Lemma 6.6.2 (006R)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 6: Sheaves on Spaces
Section 6.6: Presheaves of modules
Lemma 6.6.2 (cite)

previous definition

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Lemma 6.6.2. With $X$ , $\mathcal{O}_1$ , $\mathcal{O}_2$ , $\mathcal{F}$ and $\mathcal{G}$ as above there exists a canonical bijection

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

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

Proof. This follows from the fact that for a ring map $A \to B$ the restriction functor and the change of ring functor are adjoint to each other. $\square$

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