Lemma 18.9.2 (03CU)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 18: Modules on Sites
Section 18.9: Presheaves of modules
Lemma 18.9.2 (cite)

previous definition

numberstags

Lemma 18.9.2. With $\mathcal{C}$, $\mathcal{O}_1 \to \mathcal{O}_2$, $\mathcal{F}$ 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 defined above 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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