Rational periodic complex K-theory

Proof. By the smashing description of rationalization and Theorem 7.2.14, we have \(\KU _{\Q }\simeq \KU \otimes H\Q \), so \(\KU _{\Q }\) is canonically a commutative \(H\Q \)-algebra. Under the symmetric monoidal equivalence \(\D (\Q )\simeq \Mod _{H\Q }\) of Corollary 8.3.4, the polynomial algebra \(\Q [\beta ]\) is the free commutative \(\Q \)-algebra on a generator in degree \(2\). The rationalized Bott class therefore determines a morphism of commutative \(H\Q \)-algebras \[ H\Q [\beta ]\longrightarrow \KU _{\Q }. \] The canonical map \(H\Q [\beta ]\to H\Q [\beta ^{\pm 1}]\) exhibits its target as \((H\Q [\beta ])[\beta ^{-1}]\): indeed, the induced map from this localization is an isomorphism on homotopy groups by Lemma 8.4.6. Since the Bott class is invertible in \(\pi _*(\KU _{\Q })\), the preceding morphism therefore extends uniquely to \(H\Q [\beta ^{\pm 1}]\). It induces the isomorphism \[ \Q [\beta ^{\pm 1}]\xrightarrow {\ \cong \ }\pi _*(\KU _{\Q }) \] from Corollary 9.4.14 and Lemma 7.2.4, and hence is an isomorphism of commutative ring spectra. □