Lemma 20.37.2. Let f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y) be a morphism of ringed spaces. Then Rf_* commutes with R\mathop{\mathrm{lim}}\nolimits , i.e., Rf_* commutes with derived limits.
Proof. Let (K_ n) be an inverse system in D(\mathcal{O}_ X). Consider the defining distinguished triangle
R\mathop{\mathrm{lim}}\nolimits K_ n \to \prod K_ n \to \prod K_ n
in D(\mathcal{O}_ X). Applying the exact functor Rf_* we obtain the distinguished triangle
Rf_*(R\mathop{\mathrm{lim}}\nolimits K_ n) \to Rf_*\left(\prod K_ n\right) \to Rf_*\left(\prod K_ n\right)
in D(\mathcal{O}_ Y). Thus we see that it suffices to prove that Rf_* commutes with products in the derived category (which are not just given by products of complexes, see Injectives, Lemma 19.13.4). However, since Rf_* is a right adjoint by Lemma 20.28.1 this follows formally (see Categories, Lemma 4.24.5). Caution: Note that we cannot apply Categories, Lemma 4.24.5 directly as R\mathop{\mathrm{lim}}\nolimits K_ n is not a limit in D(\mathcal{O}_ X). \square
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