Lemma 61.20.2. Let $\Lambda $ be a Noetherian ring. Let $I \subset \Lambda $ be an ideal. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a morphism of topoi. Then

$Rf_*$ sends $D_{comp}(\mathcal{D}, \Lambda )$ into $D_{comp}(\mathcal{C}, \Lambda )$,

the map $Rf_* : D_{comp}(\mathcal{D}, \Lambda ) \to D_{comp}(\mathcal{C}, \Lambda )$ has a left adjoint $Lf_{comp}^* : D_{comp}(\mathcal{C}, \Lambda ) \to D_{comp}(\mathcal{D}, \Lambda )$ which is $Lf^*$ followed by derived completion,

$Rf_*$ commutes with derived completion,

for $K$ in $D_{comp}(\mathcal{D}, \Lambda )$ we have $Rf_*K = R\mathop{\mathrm{lim}}\nolimits Rf_*(K \otimes ^\mathbf {L}_\Lambda \underline{\Lambda /I^ n})$.

for $M$ in $D_{comp}(\mathcal{C}, \Lambda )$ we have $Lf^*_{comp}M = R\mathop{\mathrm{lim}}\nolimits Lf^*(M \otimes ^\mathbf {L}_\Lambda \underline{\Lambda /I^ n})$.

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