Proposition 19.5.3. Let \(f\colon R \to S\) be a morphism of commutative ring spectra.
- (1)
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The functor \(f^*\colon \Mod _R \to \Mod _S\) admits a right adjoint \[ f_*\colon \Mod _S \to \Mod _R. \] For every \(S\)-module \(M\), the underlying spectrum of \(f_*(M)\) is naturally isomorphic to the underlying spectrum of \(M\).
- (2)
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The functor \(f_*\colon \Mod _S \to \Mod _R\) admits a further right adjoint \[ \ihom _R(S,-)\colon \Mod _R \to \Mod _S \] whose underlying \(R\)-module is the internal hom: it satisfies \(f_*\ihom _R(S,-) \simeq \iHom _R(S,-)\) as functors \(\Mod _R \to \Mod _R\).
Proof. For part (1), Theorem 22.2.5 reduces the existence of \(f_*\) to the fact that \(f^*\) preserves small colimits. This may be tested after postcomposing with the forgetful functor \(\Mod _S \to \Sp \), where \(f^*\) is given by the relative tensor product \(S \otimes _R -\).
To identify the underlying spectrum of \(f_*(M)\), we may equivalently pass to left adjoints and show that for every \(M_0 \in \Sp \) there is a natural isomorphism \(f^*(F_R(M_0)) \simeq F_S(M_0)\). This follows from the functoriality of extension of scalars: \(F_R\) and \(F_S\) arise from the unit maps \(\eta _R\colon \S \to R\) and \(\eta _S\colon \S \to S\), and \(\eta _S=f\circ \eta _R\).
For part (2), the adjoint functor theorem applies once more because \(f_*\) preserves colimits. This can be checked on underlying spectra, since restriction of scalars does not change them and the forgetful functors create colimits. The identification \(f_*\ihom _R(S,-) \simeq \iHom _R(S,-)\) may be checked after passing to left adjoints, where it is the identity \[ f_*f^*\simeq S\otimes _R-\colon \Mod _R\longrightarrow \Mod _R. \] □
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