Proposition 7.2.16. For every rational spectrum \(X\), there is a noncanonical isomorphism \[ \bigoplus _{n \in \Z } H(\pi _n X)[n] \iso X. \]
Proof. For every integer \(n \in \Z \), we pick a basis of \(\pi _n(X)\) as a rational vector space. An element of \(\pi _n(X)\) corresponds to a map of spectra \(\S [n] \to X\). Since \(X\) is rational, this uniquely extends to a map \(H\Q [n] = \S _{\Q }[n] \to X\). By taking a direct sum over all basis vectors, this induces a map \(H(\pi _nX)[n] \cong \bigoplus H\Q [n] \to X\) which by construction induces the identity on \(\pi _n(X)\). We now take the direct sum over all \(n \in \Z \). Direct sums of spectra are filtered colimits of finite biproducts, so Lemma 4.4.28 shows that the resulting map \(\bigoplus _{n \in \Z } H(\pi _n X)[n] \to X\) induces an isomorphism on every homotopy group. It is therefore an isomorphism. □
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