The Stacks project

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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