Bounded-below projective complexes are K-projective

Proof. First observe that \([P_n,-]\colon \Aa \to \Aa \) is exact for every \(n\). Indeed, exactness may be tested by mapping out of projective objects, and for projective objects \(Q\) we have \[ \Hom _{\Aa }(Q,[P_n,-])\cong \Hom _{\Aa }(Q\otimes P_n,-), \] which is exact because \(Q\otimes P_n\) is projective. It follows that \(\uHom (P,A)\) is acyclic whenever \(A\) is acyclic: filter \(P\) by its bounded truncations and apply the preceding exactness degreewise. The transition maps between the resulting internal hom complexes are degreewise split epimorphisms, so exactness of countable products allows us to pass to their inverse limit. Applying this to the cone of a quasi-isomorphism proves the claim. □