Stable and spectral consequences

Proof. By Corollary 19.1.17, the forgetful functor \(\LMod _A(C)\to C\) creates and preserves finite limits and colimits. Thus \(\LMod _A(C)\) has a zero object, and a commutative square in \(\LMod _A(C)\) is a pullback if and only if it is a pushout, since this may be checked in \(C\). Hence \(\LMod _A(C)\) is stable. The proof for right modules is dual. □

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 \[ f_*f^*\simeq S\otimes _R-\colon \Mod _R\longrightarrow \Mod _R. \] □

Proof. Recall from Lemma 23.1.20 that a cocartesian fibration \(p\colon E \to C\) is also a cartesian fibration if and only if every induced functor between its fibers admits a right adjoint. For \(-\vert _{\Comm }\colon \Mod (\Sp ) \to \CAlg (\Sp )\), this is precisely Proposition 19.5.3(1). □

Proof. We may assume that \(D\) is small by enlarging the ambient universe if necessary. Consider the fully faithful Yoneda embedding \[ D \hookrightarrow \widehat D := \Ind (D). \] The \(\infty \)-category \(\widehat D\) is presentable and stable, and the inclusion of \(D\) is exact; see Proposition 22.3.6 and [Lurie (2017), Proposition 1.1.3.6]. In particular, it preserves mapping spectra. Being presentable and stable, \(\widehat D\) is \(\Sp \)-local, so Theorem 18.5.3, Proposition 18.5.2 equip it with a unique \(\Sp \)-module structure in \(\PrL \); in particular \(\widehat D\) is canonically left-tensored over \(\Sp \). The resulting action functor \(-\otimes M\colon \Sp \to \widehat D\) is colimit-preserving and satisfies \(\S \otimes M \simeq M\), so by Theorem 4.4.14 it agrees with the tensoring of Proposition 4.4.9; its right adjoint is therefore \(\hom _{\widehat D}(M,-)\), and for spectra \(E\) there are natural isomorphisms \[ \Hom _{\Sp }(E,\hom _D(M,M)) \iso \Hom _{\widehat D}(E \otimes M,M). \] Consequently, \(A:=\hom _D(M,M)\) is an endomorphism object of \(M \in \widehat D\). Applying Corollary 19.4.7 gives \(A\) a preferred associative algebra structure and equips \(M\) with a preferred left \(A\)-module structure. This proves (1).

Using this module structure, we may form the two-sided bar construction in \(\widehat D\), just as in Section 19.2; see also [Lurie (2017), Construction 4.4.2.7]. Its geometric realization defines a colimit-preserving functor \[ F := - \otimes _A M\colon \RMod _A \to \widehat D. \] Since both categories are presentable, Theorem 22.2.5 gives a right adjoint \[ G\colon \widehat D \to \RMod _A. \] Let \(U\colon \RMod _A \to \Sp \) denote the forgetful functor. For \(K \in \Sp \) and \(X \in \widehat D\), the free-right-module adjunction and the tensor–Hom adjunction give natural isomorphisms \begin {align*} \Hom _{\Sp }(K,UG(X)) &\simeq \Hom _{\RMod _A}(K \otimes A,G(X)) \\ &\simeq \Hom _{\widehat D}((K \otimes A) \otimes _A M,X) \\ &\simeq \Hom _{\widehat D}(K \otimes M,X) \\ &\simeq \Hom _{\Sp }(K,\hom _{\widehat D}(M,X)). \end {align*}

The Yoneda lemma therefore provides a natural isomorphism \[ UG(X) \simeq \hom _{\widehat D}(M,X). \] Restricting \(G\) along \(D \hookrightarrow \widehat D\) gives the lift in (2). □

Proof. By Corollary 18.5.6, the unit functor \(-\otimes \unit \colon \Sp \to C\) is the essentially unique colimit-preserving symmetric monoidal functor out of \(\Sp \), and its right adjoint is the mapping-spectrum functor \(\hom _C(\unit ,-)\) (Corollary 4.4.10). By Proposition 14.3.6 this right adjoint is canonically lax symmetric monoidal. In particular, it carries the commutative algebra \(\unit \in \CAlg (C)\) to the commutative algebra \[ A\coloneqq \hom _C(\unit ,\unit )\in \CAlg (\Sp ). \] Applying the functoriality of commutative module categories under lax symmetric monoidal functors gives a lax symmetric monoidal functor \[ \Phi \colon C\simeq \Mod _{\unit }(C)\longrightarrow \Mod _A(C), \qquad X\longmapsto \hom _C(\unit ,X), \] whose underlying \(A\)-module-valued functor is that of Proposition 19.5.5. Since \(\unit \) is compact, \(\Phi \) preserves filtered colimits; being exact, it preserves finite colimits, and hence all colimits.

Using the right-module equivalence of Observation 19.2.4, we may identify \(\Mod _A(C)\) with \(\RMod _A(C)\). Under this identification, the two-sided bar construction used in the proof of Proposition 19.5.5 defines a colimit-preserving functor \[ \Psi \colon \Mod _A(C)\longrightarrow C, \qquad M\longmapsto M\otimes _A\unit , \] where \(\unit \) carries the left \(A\)-module structure of Proposition 19.5.5. The same construction exhibits \(\Psi \) as a left adjoint to \(\Phi \). On the free module we have \(\Psi (A)\simeq \unit \), so the counit and unit of \(\Psi \dashv \Phi \) restrict to isomorphisms on the generators: the counit at \(\unit \) is the canonical isomorphism \(\Psi \Phi (\unit )=A\otimes _A\unit \simeq \unit \), and the unit at \(A\) is the isomorphism \(A\to \Phi \Psi (A)=\hom _C(\unit ,\unit )=A\). Both \(\Phi \) and \(\Psi \) are exact and preserve colimits, so the full subcategories of \(C\) and of \(\Mod _A(C)\) on which the counit, respectively the unit, is an isomorphism are stable and closed under colimits. Since \(\unit \) is a compact generator of \(C\) and \(A\) is a compact generator of \(\Mod _A(C)\) (Example 22.3.10), these subcategories are everything. Hence \(\Phi \) is an equivalence.

It remains to promote the lax symmetric monoidal structure on \(\Phi \) to a strong one. Its unit comparison \(A\to \Phi (\unit )\) is an isomorphism, and its multiplication comparison \[ \Phi (X)\otimes _A\Phi (Y)\longrightarrow \Phi (X\otimes Y) \] is an isomorphism for \(X=Y=\unit \), where both sides are \(A\). Fixing \(X=\unit \), the full subcategory of those \(Y\) for which the comparison is an isomorphism is stable, closed under colimits, and contains \(\unit \), hence is all of \(C\). Running the same argument in the second variable, now for each fixed \(Y\in C\), shows that the comparison is an isomorphism for all \(X\) and \(Y\). Thus \(\Phi \) is a symmetric monoidal equivalence. □