p-completion of the sphere spectrum

Example 7.3.12. All homotopy groups of the sphere spectrum are finitely generated: this is \(\pi _0(\S ) = \Z \) together with the finiteness theorem of Serre (1953) used in the proof of Theorem 7.2.14. Applying Corollary 7.3.11, we obtain \(\pi _0(\S ^{\wedge }_p) \cong \Z _p\), while for \(n \neq 0\) the group \(\pi _n(\S ^{\wedge }_p) \cong \pi _n(\S ) \otimes _{\Z } \Z _p\) is the \(p\)-primary part of the stable stem \(\pi _n(\S )\). Completing at \(p\) thus discards precisely the prime-to-\(p\) information in the stable stems.