Proposition 4.13 (0NY7) · ScalingStacks

0NY7

Proposition 4.13. Let $Y$ be a derived stack with affine diagonal, and let $f:\Spec A\rightarrow Y$ be an affine over $Y$ . Then $\Mod_{A}$ is a self-dual $\qc(Y)$ -module. In particular, for any $X\to Y$ there is a canonical equivalence

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

0NY8

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.

Since the $\infty$ -category $\Mod_{A}$ is a dualizable $\qc(Y)$ -module it follows that the functor

{(-)\otimes_{\qc(Y)}\Mod_{A}}

commutes with limits of $\qc(Y)$ -module categories. Thus we have equivalences

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

Another application of Proposition 4.1 implies the following equivalence

\lim_{A^{\prime}\in\it{Aff}/X}(\Mod_{A^{\prime}}\otimes_{\qc(Y)}\Mod_{A})\simeq\lim_{A^{\prime}\in\it{Aff}/X}(\Mod_{f^{\prime}_{*}A^{\prime}\otimes f_{*}A}(\qc(Y))).

Using that the map $f^{\prime}\times f:\Spec A^{\prime}\times_{Y}\Spec A\rightarrow Y$ is affine and that $(f^{\prime}\times f)_{*}(A^{\prime}\boxtimes A)\simeq f^{\prime}_{*}A^{\prime}\otimes f_{*}A$ , we obtain that the above is further equivalent to

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

\simeq\qc((\colim_{A^{\prime}\in\it{Aff}/X}\Spec A^{\prime})\times_{Y}\Spec A)\simeq\qc(X\times_{Y}\Spec A)

Here we have used that $\qc(-)$ sends all colimits to limits, and that $(-)\times_{Y}\Spec A$ commutes with colimits (since it is the left adjoint to the mapping stack over $Y$ ). ∎