2.1.1.

For a morphism $f : T \to S$ of affine schemes, we write

\begin{align*} \mathbf{ind} : \mathrm{QCoh}(S)\textbf{-mod} \rightleftharpoons \mathrm{QCoh}(T) \textbf{-mod} : \mathbf{res} \end{align*}

for the adjunction defined by extension and restriction of scalars along the map of symmetric monoidal categories

\begin{align*} f^* : \mathrm{QCoh}(S) \to \mathrm{QCoh}(T). \end{align*}

2.1.2.

Let us recall some features of the theory of sheaves of categories.

To every prestack $\mathscr{Y}$ , one associates the symmetric monoidal $(\infty, 2)$ -category

\begin{align*} \operatorname{ShCat}(\mathscr{Y}) = \lim_{S \to \mathscr{Y}} \mathrm{QCoh}(S)\textbf{-mod} \end{align*}

of sheaves of categories on $\mathscr{Y}$ . This limit is taken over test affine schemes mapping to $\mathscr{Y}$ , with the transition functors given by $\mathbf{ind}$ . The value of a sheaf of categories $\mathscr{C}$ at a test affine scheme S is denoted

\begin{align*} \mathbf{\Gamma}(S; \mathscr{C}) \in \mathrm{QCoh}(S)\textbf{-mod}. \end{align*}

The monoidal unit of $\operatorname{ShCat}(\mathscr{Y})$ is denoted $\mathrm{QCoh}_{/\mathscr{Y}}$ . It is given by the assignment

\begin{align*} \mathbf{\Gamma}(S; \mathrm{QCoh}_{/\mathscr{Y}}) = \mathrm{QCoh}(S). \end{align*}

To a morphism of prestacks $f : \mathscr{Z} \to \mathscr{Y}$ , one associates the adjunction

(2.1) \begin{equation} \mathbf{cores}_f : \operatorname{ShCat}(\mathscr{Y}) \rightleftharpoons \operatorname{ShCat}(\mathscr{Z}) : \mathbf{coind}_f.\end{equation}

For any cartesian diagram of prestacks as follows, the natural transformation

\begin{align*} \mathbf{cores}_{g} \circ \mathbf{coind}_{f} \to \mathbf{coind}_{f'} \circ \mathbf{cores}_{g'} \end{align*}

obtained by adjunction from the identification

\begin{align*} \mathbf{cores}_{f'} \circ \mathbf{cores}_g \overset{\sim}{\rightarrow} \mathbf{cores}_{g'} \circ \mathbf{cores}_f \end{align*}

is an isomorphism.

2.1.3.

Note that

\begin{align*} \mathbf{coind}_f(\mathrm{QCoh}_{/\mathscr{Z}}) \end{align*}

is naturally a sheaf of symmetric monoidal categories over $\mathscr{Y}$ , and the adjunction (2.1) enhances to an adjunction

\begin{align*} \mathbf{Loc}_{\mathscr{Z}/\mathscr{Y}} : \mathbf{coind}_f(\mathrm{QCoh}_{/\mathscr{Z}})\textbf{-mod} (\operatorname{ShCat}(\mathscr{Y})) \rightleftharpoons \operatorname{ShCat}(\mathscr{Z}) : \mathbf{\Gamma}_{\mathscr{Z}/\mathscr{Y}}. \end{align*}

The morphism f is called relatively 1-affine when this adjunction is an equivalence.

2.1.4.

A prestack $\mathscr{Y}$ is said to be 1-affine if the morphism $\mathscr{Y} \to \mathrm{pt}$ is relatively 1-affine. We remind the reader of the following.

• – Every (eventually coconnective, left, with affine diagonal) algebraic stack is 1-affine.

• – The de Rham prestack $Z_\mathrm{dR}$ of any finite-type scheme Z is 1-affine.

2.2.1.

