Corollary 19.2.17 (Localization of a commutative base algebra). Let \(R\in \CAlg (C)\) be as in Theorem 19.2.14, and let \[ L\colon \Mod _R(C)\rightleftarrows D\colon i \] be a symmetric monoidal Bousfield localization as in Proposition 14.5.3. Then the local \(R\)-module \(L(R)\) canonically underlies a commutative algebra in \(C\) equipped with a morphism \(R\to L(R)\). It is initial among commutative algebras under \(R\) whose underlying \(R\)-modules are local.
Proof. The object \(R\) is the monoidal unit of \(\Mod _R(C)\) and hence the initial object of \(\CAlg (\Mod _R(C))\). By Corollary 14.5.4, its reflection \(L(R)\) is the initial object of \(\CAlg (D)\), and the essential image of \[ \CAlg (D)\hookrightarrow \CAlg (\Mod _R(C)) \] consists of the commutative \(R\)-algebras with local underlying module. The equivalence \(\CAlg (\Mod _R(C))\simeq \CAlg (C)_{R/}\) from Proposition 19.2.15 gives the result. □
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