The rational sphere is an Eilenberg–MacLane spectrum

Proof. Consider the map \(\S \to H\Q \) corresponding to \(1 \in \Q \cong \pi _0(H\Q )\). All primes act invertibly on the homotopy groups of \(H\Q \), so \(H\Q \) is \(\S _{\Q }\)-local by Lemma 7.2.8, and this map therefore uniquely refines to a map \(\S _{\Q } \to H\Q \). We have to show that this map induces isomorphisms on homotopy groups. By Lemma 7.2.4, the induced map on \(\pi _0\) is the isomorphism \(\Q \cong \pi _0(\S _{\Q }) \to \pi _0(H\Q ) \cong \Q \) which sends \(1\) to \(1\). It thus remains to show that \(\pi _n(\S _{\Q }) = 0\) for all \(n \neq 0\). For \(n > 0\), the stable homotopy group \(\pi _n(\S ) = \pi _n^{\st }\) is finite by a theorem of Serre (1953), while for \(n < 0\) it vanishes since \(\S \) is connective. In either case it is killed by rationalization. □