Proposition 6.6.11. Let \(P_{\bullet }\) and \(Q_{\bullet }\) be bounded below projective chain complexes. Then any quasi-isomorphism \(f\colon P_{\bullet } \to Q_{\bullet }\) is a chain homotopy equivalence.
Proof. The mapping cone \(\Cone (f)\) fits into a short exact sequence \[ 0 \to Q_{\bullet } \to \Cone (f) \to P_{\bullet }[1] \to 0. \] By Lemma 6.1.15, the complex \(\Cone (f)\) is acyclic. It is again bounded below and projective, hence contractible by Lemma 6.6.10. Write the bottom-left component of a contracting homotopy \[ \Cone (f)_n=Q_n\oplus P_{n-1}\longrightarrow Q_{n+1}\oplus P_n=\Cone (f)_{n+1} \] as \(g_n\colon Q_n\to P_n\). The contracting-homotopy equation says that \(g\) is a chain map and that \(fg\) and \(gf\) are chain homotopic to the respective identity maps. Thus \(f\) is a chain homotopy equivalence. □
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