
## 22.8 Distinguished triangles

The following lemma produces our distinguished triangles.

Lemma 22.8.1. Let $(A, \text{d})$ be a differential graded algebra. Let $0 \to K \to L \to M \to 0$ be an admissible short exact sequence of differential graded $A$-modules. The triangle

22.8.1.1
$$\label{dga-equation-triangle-associated-to-admissible-ses} K \to L \to M \xrightarrow {\delta } K[1]$$

with $\delta$ as in Lemma 22.7.2 is, up to canonical isomorphism in $K(\text{Mod}_{(A, \text{d})})$, independent of the choices made in Lemma 22.7.2.

Proof. Namely, let $(s', \pi ')$ be a second choice of splittings as in Lemma 22.7.2. Then we claim that $\delta$ and $\delta '$ are homotopic. Namely, write $s' = s + \alpha \circ h$ and $\pi ' = \pi + g \circ \beta$ for some unique homomorphisms of $A$-modules $h : M \to K$ and $g : M \to K$ of degree $-1$. Then $g = -h$ and $g$ is a homotopy between $\delta$ and $\delta '$. The computations are done in the proof of Homology, Lemma 12.13.12. $\square$

Definition 22.8.2. Let $(A, \text{d})$ be a differential graded algebra.

1. If $0 \to K \to L \to M \to 0$ is an admissible short exact sequence of differential graded $A$-modules, then the triangle associated to $0 \to K \to L \to M \to 0$ is the triangle (22.8.1.1) of $K(\text{Mod}_{(A, \text{d})})$.

2. A triangle of $K(\text{Mod}_{(A, \text{d})})$ is called a distinguished triangle if it is isomorphic to a triangle associated to an admissible short exact sequence of differential graded $A$-modules.

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