Proposition 6.6.16. Let \(\Aa \) be an abelian category with enough projectives. Then the inclusion \(\Ch ^-_{\proj }(\Aa ) \hookrightarrow \Ch ^-(\Aa )\) induces an equivalence on localizations: \[ \Kk ^-_{\proj }(\Aa ) := \Ch ^-_{\proj }(\Aa )[\textup {chain homotopy equivalences}^{-1}] \quad \iso \quad \D ^-(\Aa ). \]
Proof. By Proposition 6.6.15 and part (1) of Corollary 2.2.9, the inclusion of the cofibrant objects induces an equivalence after localizing at the quasi-isomorphisms. By Proposition 6.6.11, the quasi-isomorphisms between bounded below projective complexes are precisely the chain homotopy equivalences, which gives the displayed equivalence. □
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