Corollary 8.4.5 (Localization of a commutative ring spectrum). Let \(R\) be a commutative ring spectrum and let \(S \subseteq \pi _*(R)\) be a multiplicative subset of homogeneous elements. Then the \(R\)-module \(S^{-1}R\) carries a canonical structure of a commutative ring spectrum \(R[S^{-1}]\), and the localization map refines to a morphism \[ \eta \colon R \longrightarrow R[S^{-1}] \] in \(\CAlg (\Sp )\). It has the following universal property: for every commutative ring spectrum \(A\), precomposition with \(\eta \) induces a monomorphism of animae \[ -\circ \eta \colon \Hom _{\CAlg (\Sp )}(R[S^{-1}],A) \hookrightarrow \Hom _{\CAlg (\Sp )}(R,A) \] whose image consists of those maps \(f\colon R \to A\) for which \(\pi _*(f)\) carries every element of \(S\) to a unit of \(\pi _*(A)\).
Proof. By Proposition 8.4.4, the reflection \[ S^{-1}(-)\colon \Mod _R \longrightarrow \Mod _R^{\Loc (S)} \] is a symmetric monoidal Bousfield localization. The Part II result Corollary 19.2.17 therefore equips \(S^{-1}R\) with a commutative \(R\)-algebra structure and characterizes it as initial among commutative \(R\)-algebras whose underlying modules are \(S\)-local. Under the equivalence \[ \CAlg (\Mod _R)\simeq \CAlg (\Sp )_{R/}, \] this gives the asserted morphism \(\eta \colon R\to R[S^{-1}]\) of commutative ring spectra.
Let \(f\colon R\to A\) be a morphism of commutative ring spectra. Its underlying \(R\)-module is \(S\)-local if and only if \(\pi _*(f)\) sends every \(s\in S\) to a unit. One direction is immediate. Conversely, suppose that \(A\) is \(S\)-local and let \(s'=\pi _*(f)(s)\) have degree \(d\). Since multiplication by \(s'\) is bijective, there is an element \(u\in \pi _{-d}(A)\) such that \(s'u=1\). Graded commutativity gives \(us'=(-1)^d\), so \((-1)^du\) is a left inverse of \(s'\), while \(u\) is a right inverse. Since left and right inverses coincide, \(s'\) is a unit.
The initial property of \(S^{-1}R\) among commutative \(R\)-algebras with local underlying module now gives the claimed factorization property. Conversely, every morphism that factors through \(R[S^{-1}]\) inverts \(S\), since the image of \(S\) in \(\pi _*(R[S^{-1}])\) consists of units and ring morphisms preserve units. □
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