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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