Lemma 21.17.10 (077J)—The Stacks project

Lemma 21.17.10. 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

\[ \xymatrix{ \mathcal{K}_1^\bullet \ar[d] \ar[r] & \mathcal{K}_2^\bullet \ar[d] \ar[r] & \ldots \\ \tau _{\leq 1}\mathcal{G}^\bullet \ar[r] & \tau _{\leq 2}\mathcal{G}^\bullet \ar[r] & \ldots } \]

with the following properties: (1) the vertical arrows are quasi-isomorphisms and termwise surjective, (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.

Proof. The existence of the diagram and properties (1), (2), (3) follows immediately from Modules on Sites, Lemma 18.28.8 and Derived Categories, Lemma 13.29.1. The induced map $\mathop{\mathrm{colim}}\nolimits \mathcal{K}_ n^\bullet \to \mathcal{G}^\bullet $ is a quasi-isomorphism because filtered colimits are exact. $\square$

The Stacks project

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