The Stacks project

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 18.28.6. 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 18.28.4 it has flat terms. By Modules on Sites, Lemma 18.28.9 $\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 18.28.9 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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