Lemma 21.19.6. Let f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) be a morphism of ringed topoi. Let \mathcal{K}^\bullet be a complex of \mathcal{O}_\mathcal {C}-modules. The diagram
\xymatrix{ Lf^*f_*\mathcal{K}^\bullet \ar[r] \ar[d] & f^*f_*\mathcal{K}^\bullet \ar[d] \\ Lf^*Rf_*\mathcal{K}^\bullet \ar[r] & \mathcal{K}^\bullet }
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}_\mathcal {C}).
Proof.
We will use the existence of K-flat resolutions and K-injective resolutions, see Lemmas 21.17.11, 21.18.2, and 21.18.1 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}_\mathcal {C}-modules. Choose a quasi-isomorphism \mathcal{Q}^\bullet \to f_*\mathcal{I}^\bullet where \mathcal{Q}^\bullet is a K-flat complex of \mathcal{O}_\mathcal {D}-modules with flat terms. We can choose a K-flat complex of \mathcal{O}_\mathcal {D}-modules \mathcal{P}^\bullet with flat terms and a diagram of morphisms of complexes
\xymatrix{ \mathcal{P}^\bullet \ar[r] \ar[d] & f_*\mathcal{K}^\bullet \ar[d] \\ \mathcal{Q}^\bullet \ar[r] & f_*\mathcal{I}^\bullet }
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 choose a resolution of \mathcal{P}^\bullet by a K-flat complex with flat terms. Taking pullbacks we obtain a diagram of morphisms of complexes
\xymatrix{ f^*\mathcal{P}^\bullet \ar[r] \ar[d] & f^*f_*\mathcal{K}^\bullet \ar[d] \ar[r] & \mathcal{K}^\bullet \ar[d] \\ f^*\mathcal{Q}^\bullet \ar[r] & f^*f_*\mathcal{I}^\bullet \ar[r] & \mathcal{I}^\bullet }
commutative up to homotopy. The outer rectangle witnesses the truth of the statement in the lemma.
\square
Comments (1)
Comment #7138 by Hao Peng on