O})}(f_ p\mathcal{G}, \mathcal{F}) \\ & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}). \end{eqnarray*}$

where the second is Lemmas 6.24.3 and the first is by Lemma 6.20.1. $\square$

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