Homotopy groups of p-completion

Proof. Let \(F := \bigoplus _{k \geq 0}\S \), with summand inclusions \(\iota _k\colon \S \to F\). Splicing the telescope sequence for \(\S [\frac {1}{p}]\) from Lemma 4.4.34 with the defining exact sequence \(\S \to \S [\frac {1}{p}] \to \S /p^{\infty }\), and then reindexing the source, gives an exact sequence \[ F \xrightarrow {\ d\ } F \longrightarrow \S /p^{\infty }, \] where \(d \circ \iota _0 = \iota _0\) and \(d \circ \iota _{k+1} = \iota _k-p\iota _{k+1}\) for \(k \geq 0\). On \(\pi _0\) this realizes the free resolution \[ 0 \longrightarrow \bigoplus _{k \geq 0}\Z \xrightarrow {\ d_*\ } \bigoplus _{k \geq 0}\Z \longrightarrow \Z /p^{\infty } \longrightarrow 0, \] in which the \(k\)-th basis vector of the middle group maps to \(p^{-k}+\Z \). Thus the exact sequence of spectra is obtained by realizing this free resolution by direct sums of spheres.

Apply \(\hom (-,X)\) to the exact sequence above. For every \(m \in \Z \), the universal property of the coproduct identifies \[ \pi _m\hom (F,X) \cong \Hom _{\Z }\Big (\bigoplus _{k \geq 0}\Z ,\pi _m X\Big ), \] and the map induced by \(d\) agrees with precomposition by \(d_*\). By Theorem 7.3.5, we have \(X_p^{\wedge } \simeq \hom (\S /p^{\infty },X)[1]\), so that \(\pi _n(X_p^{\wedge }) \cong \pi _{n-1}\hom (\S /p^{\infty },X)\). Write \(d_m^*\) for the endomorphism induced by \(d^*\) on \(\pi _m\hom (F,X)\). The long exact sequence associated to the exact sequence \(\hom (\S /p^{\infty },X) \to \hom (F,X) \xrightarrow {d^*} \hom (F,X)\) then gives a short exact sequence \[ 0 \to \coker (d_n^*) \to \pi _n(X_p^{\wedge }) \to \ker (d_{n-1}^*) \to 0. \] By Proposition 6.4.14, its outer terms are \[ \Ext ^1_{\Z }(\Z /p^{\infty },\pi _n X) = L_0^p(\pi _n X) \qquad \text {and}\qquad \Hom _{\Z }(\Z /p^{\infty },\pi _{n-1}X) = L_1^p(\pi _{n-1}X), \] respectively. This gives the asserted short exact sequence. □