Lemma 109.63.1. There exists a ring $R$, a distinguished triangle $(K, L, M, \alpha , \beta , \gamma )$ in the homotopy category $K(R)$, and an endomorphism $(a, b, c)$ of this distinguished triangle, such that $K$, $L$, $M$ are perfect complexes and $\text{Tr}_ K(a) + \text{Tr}_ M(c) \not= \text{Tr}_ L(b)$.

Proof. Consider the example above. The map $\gamma : C^\bullet \to A^\bullet [1]$ is given by multiplication by $\epsilon$ in degree $0$, see Derived Categories, Definition 13.10.1. Hence it is also true that

$\xymatrix{ C^\bullet \ar[d] \ar[r]_\gamma & A^\bullet [1] \ar[d] \\ C^\bullet \ar[r]^\gamma & A^\bullet [1] }$

commutes in $K(R)$ as $\epsilon (1 + \epsilon ) = \epsilon$. Thus we indeed have a morphism of distinguished triangles. $\square$

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