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The Stacks project

Lemma 21.18.2. Let f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') be a morphism of ringed topoi. There exists an exact functor

Lf^* : D(\mathcal{O}') \longrightarrow D(\mathcal{O})

of triangulated categories so that Lf^*\mathcal{K}^\bullet = f^*\mathcal{K}^\bullet for any K-flat complex \mathcal{K}^\bullet with flat terms and in particular for any bounded above complex of flat \mathcal{O}'-modules.

Proof. To see this we use the general theory developed in Derived Categories, Section 13.14. Set \mathcal{D} = K(\mathcal{O}') and \mathcal{D}' = D(\mathcal{O}). Let us write F : \mathcal{D} \to \mathcal{D}' the exact functor of triangulated categories defined by the rule F(\mathcal{G}^\bullet ) = f^*\mathcal{G}^\bullet . We let S be the set of quasi-isomorphisms in \mathcal{D} = K(\mathcal{O}'). This gives a situation as in Derived Categories, Situation 13.14.1 so that Derived Categories, Definition 13.14.2 applies. We claim that LF is everywhere defined. This follows from Derived Categories, Lemma 13.14.15 with \mathcal{P} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}) the collection of K-flat complexes \mathcal{K}^\bullet with flat terms. Namely, (1) follows from Lemma 21.17.11 and to see (2) we have to show that for a quasi-isomorphism \mathcal{K}_1^\bullet \to \mathcal{K}_2^\bullet between elements of \mathcal{P} the map f^*\mathcal{K}_1^\bullet \to f^*\mathcal{K}_2^\bullet is a quasi-isomorphism. To see this write this as

f^{-1}\mathcal{K}_1^\bullet \otimes _{f^{-1}\mathcal{O}'} \mathcal{O} \longrightarrow f^{-1}\mathcal{K}_2^\bullet \otimes _{f^{-1}\mathcal{O}'} \mathcal{O}

The functor f^{-1} is exact, hence the map f^{-1}\mathcal{K}_1^\bullet \to f^{-1}\mathcal{K}_2^\bullet is a quasi-isomorphism. The complexes f^{-1}\mathcal{K}_1^\bullet and f^{-1}\mathcal{K}_2^\bullet are K-flat complexes of f^{-1}\mathcal{O}'-modules by Lemma 21.18.1 because we can consider the morphism of ringed topoi (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), f^{-1}\mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}'). Hence Lemma 21.17.12 guarantees that the displayed map is a quasi-isomorphism. Thus we obtain a derived functor

LF : D(\mathcal{O}') = S^{-1}\mathcal{D} \longrightarrow \mathcal{D}' = D(\mathcal{O})

see Derived Categories, Equation (13.14.9.1). Finally, Derived Categories, Lemma 13.14.15 also guarantees that LF(\mathcal{K}^\bullet ) = F(\mathcal{K}^\bullet ) = f^*\mathcal{K}^\bullet when \mathcal{K}^\bullet is in \mathcal{P}. The proof is finished by observing that bounded above complexes of flat modules are in \mathcal{P} by Lemma 21.17.8. \square


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