Proposition 8.4.4 (Commutative localization is smashing). Let \(S \subseteq \pi _*(R)\) be a multiplicative subset of homogeneous elements.
- (1)
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The inclusion \(\Mod _R^{\Loc (S)} \hookrightarrow \Mod _R\) admits a left adjoint \[ S^{-1}(-)\colon \Mod _R \longrightarrow \Mod _R^{\Loc (S)}. \] The subcategory \(\Mod _R^{\Loc (S)}\) inherits a symmetric monoidal structure for which this localization functor is symmetric monoidal and the inclusion is lax symmetric monoidal.
- (2)
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The localization is smashing: for every \(M \in \Mod _R\), the map \[ M \simeq M \otimes _R R \xrightarrow {\ \id _M \otimes \,\eta _R\ } M \otimes _R S^{-1}R \] exhibits \(M \otimes _R S^{-1}R\) as the \(S\)-localization of \(M\). In particular, \[ S^{-1}M \simeq M \otimes _R S^{-1}R \] naturally in \(M\).
Proof. For every \(s \in S\) of degree \(d\) and every \(n \in \Z \), consider the shifted multiplication map \[ s[n]\colon R[d+n] \longrightarrow R[n]. \] An \(R\)-module is local with respect to this small collection of morphisms precisely when it is \(S\)-local: allowing all shifts means that locality tests multiplication by \(s\) on every homotopy group. Since \(\Mod _R\) is presentable by Proposition 22.5.5, Theorem 22.2.2(4) provides the asserted reflection \(S^{-1}(-)\).
The symmetric monoidal \(\infty \)-category \(\Mod _R\) is closed, as recalled after Corollary 19.2.17. If \(N\) is \(S\)-local and \(P\) is any \(R\)-module, then \(\iHom _R(P,N)\) is again \(S\)-local. Indeed, under the natural isomorphism \[ \iHom _R(P,N)[d] \iso \iHom _R(P,N[d]), \] multiplication by \(s\) is obtained by applying \(\iHom _R(P,-)\) to the isomorphism \(s\colon N[d] \iso N\). The internal-hom criterion Lemma 14.5.6 therefore gives the symmetric monoidal structure and the symmetric monoidality of the localization.
It remains to prove the smashing formula. The unit \(\eta _R\colon R \to S^{-1}R\) is an \(S\)-local equivalence, and the proof of the internal-hom criterion shows that \(S\)-local equivalences remain so after tensoring with any \(R\)-module. Consequently, \[ \id _M\otimes \eta _R\colon M\longrightarrow M\otimes _R S^{-1}R \] is an \(S\)-local equivalence. Its target is \(S\)-local: multiplication by \(s\) on \(M\otimes _R S^{-1}R\) is obtained by tensoring \(\id _M\) with the isomorphism \(s\colon (S^{-1}R)[\abs {s}]\iso S^{-1}R\). A local equivalence from \(M\) to a local object exhibits that object as the reflection of \(M\), which proves the claim. □
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