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The Stacks project

Lemma 13.33.7. Let \mathcal{A} be an abelian category. Let L_ n^\bullet be a system of complexes of \mathcal{A}. Assume colimits over \mathbf{N} exist and are exact in \mathcal{A}. Then the termwise colimit L^\bullet = \mathop{\mathrm{colim}}\nolimits L_ n^\bullet is a homotopy colimit of the system in D(\mathcal{A}).

Proof. We have an exact sequence of complexes

0 \to \bigoplus L_ n^\bullet \to \bigoplus L_ n^\bullet \to L^\bullet \to 0

by Lemma 13.33.6. The direct sums are direct sums in D(\mathcal{A}) by Lemma 13.33.5. Thus the result follows from the definition of derived colimits in Definition 13.33.1 and the fact that a short exact sequence of complexes gives a distinguished triangle (Lemma 13.12.1). \square


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