$R$ 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 $$\require{AMScd} \begin{CD} &&0&&0&&0\\ &&@VVV&@VVV&@VVV\\ 0@>>>0@>>>\mathbb{Z}@>\text{id}>>\mathbb{Z}@>>>0\\ &&@VVV&@VV\pmatrix{1\\2}V&@VV\times2V\\ 0@>>>\mathbb{Z}@>>>\mathbb{Z}\oplus\mathbb{Z}@>>>\mathbb{Z}@>>>0\\ &&@VVV&@VVV&@VVV\\ &&0&&0&&0\\ \end{CD}$$

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