The Stacks project

Lemma 24.26.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. The full subcategory $\text{Ac}$ of the homotopy category $K(\textit{Mod}(\mathcal{A}, \text{d}))$ consisting of acyclic modules is a strictly full saturated triangulated subcategory of $K(\textit{Mod}(\mathcal{A}, \text{d}))$.

Proof. Of course an object $\mathcal{M}$ of $K(\textit{Mod}(\mathcal{A}, \text{d}))$ is in $\text{Ac}$ if and only if $H^ i(\mathcal{M}) = H^0(\mathcal{M}[i])$ is zero for all $i$. The lemma follows from this, Lemma 24.26.1, and Derived Categories, Lemma 13.6.3. See also Derived Categories, Definitions 13.6.1 and 13.3.4 and Lemma 13.4.16. $\square$

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