The Stacks project

Remark 46.5.9. Let $S$ be a scheme. We have functors $u : \mathit{QCoh}(\mathcal{O}_ S) \to \textit{Adeq}(\mathcal{O})$ and $v : \textit{Adeq}(\mathcal{O}) \to \mathit{QCoh}(\mathcal{O}_ S)$. Namely, the functor $u : \mathcal{F} \mapsto \mathcal{F}^ a$ comes from taking the associated $\mathcal{O}$-module which is adequate by Lemma 46.5.5. Conversely, the functor $v$ comes from restriction $v : \mathcal{G} \mapsto \mathcal{G}|_{S_{Zar}}$, see Lemma 46.5.8. Since $\mathcal{F}^ a$ can be described as the pullback of $\mathcal{F}$ under a morphism of ringed topoi $((\mathit{Sch}/S)_\tau , \mathcal{O}) \to (S_{Zar}, \mathcal{O}_ S)$, see Descent, Remark 35.8.6 and since restriction is the pushforward we see that $u$ and $v$ are adjoint as follows

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ S}(\mathcal{F}, v\mathcal{G}) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(u\mathcal{F}, \mathcal{G}) \]

where $\mathcal{O}$ denotes the structure sheaf on the big site. It is immediate from the description that the adjunction mapping $\mathcal{F} \to vu\mathcal{F}$ is an isomorphism for all quasi-coherent sheaves.

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