The Stacks project

Lemma 24.27.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{M}_ n$ be a system of differential graded $\mathcal{A}$-modules. Then the derived colimit $\text{hocolim} \mathcal{M}_ n$ in $D(\mathcal{A}, \text{d})$ is represented by the differential graded module $\mathop{\mathrm{colim}}\nolimits \mathcal{M}_ n$.

Proof. Set $\mathcal{M} = \mathop{\mathrm{colim}}\nolimits \mathcal{M}_ n$. We have an exact sequence of differential graded $\mathcal{A}$-modules

\[ 0 \to \bigoplus \mathcal{M}_ n \to \bigoplus \mathcal{M}_ n \to \mathcal{M} \to 0 \]

by Derived Categories, Lemma 13.33.6 (applied the underlying complexes of $\mathcal{O}$-modules). The direct sums are direct sums in $D(\mathcal{A}, \text{d})$ by Lemma 24.26.8. 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 24.27.1). $\square$

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