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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