The Stacks project

Lemma 24.27.1. 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 localization functor $\textit{Mod}(\mathcal{A}, \text{d}) \to D(\mathcal{A}, \text{d})$ has the natural structure of a $\delta $-functor, with

\[ \delta _{\mathcal{K} \to \mathcal{L} \to \mathcal{M}} = - p \circ q^{-1} \]

with $p$ and $q$ as explained above.

Proof. We have already seen that this choice leads to a distinguished triangle whenever given a short exact sequence of complexes. We have to show functoriality of this construction, see Derived Categories, Definition 13.3.6. This follows from Differential Graded Algebra, Lemma 22.27.3 (which we may use by the discussion in Section 24.22) with a bit of work. Compare with Derived Categories, Lemma 13.12.1. $\square$

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