Lemma 21.18.9. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. For any complex $\mathcal{G}^\bullet $ of $\mathcal{O}$-modules there exists a commutative diagram of complexes of $\mathcal{O}$-modules

with the following properties: (1) the vertical arrows are quasi-isomorphisms, (2) each $\mathcal{K}_ n^\bullet $ is a bounded above complex whose terms are direct sums of $\mathcal{O}$-modules of the form $j_{U!}\mathcal{O}_ U$, and (3) the maps $\mathcal{K}_ n^\bullet \to \mathcal{K}_{n + 1}^\bullet $ are termwise split injections whose cokernels are direct sums of $\mathcal{O}$-modules of the form $j_{U!}\mathcal{O}_ U$. Moreover, the map $\mathop{\mathrm{colim}}\nolimits \mathcal{K}_ n^\bullet \to \mathcal{G}^\bullet $ is a quasi-isomorphism.

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