The Stacks project

Proposition 24.22.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. The homotopy category $K(\textit{Mod}(\mathcal{A}, \text{d}))$ is a triangulated category where

  1. the shift functors are those constructed in Section 24.20,

  2. the distinghuished triangles are those triangles in $K(\textit{Mod}(\mathcal{A}, \text{d}))$ which are isomorphic as a triangle to a triangle

    \[ \mathcal{K} \to \mathcal{L} \to \mathcal{N} \xrightarrow {\delta } \mathcal{K}[1],\quad \quad \delta = \pi \circ \text{d}_\mathcal {L} \circ s \]

    constructed from an admissible short exact sequence $0 \to \mathcal{K} \to \mathcal{L} \to \mathcal{N} \to 0$ in $\textit{Mod}(\mathcal{A}, \text{d})$ above.

Proof. Recall that $K(\textit{Mod}(\mathcal{A}, \text{d})) = K(\textit{Mod}^{dg}(\mathcal{A}, \text{d}))$, see Section 24.21. Having said this, the proposition follows from Lemmas 24.22.1 and 24.22.3 and Differential Graded Algebra, Proposition 22.27.16. $\square$

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