The Stacks project

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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