# Split cofibrations up to quasi-isomorphism

$$R$$ a ring $$(1\neq 0)$$, $$\mathbf{Perf}(R)$$ is the category of perfect complexes (of right $$R$$-modules).

Suppose that $$A_{\bullet}\rightarrow B_{\bullet}\rightarrow B_{\bullet}/A_{\bullet}$$ a short exact sequence in $$\mathbf{Perf}(R)$$ such that

1. $$A_{\bullet}\rightarrow B_{\bullet}$$ is a cofibration in the projective model structure of chain complexes $$\mathbf{Ch}_{R}$$.
2. The homology of the chain complexes $$A_{\bullet}$$, $$B_{\bullet}$$ and $$B_{\bullet}/A_{\bullet}$$ is concentrated at level $$n$$ (the same $$n$$ for all $$A_{\bullet}$$, $$B_{\bullet}$$ and $$B_{\bullet}/A_{\bullet}$$ complexes).

Here is my question: Is it true that $$A_{\bullet}\oplus B_{\bullet}/A_{\bullet}$$ is isomorphic to $$B_{\bullet}$$ in homotopy category $$Ho(\mathbf{Ch}_{R})$$.

No. You can construct counterexamples by taking projective resolutions of modules in a nonsplit short exact sequence. For example, from the short exact sequence $$0\to\mathbb{Z}\stackrel{2}{\to}\mathbb{Z}\to\mathbb{Z}/2\mathbb{Z}\to0$$ of abelian groups you get $$\require{AMScd} \begin{CD} &&0&&0&&0\\ &&@VVV&@VVV&@VVV\\ 0@>>>0@>>>\mathbb{Z}@>\text{id}>>\mathbb{Z}@>>>0\\ &&@VVV&@VV\pmatrix{1\\2}V&@VV\times2V\\ 0@>>>\mathbb{Z}@>>>\mathbb{Z}\oplus\mathbb{Z}@>>>\mathbb{Z}@>>>0\\ &&@VVV&@VVV&@VVV\\ &&0&&0&&0\\ \end{CD}$$