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