ScalingStacks

Proof. Since $Y$ has affine diagonal, the map $f:\Spec A\rightarrow Y$ is a relative affine, which implies that $f_{*}$ is colimit preserving and conservative. Hence $f_{*}$ satisfies the monadic Barr-Beck criteria [L4, Theorem 3.4.5], implying that the natural map $\Mod_{A}\rightarrow\Mod_{f_{*}f^{*}}(\qc(Y))$ is an equivalence. By the projection formula, the monad $f_{*}f^{*}(-)$ is equivalent to the functor $f_{*}A\otimes(-)$, with monad structure given by the algebra structure on $A$. As a consequence, we see that $\Mod_{A}$ is equivalent to $\Mod_{f_{*}A}(\qc(Y))$. Now Proposition 4.1 gives that $\Mod_{f_{*}A}(\qc(Y))$ is self-dual as a $\qc(Y)$-module.

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

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

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

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

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

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

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

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

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

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

and so in particular we obtain equivalences

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

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

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