The Stacks project

Lemma 24.26.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Consider the subclass $\text{Qis} \subset \text{Arrows}(K(\textit{Mod}(\mathcal{A}, \text{d})))$ consisting of quasi-isomorphisms. This is a saturated multiplicative system compatible with the triangulated structure on $K(\textit{Mod}(\mathcal{A}, \text{d}))$.

Proof. Observe that if $f , g : \mathcal{M} \to \mathcal{N}$ are morphisms of $\textit{Mod}(\mathcal{A}, \text{d})$ which are homotopic, then $f$ is a quasi-isomorphism if and only if $g$ is a quasi-isomorphism. Namely, the maps $H^ i(f) = H^0(f[i])$ and $H^ i(g) = H^0(g[i])$ are the same by Lemma 24.26.1. Thus it is unambiguous to say that a morphism of the homotopy category $K(\textit{Mod}(\mathcal{A}, \text{d}))$ is a quasi-isomorphism. For definitions of “multiplicative system”, “saturated”, and “compatible with the triangulated structure” see Derived Categories, Definition 13.5.1 and Categories, Definitions 4.27.1 and 4.27.20.

To actually prove the lemma consider the composition of exact functors of triangulated categories

\[ K(\textit{Mod}(\mathcal{A}, \text{d})) \longrightarrow K(\textit{Mod}(\mathcal{O})) \longrightarrow D(\mathcal{O}) \]

and observe that a morphism $f : \mathcal{M} \to \mathcal{N}$ of $K(\textit{Mod}(\mathcal{A}, \text{d}))$ is in $\text{Qis}$ if and only if it maps to an isomorphism in $D(\mathcal{O})$. Thus the lemma follows from Derived Categories, Lemma 13.5.4. $\square$

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