Lemma 7.2.4. Let \(X\) be a spectrum. Then there is an isomorphism \[ \pi _*(X[P^{-1}]) \cong \pi _*(X)[P^{-1}]. \]
Proof. In each case of the definition, \(X[P^{-1}]\) is a filtered colimit whose transition maps are multiplication by products of primes in \(P\). By Exercise 7.2.2, the functors \(\pi _n\colon \Sp \to \Ab \) carry these to the corresponding multiplication maps of abelian groups. Since they preserve filtered colimits by Lemma 4.4.28, and since the resulting colimit of abelian groups has the universal property defining localization, we obtain \[ \pi _*(X[P^{-1}]) \cong \pi _*(X)[P^{-1}]. \qedhere \] □
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