Lemma 20.28.6. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Let $\mathcal{K}^\bullet $ be a complex of $\mathcal{O}_ X$-modules. The diagram
coming from $Lf^* \to f^*$ on complexes, $f_* \to Rf_*$ on complexes, and adjunction $Lf^* \circ Rf_* \to \text{id}$ commutes in $D(\mathcal{O}_ X)$.
Proof. We will use the existence of K-flat resolutions and K-injective resolutions, see Lemma 20.26.8 and the discussion above. Choose a quasi-isomorphism $\mathcal{K}^\bullet \to \mathcal{I}^\bullet $ where $\mathcal{I}^\bullet $ is K-injective as a complex of $\mathcal{O}_ X$-modules. Choose a quasi-isomorphism $\mathcal{Q}^\bullet \to f_*\mathcal{I}^\bullet $ where $\mathcal{Q}^\bullet $ is K-flat as a complex of $\mathcal{O}_ Y$-modules. We can choose a K-flat complex of $\mathcal{O}_ Y$-modules $\mathcal{P}^\bullet $ and a diagram of morphisms of complexes
commutative up to homotopy where the top horizontal arrow is a quasi-isomorphism. Namely, we can first choose such a diagram for some complex $\mathcal{P}^\bullet $ because the quasi-isomorphisms form a multiplicative system in the homotopy category of complexes and then we can replace $\mathcal{P}^\bullet $ by a K-flat complex. Taking pullbacks we obtain a diagram of morphisms of complexes
commutative up to homotopy. The outer rectangle witnesses the truth of the statement in the lemma. $\square$
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