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The Stacks project

Lemma 52.6.18. Let f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) be a morphism of ringed topoi. Let \mathcal{I} \subset \mathcal{O} and \mathcal{I}' \subset \mathcal{O}' be finite type sheaves of ideals such that f^\sharp sends f^{-1}\mathcal{I} into \mathcal{I}'. Then Rf_* sends D_{comp}(\mathcal{O}', \mathcal{I}') into D_{comp}(\mathcal{O}, \mathcal{I}) and has a left adjoint Lf_{comp}^* which is Lf^* followed by derived completion.

Proof. The first statement we have seen in Lemma 52.6.7. Note that the second statement makes sense as we have a derived completion functor D(\mathcal{O}') \to D_{comp}(\mathcal{O}', \mathcal{I}') by Proposition 52.6.12. OK, so now let K \in D_{comp}(\mathcal{O}, \mathcal{I}) and M \in D_{comp}(\mathcal{O}', \mathcal{I}'). Then we have

\mathop{\mathrm{Hom}}\nolimits (K, Rf_*M) = \mathop{\mathrm{Hom}}\nolimits (Lf^*K, M) = \mathop{\mathrm{Hom}}\nolimits (Lf_{comp}^*K, M)

by the universal property of derived completion. \square


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