Lemma 6.1.15 (Mapping-cone criterion). A chain map \(f\colon C_{\bullet }\to D_{\bullet }\) is a quasi-isomorphism if and only if its mapping cone \(\Cone (f)\) is acyclic.
Proof. The long exact sequence in homology associated with the short exact sequence \[ 0\longrightarrow D_{\bullet }\longrightarrow \Cone (f)\longrightarrow C_{\bullet }[1]\longrightarrow 0 \] identifies the connecting maps with the maps on homology induced by \(f\). The claim follows immediately from exactness. □
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