Proposition 6.2.3. For a complex \(A \in \D (\Z )\) and integers \(n \in \Z \), \(k \geq 0\), there are natural isomorphisms of abelian groups \[ \pi _k(K(A,n)) \quad \cong \quad H_{k-n}(A) \qquadtext { and } \pi _n(HA) \quad \cong \quad H_n(A). \]
Proof of Proposition 6.2.3. Combining Corollary 6.2.9 and Lemma 6.2.10, we see that for every complex \(A \in \Kk (\Ab )\) and \(n \in \Z \), the canonical map \[ \Hom _{\Kk (\Ab )}(\Z [-n],A) \to \Hom _{\D (\Z )}(\Z [-n], A) = K(A,n) \] is an isomorphism of animae. In particular, we have \[ \pi _k(K(A,n)) \cong \pi _k\Hom _{\Kk (\Ab )}(\Z [-n],A) \cong \pi _0\Hom _{\Kk (\Ab )}(\Z [k-n], A) \cong H_{k-n}(A) \] and thus \[ \pi _n(HA) \cong \pi _0 H(A[-n]) \cong \pi _0 K(A[-n],0) \cong H_{0}(A[-n]) \cong H_n(A). \qedhere \] □
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