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 ( 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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