Generalization of [Lemma 6.5.9 (2), BS]. Compare with [Theorem 6.5, HL-P] in the setting of quasi-coherent modules and morphisms of (derived) algebraic stacks.

Lemma 52.6.19. 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 $\mathcal{I} \subset \mathcal{O}$ be a finite type sheaf of ideals. Let $\mathcal{I}' \subset \mathcal{O}'$ be the ideal generated by $f^\sharp (f^{-1}\mathcal{I})$. Then $Rf_*$ commutes with derived completion, i.e., $Rf_*(K^\wedge ) = (Rf_*K)^\wedge$.

Proof. By Proposition 52.6.12 the derived completion functors exist. By Lemma 52.6.7 the object $Rf_*(K^\wedge )$ is derived complete, and hence we obtain a canonical map $(Rf_*K)^\wedge \to Rf_*(K^\wedge )$ by the universal property of derived completion. We may check this map is an isomorphism locally on $\mathcal{C}$. Thus, since derived completion commutes with localization (Remark 52.6.14) we may assume that $\mathcal{I}$ is generated by global sections $f_1, \ldots , f_ r$. Then $\mathcal{I}'$ is generated by $g_ i = f^\sharp (f_ i)$. By Lemma 52.6.9 we have to prove that

$R\mathop{\mathrm{lim}}\nolimits \left( Rf_*K \otimes ^\mathbf {L}_\mathcal {O} K(\mathcal{O}, f_1^ n, \ldots , f_ r^ n) \right) = Rf_*\left( R\mathop{\mathrm{lim}}\nolimits K \otimes ^\mathbf {L}_{\mathcal{O}'} K(\mathcal{O}', g_1^ n, \ldots , g_ r^ n) \right)$

Because $Rf_*$ commutes with $R\mathop{\mathrm{lim}}\nolimits$ (Cohomology on Sites, Lemma 21.22.3) it suffices to prove that

$Rf_*K \otimes ^\mathbf {L}_\mathcal {O} K(\mathcal{O}, f_1^ n, \ldots , f_ r^ n) = Rf_*\left( K \otimes ^\mathbf {L}_{\mathcal{O}'} K(\mathcal{O}', g_1^ n, \ldots , g_ r^ n) \right)$

This follows from the projection formula (Cohomology on Sites, Lemma 21.48.1) and the fact that $Lf^*K(\mathcal{O}, f_1^ n, \ldots , f_ r^ n) = K(\mathcal{O}', g_1^ n, \ldots , g_ r^ n)$. $\square$

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