ScalingStacks

Proof. Suppose $G$ is not conservative. Then to prove the lemma, it suffices to exhibit a presentable $\infty$-category $\mathcal{D}$ also cotensored over $\mathcal{C}$ and a nontrivial right adjoint $j:\mathcal{D}\rightarrow\mathcal{M}^{\prime}$ cotensored over $\mathcal{C}$ such that $j\circ G$ is trivial.

Define $\mathcal{D}$ to be the full $\infty$-subcategory of $\mathcal{M}^{\prime}$ of $G$-acyclic objects, that is, objects $m\in\mathcal{M}$ such that $G(m)$ is trivial. Our first task is to show that $\mathcal{D}$ is indeed presentable.

Observe that $\mathcal{D}$ is equivalent to the fiber product $\mathcal{D}\simeq 0\times_{\mathcal{M}}\mathcal{M}^{\prime}$, where the limit is computed in the $\infty$-category $\rm Cat_{\infty}$ of $\infty$-categories. Recall by [L2, Proposition 5.5.3.13], the natural functor $\mathcal{P}r^{\rm L}\rightarrow{\rm Cat}_{\infty}$ preserves limits. Furthermore, the forgetful functor $\mathrm{Mod}_{\mathcal{C}}(\mathcal{P}r^{\rm L})\to\mathcal{P}r^{\rm L}$ also preserves limits since it has a left adjoint (given by induction).

Since the functor $G$ preserves colimits and is $\mathcal{C}$-linear, we may regard it as a morphism in $\mathrm{Mod}_{\mathcal{C}}(\mathcal{P}r^{\rm L})$. Thus $\mathcal{D}$ can be computed as a limit in $\mathrm{Mod}_{\mathcal{C}}(\mathcal{P}r^{\rm L})$, and so can be regarded as an object of $\mathcal{P}r^{\rm L}$. In other words, $\mathcal{D}$ is presentable and furthermore tensored over $\mathcal{C}$. Finally, since $\mathcal{D}$ is tensored over $\mathcal{C}$, it is automatically cotensored as well.

Now it remains to show that the inclusion $j:\mathcal{D}\rightarrow\mathcal{M}^{\prime}$ is indeed a right adjoint and cotensored over $\mathcal{C}$. Since $j$ preserves all limits and colimits (and in particular $\kappa$-filtered colimits), the adjoint functor theorem applies. Finally, since $G$ is cotensored over $\mathcal{C}$, $j$ is as well. ∎

Proof. Observe that for any $\mathcal{D}$ tensored over $\mathcal{C}$, the pullback

induced by the induction $F:\mathcal{M}\to\mathrm{Mod}_{A}\otimes_{\mathcal{C}}\mathcal{M}$ is conservative. In other words, if a functor out of $\mathrm{Mod}_{A}\times\mathcal{M}$ (which preserves colimits in each variable) is trivial when restricted to $\mathcal{M}$, then it is necessarily trivial.

Consequently, switching to opposite categories, we have that the corresponding functor

induced by the forgetful functor $G:\mathrm{Mod}_{A}\otimes_{\mathcal{C}}\mathcal{M}\to\mathcal{M}$ is conservative.

Now we can apply Lemma 4.2 with $\mathcal{M}^{\prime}=\mathrm{Mod}_{A}\otimes\mathcal{M}$ to obtain that $G$ is conservative. ∎

Proof of Proposition 4.1. We first prove that $\mathrm{Mod}_{A}(\mathcal{C})\otimes_{\mathcal{C}}\mathcal{M}$ is equivalent to $\mathrm{Mod}_{A}(\mathcal{M})$ by the natural evaluation functor. Consider the adjunction

where $F(-)=A\otimes-$ is the induction, and $G$ is the forgetful functor.

The above adjunction induces an adjunction

and thus a functor to modules over the monad $T=(G\otimes{\rm id})\circ(F\otimes{\rm id})$ acting on $\mathcal{M}$. The functor underlying $T$ is given by tensoring with $A$, so we also have an equivalence $\mathrm{Mod}_{T}(\mathcal{M})\simeq\mathrm{Mod}_{A}(\mathcal{M})$.

By its universal characterization, the functor $G\otimes{\rm id}$ is colimit preserving. Note as well that $G$ and hence $G\otimes{\rm id}$ is also $\mathcal{C}$-linear (or in other words, the adjunction satisfies an analogue of the projection formula). Thus it follows from Lemma 4.3 that $G\otimes{\rm id}$ is also conservative. Thus $G\otimes{\rm id}$ satisfies the monadic Barr-Beck conditions, and we obtain the desired equivalence $\mathrm{Mod}_{A}(\mathcal{C})\otimes_{\mathcal{C}}\mathcal{M}\simeq\mathrm{Mod}_{A}(\mathcal{M})$.

Next, we can apply this to the instance where $\mathcal{M}$ is the $\infty$-category of left modules over another associative algebra $A^{\prime}$ to conclude that there is a natural equivalence $\mathrm{Mod}_{A}(\mathcal{C})\otimes_{\mathcal{C}}\mathrm{Mod}_{A^{\prime}}(\mathcal{C})\simeq\mathrm{Mod}_{A}(\mathrm{Mod}_{A^{\prime}}(\mathcal{C}))$. We now have a chain of adjunctions

in which the composite $G^{\prime}\circ G^{\prime\prime}$ is colimit preserving and conservative, and hence satisfies the monadic Barr-Beck conditions.

Furthermore, the above adjunction naturally extends to a diagram in which the cycle of left adjoints (denoted by bowed arrows), and hence also the cycle of right adjoints (denoted by straight arrows), commute

Here $F(-)=A_{1}\otimes A_{2}\otimes-$ is the induction, $G$ is the forgetful functor, $f$ is the natural functor factoring through $\mathrm{Mod}_{A}(\mathcal{C})\otimes_{\mathcal{C}}\mathrm{Mod}_{A^{\prime}}(\mathcal{C})$, and $g$ is its right adjoint. From this diagram, we obtain a morphism of monads

Now the underlying functors of the monads $GF(-)$ and $G^{\prime}G^{\prime\prime}F^{\prime\prime}F^{\prime}(-)$ are both equivalent to the tensor $A\otimes A^{\prime}\otimes(-)$, so the above morphism of monads is an equivalence. Thus we obtain the promised equivalence $\mathrm{Mod}_{A}(\mathcal{C})\otimes_{\mathcal{C}}\mathrm{Mod}_{A^{\prime}}(\mathcal{C})\simeq\mathrm{Mod}_{A}(\mathrm{Mod}_{A^{\prime}}(\mathcal{C}))\simeq\mathrm{Mod}_{A\otimes A^{\prime}}(\mathcal{C})$.

Finally, we show that the $\infty$-category of left $A$-modules $\mathrm{Mod}_{A}(\mathcal{C})$ is a dualizable $\mathcal{C}$-module by directly exhibiting the $\infty$-category of right $A$-modules $\mathrm{Mod}_{A^{\rm op}}(\mathcal{C})$ as its dual. The trace map is given by the two-sided bar construction

The unit map is given by the induction

where we regard $A\otimes c$ as an $A$-bimodule.

One can verify directly that the composition

is equivalent to the identity. First, $({\rm id}\otimes u)(M)$ is equivalent to $A\otimes M$ regarded as an $A\otimes A^{\rm op}\otimes A$-module, and second, $(\tau\otimes{\rm id})(A\otimes M)$ is equivalent to $A\otimes_{A}M\simeq M$. ∎