The Stacks project

Proposition 22.10.3. Let $(A, \text{d})$ be a differential graded algebra. The homotopy category $K(\text{Mod}_{(A, \text{d})})$ of differential graded $A$-modules with its natural translation functors and distinguished triangles is a triangulated category.

Proof. We know that $K(\text{Mod}_{(A, \text{d})})$ is a pre-triangulated category. Hence it suffices to prove TR4 and to prove it we can use Derived Categories, Lemma 13.4.15. Let $K \to L$ and $L \to M$ be composable morphisms of $K(\text{Mod}_{(A, \text{d})})$. By Lemma 22.7.5 we may assume that $K \to L$ and $L \to M$ are admissible monomorphisms. In this case the result follows from Lemma 22.10.2. $\square$

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