Lemma 8.4.6 (Inverting a single element). Let \(R\) be a commutative ring spectrum, let \(s \in \pi _d(R)\) be a homogeneous element, and let \(S := \{1,s,s^2,\dots \}\) be the multiplicative subset it generates. Then for every \(M \in \Mod _R\) the canonical map exhibits the telescope \[ M[s^{-1}] := \colim \bigl (M \xrightarrow {\ s\ } M[-d] \xrightarrow {\ s\ } M[-2d] \xrightarrow {\ s\ } \cdots \bigr ) \] as the \(S\)-localization of \(M\). On homotopy groups it is given by graded localization: \[ \pi _*(M[s^{-1}]) \cong \pi _*(M)[s^{-1}]. \]
Proof. The transition maps are shifts of the module map of Observation 8.4.2, so the telescope is a colimit in \(\Mod _R\). Since homotopy groups preserve filtered colimits by Lemma 4.4.28, we have \[ \pi _k(M[s^{-1}]) \cong \colim \bigl (\pi _k(M)\xrightarrow {\ s\cdot \ }\pi _{k+d}(M) \xrightarrow {\ s\cdot \ }\pi _{k+2d}(M)\longrightarrow \cdots \bigr ), \] which is the degree-\(k\) part of \(\pi _*(M)[s^{-1}]\). In particular, multiplication by \(s\) on \(M[s^{-1}]\) is an isomorphism, so the telescope is \(S\)-local.
Let \(N\) be any \(S\)-local \(R\)-module. Mapping out of the telescope gives \[ \Hom _{\Mod _R}(M[s^{-1}],N) \simeq \lim _j\Hom _{\Mod _R}(M[-jd],N). \] Every transition map in this limit is an equivalence, because under the shift adjunction it is induced by the isomorphism \(s\colon N[d]\iso N\). Evaluation at the zeroth term therefore gives an equivalence \[ \Hom _{\Mod _R}(M[s^{-1}],N) \iso \Hom _{\Mod _R}(M,N), \] induced by precomposition with the canonical map \(M\to M[s^{-1}]\). This is the universal property of the \(S\)-localization. □
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