Corollary 6.2.9. For a chain complex \(C_{\bullet } \in \Ch (\Ab )\) and \(n \in \Z \) there is a natural isomorphism \[ \pi _0 \Hom _{\Kk (\Ab )}(\Z [n],C_{\bullet }) \cong H_n(C). \] In particular, the functor \(\Hom _{\Kk (\Ab )}(\Z [n],-)\colon \Kk (\Ab ) \to \An \) corepresented by \(\Z [n]\) inverts quasi-isomorphisms.
Proof. By Lemma 6.2.8, the left-hand side is the set of chain maps \(\Z [n] \to C_{\bullet }\) up to chain homotopy. The first claim follows, since a chain map \(\Z [n] \to C_{\bullet }\) is nothing but an element of \(\ker (d_n\colon C_n \to C_{n-1})\) and two chain maps are chain homotopic if and only if their difference lies in the image of \(d_{n+1}\colon C_{n+1} \to C_n\).
The second claim now follows from the stability of \(\Kk (\Ab )\) established in Proposition 6.2.7: for \(k \geq 0\) we have \[ \pi _k\Hom _{\Kk (\Ab )}(\Z [n],A) \cong \pi _0\Hom _{\Kk (\Ab )}(\Z [n+k], A) \cong H_{n+k}(A). \qedhere \] □
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