Lemma 22.9.2. Let (A, \text{d}) be a differential graded algebra. Let \alpha : K \to L and \beta : L \to M define an admissible short exact sequence
0 \to K \to L \to M \to 0
of differential graded A-modules. Let (K, L, M, \alpha , \beta , \delta ) be the associated triangle. Then the triangles
(M[-1], K, L, \delta [-1], \alpha , \beta ) \quad \text{and}\quad (M[-1], K, C(\delta [-1]), \delta [-1], i, p)
are isomorphic.
Proof.
Using a choice of splittings we write L = K \oplus M and we identify \alpha and \beta with the natural inclusion and projection maps. By construction of \delta we have
d_ B = \left( \begin{matrix} d_ K
& \delta
\\ 0
& d_ M
\end{matrix} \right)
On the other hand the cone of \delta [-1] : M[-1] \to K is given as C(\delta [-1]) = K \oplus M with differential identical with the matrix above! Whence the lemma.
\square
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