Lemma 21.44.7. Let (\mathcal{C}, \mathcal{O}) be a ringed site. Let U be an object of \mathcal{C}. Given a solid diagram of complexes of \mathcal{O}_ U-modules
\xymatrix{ \mathcal{E}^\bullet \ar@{..>}[dr] \ar[r]_\alpha & \mathcal{F}^\bullet \\ & \mathcal{G}^\bullet \ar[u]_ f }
with \mathcal{E}^\bullet strictly perfect, \mathcal{E}^ j = 0 for j < a and H^ j(f) an isomorphism for j > a and surjective for j = a, then there exists a covering \{ U_ i \to U\} and for each i a dotted arrow over U_ i making the diagram commute up to homotopy.
Proof.
Our assumptions on f imply the cone C(f)^\bullet has vanishing cohomology sheaves in degrees \geq a. Hence Lemma 21.44.6 guarantees there is a covering \{ U_ i \to U\} such that the composition \mathcal{E}^\bullet \to \mathcal{F}^\bullet \to C(f)^\bullet is homotopic to zero over U_ i. Since
\mathcal{G}^\bullet \to \mathcal{F}^\bullet \to C(f)^\bullet \to \mathcal{G}^\bullet [1]
restricts to a distinguished triangle in K(\mathcal{O}_{U_ i}) we see that we can lift \alpha |_{U_ i} up to homotopy to a map \alpha _ i : \mathcal{E}^\bullet |_{U_ i} \to \mathcal{G}^\bullet |_{U_ i} as desired.
\square
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