Proposition 6.6.13 (Existence of projective resolutions). Let \(\Aa \) be an abelian category with enough projectives. Every bounded below chain complex \(C_{\bullet }\) admits a degreewise epimorphic quasi-isomorphism \[ q\colon P_{\bullet }\xrightarrow {\sim }C_{\bullet } \] from a bounded below projective chain complex \(P_{\bullet }\). Moreover, if \(C_n=0\) for all \(n<m\), then \(P_{\bullet }\) may be chosen with \(P_n=0\) for all \(n<m\).
Proof. This is the standard bounded-below projective-replacement theorem. One construction takes the total complex of a Cartan–Eilenberg projective resolution; it produces a degreewise epimorphic quasi-isomorphism with the stated lower bound. See [Weibel (1994), Theorem 10.4.8 and its proof]. □
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