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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