The Stacks project

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

I think this lemma can be genralized to the case of two adjoint functor between Abelian categories. I can prove it using the universal properties of derived functors in the sence of Gelfand's book "methods in homological algebra", assuming both functor have enough acyclic objects. Not sure about the most genral case.

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