The Stacks project

Remark 52.6.13 (Comparison with 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. For any $n \geq 1$ the object $K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n$ is derived complete as it is annihilated by powers of local sections of $\mathcal{I}$. Hence there is a canonical factorization

\[ K \to K^\wedge \to K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n \]

of the canonical map $K \to K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n$. These maps are compatible for varying $n$ and we obtain a comparison map

\[ K^\wedge \longrightarrow R\mathop{\mathrm{lim}}\nolimits \left(K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n\right) \]

The right hand side is more recognizable as a kind of completion. In general this comparison map is not an isomorphism.

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