Lemma 91.16.1. Let \mathcal{C} be a site. Let \mathcal{O} \to \mathcal{O}_0 be a surjection of sheaves of rings. Assume given the following data
flat \mathcal{O}-modules \mathcal{G}^ n,
maps of \mathcal{O}-modules \mathcal{G}^ n \to \mathcal{G}^{n + 1},
a complex \mathcal{K}_0^\bullet of \mathcal{O}_0-modules,
maps of \mathcal{O}-modules \mathcal{G}^ n \to \mathcal{K}_0^ n
such that
H^ n(\mathcal{K}_0^\bullet ) = 0 for n \gg 0,
\mathcal{G}^ n = 0 for n \gg 0,
with \mathcal{G}^ n_0 = \mathcal{G}^ n \otimes _\mathcal {O} \mathcal{O}_0 the induced maps determine a complex \mathcal{G}_0^\bullet and a map of complexes \mathcal{G}_0^\bullet \to \mathcal{K}_0^\bullet .
Then there exist
flat \mathcal{O}-modules \mathcal{F}^ n,
maps of \mathcal{O}-modules \mathcal{F}^ n \to \mathcal{F}^{n + 1},
maps of \mathcal{O}-modules \mathcal{F}^ n \to \mathcal{K}_0^ n,
maps of \mathcal{O}-modules \mathcal{G}^ n \to \mathcal{F}^ n,
such that \mathcal{F}^ n = 0 for n \gg 0, such that the diagrams
\xymatrix{ \mathcal{G}^ n \ar[r] \ar[d] & \mathcal{G}^{n + 1} \ar[d] \\ \mathcal{F}^ n \ar[r] & \mathcal{F}^{n + 1} }
commute for all n, such that the composition \mathcal{G}^ n \to \mathcal{F}^ n \to \mathcal{K}_0^ n is the given map \mathcal{G}^ n \to \mathcal{K}_0^ n, and such that with \mathcal{F}^ n_0 = \mathcal{F}^ n \otimes _\mathcal {O} \mathcal{O}_0 we obtain a complex \mathcal{F}_0^\bullet and map of complexes \mathcal{F}_0^\bullet \to \mathcal{K}_0^\bullet which is a quasi-isomorphism.
Proof.
We will prove by descending induction on e that we can find \mathcal{F}^ n, \mathcal{G}^ n \to \mathcal{F}^ n, and \mathcal{F}^ n \to \mathcal{F}^{n + 1} for n \geq e fitting into a commutative diagram
\xymatrix{ \ldots \ar[r] & \mathcal{G}^{e - 1} \ar[r] \ar@/_2pc/[dd] & \mathcal{G}^ e \ar[d] \ar[r] \ar@/_2pc/[dd] & \mathcal{G}^{e + 1} \ar[d] \ar[r] \ar@/_2pc/[dd]|\hole & \ldots \\ & & \mathcal{F}^ e \ar[d] \ar[r] & \mathcal{F}^{e + 1} \ar[d] \ar[r] & \ldots \\ \ldots \ar[r] & \mathcal{K}_0^{e - 1} \ar[r] & \mathcal{K}_0^ e \ar[r] & \mathcal{K}_0^{e + 1} \ar[r] & \ldots }
such that \mathcal{F}_0^\bullet is a complex, the induced map \mathcal{F}_0^\bullet \to \mathcal{K}_0^\bullet induces an isomorphism on H^ n for n > e and a surjection for n = e. For e \gg 0 this is true because we can take \mathcal{F}^ n = 0 for n \geq e in that case by assumptions (a) and (b).
Induction step. We have to construct \mathcal{F}^{e - 1} and the maps \mathcal{G}^{e - 1} \to \mathcal{F}^{e - 1}, \mathcal{F}^{e - 1} \to \mathcal{F}^ e, and \mathcal{F}^{e - 1} \to \mathcal{K}_0^{e - 1}. We will choose \mathcal{F}^{e - 1} = A \oplus B \oplus C as a direct sum of three pieces.
For the first we take A = \mathcal{G}^{e - 1} and we choose our map \mathcal{G}^{e - 1} \to \mathcal{F}^{e - 1} to be the inclusion of the first summand. The maps A \to \mathcal{K}^{e - 1}_0 and A \to \mathcal{F}^ e will be the obvious ones.
To choose B we consider the surjection (by induction hypothesis)
\gamma : \mathop{\mathrm{Ker}}(\mathcal{F}^ e_0 \to \mathcal{F}^{e + 1}_0) \longrightarrow \mathop{\mathrm{Ker}}(\mathcal{K}^ e_0 \to \mathcal{K}^{e + 1}_0)/ \mathop{\mathrm{Im}}(\mathcal{K}^{e - 1}_0 \to \mathcal{K}^ e_0)
We can choose a set I, for each i \in I an object U_ i of \mathcal{C}, and sections s_ i \in \mathcal{F}^ e(U_ i), t_ i \in \mathcal{K}^{e - 1}_0(U_ i) such that
s_ i maps to a section of \mathop{\mathrm{Ker}}(\gamma ) \subset \mathop{\mathrm{Ker}}(\mathcal{F}^ e_0 \to \mathcal{F}^{e + 1}_0),
s_ i and t_ i map to the same section of \mathcal{K}^ e_0,
the sections s_ i generate \mathop{\mathrm{Ker}}(\gamma ) as an \mathcal{O}_0-module.
We omit giving the full justification for this; one uses that \mathcal{F}^ e \to \mathcal{F}^ e_0 is a surjective maps of sheaves of sets. Then we set to put
B = \bigoplus \nolimits _{i \in I} j_{U_ i!}\mathcal{O}_{U_ i}
and define the maps B \to \mathcal{F}^ e and B \to \mathcal{K}_0^{e - 1} by using s_ i and t_ i to determine where to send the summand j_{U_ i!}\mathcal{O}_{U_ i}.
With \mathcal{F}^{e - 1} = A \oplus B and maps as above, this produces a diagram as above for e - 1 such that \mathcal{F}_0^\bullet \to \mathcal{K}_0^\bullet induces an isomorphism on H^ n for n \geq e. To get the map to be surjective on H^{e - 1} we choose the summand C as follows. Choose a set J, for each j \in J an object U_ j of \mathcal{C} and a section t_ j of \mathop{\mathrm{Ker}}(\mathcal{K}^{e - 1}_0 \to \mathcal{K}^ e_0) over U_ j such that these sections generate this kernel over \mathcal{O}_0. Then we put
C = \bigoplus \nolimits _{j \in J} j_{U_ j!}\mathcal{O}_{U_ j}
and the zero map C \to \mathcal{F}^ e and the map C \to \mathcal{K}_0^{e - 1} by using s_ j to determine where to the summand j_{U_ j!}\mathcal{O}_{U_ j}. This finishes the induction step by taking \mathcal{F}^{e - 1} = A \oplus B \oplus C and maps as indicated.
\square
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