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The Stacks project

Lemma 22.23.2. Let (A, \text{d}) be a differential graded algebra. Let M_ n be a system of differential graded modules. Then the derived colimit \text{hocolim} M_ n in D(A, \text{d}) is represented by the differential graded module \mathop{\mathrm{colim}}\nolimits M_ n.

Proof. Set M = \mathop{\mathrm{colim}}\nolimits M_ n. We have an exact sequence of differential graded modules

0 \to \bigoplus M_ n \to \bigoplus M_ n \to M \to 0

by Derived Categories, Lemma 13.33.6 (applied the underlying complexes of abelian groups). The direct sums are direct sums in D(\mathcal{A}) by Lemma 22.22.4. Thus the result follows from the definition of derived colimits in Derived Categories, Definition 13.33.1 and the fact that a short exact sequence of complexes gives a distinguished triangle (Lemma 22.23.1). \square


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