The Stacks project

Remark 52.6.21. 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 of Proposition 52.6.12. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be an object such that $\mathcal{I}$ is generated as an ideal sheaf by $f_1, \ldots , f_ r \in \mathcal{I}(U)$. Set $A = \mathcal{O}(U)$ and $I = (f_1, \ldots , f_ r) \subset A$. Warning: it may not be the case that $I = \mathcal{I}(U)$. Then we have

\[ R\Gamma (U, K^\wedge ) = R\Gamma (U, K)^\wedge \]

where the right hand side is the derived completion of the object $R\Gamma (U, K)$ of $D(A)$ with respect to $I$. This is true because derived completion commutes with localization (Remark 52.6.14) and Lemma 52.6.20.

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