Lemma 6.20.2 (008A)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 6: Sheaves on Spaces
Section 6.20: Sheafification of presheaves of modules
Lemma 6.20.2 (cite)

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Lemma 6.20.2. With $X$, $\mathcal{O}_1$, $\mathcal{O}_2$, $\mathcal{F}$ and $\mathcal{G}$ as above there exists a canonical bijection

\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_1}(\mathcal{G}, \mathcal{F}_{\mathcal{O}_1}) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_2}( \mathcal{O}_2 \otimes _{\mathcal{O}_1} \mathcal{G}, \mathcal{F} ) \]

In other words, the restriction and change of rings functors are adjoint to each other.

Proof. This follows from Lemma 6.6.2 and the fact that $\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_2}( \mathcal{O}_2 \otimes _{\mathcal{O}_1} \mathcal{G}, \mathcal{F} ) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_2}( \mathcal{O}_2 \otimes _{p, \mathcal{O}_1} \mathcal{G}, \mathcal{F} )$ because $\mathcal{F}$ is a sheaf. $\square$

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