The Stacks project

Remark 52.6.14 (Localization and derived completion). Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I} \subset \mathcal{O}$ be a finite type sheaf of ideals. Let $K \mapsto K^\wedge $ be the derived completion functor of Proposition 52.6.12. It follows from the construction in the proof of the proposition that $K^\wedge |_ U$ is the derived completion of $K|_ U$ for any $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. But we can also prove this as follows. From the definition of derived complete objects it follows that $K^\wedge |_ U$ is derived complete. Thus we obtain a canonical map $a : (K|_ U)^\wedge \to K^\wedge |_ U$. On the other hand, if $E$ is a derived complete object of $D(\mathcal{O}_ U)$, then $Rj_*E$ is a derived complete object of $D(\mathcal{O})$ by Lemma 52.6.7. Here $j$ is the localization morphism (Modules on Sites, Section 18.19). Hence we also obtain a canonical map $b : K^\wedge \to Rj_*((K|_ U)^\wedge )$. We omit the (formal) verification that the adjoint of $b$ is the inverse of $a$.

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