Monomorphic quasi-isomorphisms and acyclic cokernels

Corollary 6.1.18. Given a short exact sequence \(0 \to C_{\bullet } \xhookrightarrow {f} D_{\bullet } \overset {g}{\twoheadrightarrow } E_{\bullet } \to 0\), the map \(f\) is a quasi-isomorphism if and only if \(E_{\bullet }\) has trivial homology. □