ScalingStacks

Proof. Consider the Cartesian diagram

We define a functor $\qc({X\times_{Y}X^{\prime}})\rightarrow\Fun(\qc(X),\qc({X^{\prime}}))$ by sending a quasi-coherent sheaf $M\in\qc(X\times_{Y}X^{\prime})$ to the functor $\tilde{f}_{*}(M\otimes\tilde{g}^{*}-)$. This is a colimit preserving functor, since $\tilde{f}$ is perfect. Furthermore, using the projection formula for the map $\tilde{f}$, this functor naturally admits the extra structure of $\qc(Y)$-linearity as follows

$$\tilde{f}_{*}(M\otimes\tilde{g}^{*}-)\otimes g^{*}V\simeq\tilde{f}_{*}(M\otimes\tilde{g}^{*}(-)\otimes\tilde{f}^{*}g^{*}V)$$

$$\simeq\tilde{f}_{*}(M\otimes\tilde{g}^{*}(-)\otimes\tilde{g}^{*}f^{*}V)\simeq\tilde{f}_{*}(M\otimes\tilde{g}^{*}(-\otimes f^{*}V)),$$

for any $V\in\qc(Y)$. Therefore, we in fact obtain a functor $\qc(X\times_{Y}X^{\prime})\rightarrow\Fun_{Y}(\qc(X),\qc(X^{\prime}))$, and the rest of this proof will be devoted to showing it is an equivalence.

Recall that the $\infty$-category of quasi-coherent sheaves on $X^{\prime}$ is given by the limit

$$\qc({X^{\prime}})\simeq\lim_{\it{Aff}/X^{\prime}}\Mod_{A}.$$

By working locally in the target, this provides a description of the $\infty$-category of $\qc(Y)$-linear functors with values in $\qc({X^{\prime}})$ as the limit

$$\Fun_{Y}(\qc(X),\qc({X^{\prime}}))\simeq\Fun_{Y}(\qc(X),\lim_{\it{Aff}/X^{\prime}}\Mod_{A})\simeq\lim_{\it{Aff}/X^{\prime}}\Fun_{Y}(\qc(X),\Mod_{A})$$

Likewise, we have a description of the $\infty$-category of quasi-coherent sheaves on the fiber product $X\times_{Y}X^{\prime}$ as a limit

$$\qc({X\times_{Y}X^{\prime}})\simeq\lim_{\it{Aff}/X^{\prime}}\qc({X\times_{Y}\Spec A}).$$

This follows from the fact that the functor $\qc(-)$ takes colimits to limits, and the fiber product functor $X\times_{Y}(-)$ commutes with all colimits (because it has a right adjoint).

Now one can analyze the functor $\qc({X\times_{Y}X^{\prime}})\rightarrow\Fun_{Y}(\qc(X),\qc(X^{\prime}))$ by considering the terms in the above two limits. That is, to prove the theorem, it suffices to prove it locally in the target $X^{\prime}$: for any $\Spec A\rightarrow X^{\prime}$, we must show that the functor

$$\qc({X\times_{Y}\Spec A})\rightarrow\Fun_{Y}(\qc(X),\Mod_{A})$$

is an equivalence.

We will prove this in two steps. First, we will deal with case that the base $Y$ is affine. Afterward, we will use this case to deal with a general base $Y$.

So assume for the time being that $Y=\Spec B$. Then $X$ is a perfect stack over $B$, and by Corollary 4.8, $\qc(X)$ is a self-dual $\Mod_{B}$-module. Thus we have equivalences

$$\Fun_{B}(\qc(X),\Mod_{A})\simeq\Fun_{B}(\Mod_{B},\qc(X)^{\vee}\otimes_{B}\Mod_{A})\simeq\qc(X)\otimes_{B}\Mod_{A}.$$

By Proposition 4.13 we know that the functor $\qc(-)$ takes affine base change to tensor product of $\infty$-categories. Therefore we have an equivalence

$$\qc({X\times_{B}\Spec A})\simeq\qc(X)\otimes_{B}\Mod_{A}.$$

Putting together the above equivalences, we conclude that we have equivalences

$$\qc({X\times_{B}\Spec A})\simeq\qc(X)\otimes_{B}\Mod_{A}\simeq\Fun_{B}(\qc(X),\Mod_{A}),$$

This proves the theorem when $Y$ is affine.

Working locally in the base $Y$, we will now use the above discussion to prove the theorem in general. As above, since $\qc(-)$ and fiber products behave well with respect to colimits we can calculate the $\infty$-category of quasi-coherent sheaves on the fiber product $X\times_{Y}\Spec A$ as the limit

$$\qc(X\times_{Y}\Spec A)\simeq\lim_{B\in\it{Aff}/Y}\qc(X\times_{Y}\Spec B\times_{Y}\Spec A).$$

To calculate the $\infty$-category of functors, we use the following: by Proposition 4.13, the $\infty$-category $\Mod_{A}$ is a dualizable $\qc(Y)$-module and hence the functor $\Mod_{A}\otimes_{\qc(Y)}(-)$ commutes with all limits. Therefore we have an equivalence

$$\Mod_{A}\simeq\Mod_{A}\otimes_{\qc(Y)}(\lim_{B\in\it{Aff}/Y}\Mod_{B})\simeq\lim_{B\in\it{Aff}/Y}\Mod_{A}\otimes_{\qc(Y)}\Mod_{B},$$

and so in particular we obtain equivalences

$$\Fun_{Y}(\qc(X),\Mod_{A})\simeq\Fun_{Y}(\qc({X}),\lim_{B\in\it{Aff}/Y}\Mod_{A}\otimes_{\qc(Y)}\Mod_{B})$$

$$\simeq\lim_{B\in\it{Aff}/Y}\Fun_{Y}(\qc({X}),\Mod_{A}\otimes_{\qc(Y)}\Mod_{B}).$$

By the adjunction between induction and restriction, and a repeated application of Proposition 4.13, we also have equivalences

$$\Fun_{Y}(\qc({X}),\Mod_{A}\otimes_{\qc(Y)}\Mod_{B})\simeq\Fun_{B}(\qc(X)\otimes_{\qc(Y)}\Mod_{B},\Mod_{A}\otimes_{\qc(Y)}\Mod_{B}).$$

$$\simeq\Fun_{B}(\qc({X\times_{Y}\Spec B}),\qc({\Spec A\times_{Y}\Spec B}))$$

Finally, by the above discussion and the affine case of the theorem with base $\Spec B$, we obtain the following chain of equivalences

$$\qc({X\times_{Y}\Spec A})\simeq\lim_{B\in\it{Aff}/Y}\qc({X\times_{Y}\Spec B\times_{Y}\Spec A})$$

$$\simeq\lim_{B\in\it{Aff}/Y}\Fun_{B}(\qc({X\times_{Y}B}),\qc({A\times_{Y}\Spec B}))\simeq\Fun_{Y}(\qc(X),\Mod_{A}).$$

Since we previously reduced the theorem to the case when $X^{\prime}=\Spec A$, this completes the proof. ∎