Lemma 21.18.1. 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. Let \mathcal{K}^\bullet be a K-flat complex of \mathcal{O}-modules whose terms are flat \mathcal{O}-modules. Then f^*\mathcal{K}^\bullet is a K-flat complex of \mathcal{O}'-modules whose terms are flat \mathcal{O}'-modules.
Proof. The terms f^*\mathcal{K}^ n are flat \mathcal{O}'-modules by Modules on Sites, Lemma 18.39.1. Choose a diagram
\xymatrix{ \mathcal{K}_1^\bullet \ar[d] \ar[r] & \mathcal{K}_2^\bullet \ar[d] \ar[r] & \ldots \\ \tau _{\leq 1}\mathcal{K}^\bullet \ar[r] & \tau _{\leq 2}\mathcal{K}^\bullet \ar[r] & \ldots }
as in Lemma 21.17.10. We will use all of the properties stated in the lemma without further mention. Each \mathcal{K}_ n^\bullet is a bounded above complex of flat modules, see Modules on Sites, Lemma 18.28.7. Consider the short exact sequence of complexes
0 \to \mathcal{M}^\bullet \to \mathop{\mathrm{colim}}\nolimits \mathcal{K}_ n^\bullet \to \mathcal{K}^\bullet \to 0
defining \mathcal{M}^\bullet . By Lemmas 21.17.8 and 21.17.9 the complex \mathop{\mathrm{colim}}\nolimits \mathcal{K}_ n^\bullet is K-flat and by Modules on Sites, Lemma 18.28.5 it has flat terms. By Modules on Sites, Lemma 18.28.10 \mathcal{M}^\bullet has flat terms, by Lemma 21.17.7 \mathcal{M}^\bullet is K-flat, and by the long exact cohomology sequence \mathcal{M}^\bullet is acyclic (because the second arrow is a quasi-isomorphism). The pullback f^*(\mathop{\mathrm{colim}}\nolimits \mathcal{K}_ n^\bullet ) = \mathop{\mathrm{colim}}\nolimits f^*\mathcal{K}_ n^\bullet is a colimit of bounded below complexes of flat \mathcal{O}'-modules and hence is K-flat (by the same lemmas as above). The pullback of our short exact sequence
0 \to f^*\mathcal{M}^\bullet \to f^*(\mathop{\mathrm{colim}}\nolimits \mathcal{K}_ n^\bullet ) \to f^*\mathcal{K}^\bullet \to 0
is a short exact sequence of complexes by Modules on Sites, Lemma 18.39.4. Hence by Lemma 21.17.7 it suffices to show that f^*\mathcal{M}^\bullet is K-flat. This reduces us to the case discussed in the next paragraph.
Assume \mathcal{K}^\bullet is acyclic as well as K-flat and with flat terms. Then Lemma 21.17.16 guarantees that all terms of \tau _{\leq n}\mathcal{K}^\bullet are flat \mathcal{O}-modules. We choose a diagram as above and we will use all the properties proven above for this diagram. Denote \mathcal{M}_ n^\bullet the kernel of the map of complexes \mathcal{K}_ n^\bullet \to \tau _{\leq n}\mathcal{K}^\bullet so that we have short exact sequences of complexes
0 \to \mathcal{M}_ n^\bullet \to \mathcal{K}_ n^\bullet \to \tau _{\leq n}\mathcal{K}^\bullet \to 0
By Modules on Sites, Lemma 18.28.10 we see that the terms of the complex \mathcal{M}_ n^\bullet are flat. Hence we see that \mathcal{M} = \mathop{\mathrm{colim}}\nolimits \mathcal{M}_ n^\bullet is a filtered colimit of bounded below complexes of flat modules in this case. Thus f^*\mathcal{M}^\bullet is K-flat (same argument as above) and we win. \square
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