Remark 13.9.11. Let $\mathcal{A}$ be an additive category. Let $0 \to A_ i^\bullet \to B_ i^\bullet \to C_ i^\bullet \to 0$, $i = 1, 2$ be termwise split exact sequences. Suppose that $a : A_1^\bullet \to A_2^\bullet $, $b : B_1^\bullet \to B_2^\bullet $, and $c : C_1^\bullet \to C_2^\bullet $ are morphisms of complexes such that

commutes in $K(\mathcal{A})$. In general, there does **not** exist a morphism $b' : B_1^\bullet \to B_2^\bullet $ which is homotopic to $b$ such that the diagram above commutes in the category of complexes. Namely, consider Examples, Equation (109.63.0.1). If we could replace the middle map there by a homotopic one such that the diagram commutes, then we would have additivity of traces which we do not.

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