Let ${\mathbf{G}_m}\textbf{-mod}$ denote the symmetric monoidal $(\infty, 2)$ -category of dg categories with a weak action of ${\mathbf{G}_m}$ . Recall from [Reference Arinkin and GaitsgoryAG15, § A.2] that this category is equipped with a symmetric monoidal auto-equivalence, known as shearing:

\begin{align*} (-)^\Rightarrow : {\mathbf{G}_m}\textbf{-mod} \to {\mathbf{G}_m}\textbf{-mod}. \end{align*}

We write $(-)^\Leftarrow$ for its inverse.

2.2.2.

Let $\mathscr{A}$ be a symmetric monoidal category acted on by ${\mathbf{G}_m}$ (i.e. $\mathscr{A}$ is a commutative algebra object of ${\mathbf{G}_m}\textbf{-mod}$ ). Since shearing is symmetric monoidal, $\mathscr{A}^\Rightarrow$ is another symmetric monoidal category acted on by ${\mathbf{G}_m}$ .

The goal of this subsection is to give an answer to the question of what it takes to equip a dg category $\mathscr{M}$ with an action of $\mathscr{A}^\Rightarrow$ (without regard for its ${\mathbf{G}_m}$ -action).

That is, we will give a description of

\begin{align*} \mathscr{A}^\Rightarrow\textbf{-mod} = \mathscr{A}^\Rightarrow\textbf{-mod}(\operatorname{dgCat}). \end{align*}

2.2.3 A toy model

The construction we are about to give has an analogue one category level down. Namely, for a graded algebra A, and a vector space M, the following data are equivalent:

1. (a) a non-graded A-module structure on M;

2. (b) a graded $A[t^{\pm 1}]$ -module structure on $M[t^{\pm 1}]$ .

Indeed, if M is given a non-graded A-module structure, then we endow $M[t^{\pm 1}]$ with the structure of a graded A-module, where a homogeneous element a of degree n acts by

\begin{align*} a \cdot (m t^k) = (am) t^{n + k}. \end{align*}

This action commutes with the natural graded action of $k[t^{\pm 1}]$ on $M[t^{\pm 1}]$ and hence makes $M[t^{\pm 1}]$ into a graded $A[t^{\pm 1}]$ -module. To recover the original non-graded action of A on M, observe that A is the degree-zero part of $A[t^{\pm 1}]$ and M is the degree-zero part of $M[t^{\pm 1}]$ .

2.2.4.

Since the morphism $f: \mathrm{pt} \to B{\mathbf{G}_m}$ is relatively 1-affine, we obtain an identification

\begin{align*} \operatorname{dgCat} \simeq \mathrm{QCoh}({\mathbf{G}_m})\textbf{-mod}({\mathbf{G}_m}\textbf{-mod}){,} \quad \mathscr{M} \mapsto \mathscr{M} \otimes \mathrm{QCoh}({\mathbf{G}_m}), \end{align*}

where $\mathrm{QCoh}({\mathbf{G}_m})$ is viewed as a ${\mathbf{G}_m}$ -equivariant symmetric monoidal category via its pointwise tensor product.

Now, let $\mathscr{B}$ be any symmetric monoidal category. From the monadic adjunction

\begin{align*} \mathbf{ind} : \operatorname{dgCat} \rightleftharpoons \mathscr{B} \textbf{-mod}(\operatorname{dgCat}) : \mathbf{oblv}, \end{align*}

we obtain the following commutative diagram in which both horizontal arrows are equivalences.

(2.2)

2.2.5.

Now, let $\mathscr{B}$ be the underlying symmetric monoidal category of $\mathscr{A}^\Rightarrow$ . By the projection formula for sheaves of categories, we obtain the equivalences of commutative algebras

\begin{align*} \mathscr{B} \otimes \mathrm{QCoh}({\mathbf{G}_m}) & \simeq \mathbf{coind}_f \circ \mathbf{cores}_f(\mathscr{A}^\Rightarrow) \\ & \simeq \mathscr{A}^\Rightarrow \otimes \mathbf{coind}_f(\mathrm{Vect}) \simeq \mathscr{A}^\Rightarrow \otimes \mathrm{QCoh}({\mathbf{G}_m})\end{align*}

in ${\mathbf{G}_m} \textbf{-mod}$ .

As a result, for our choice of $\mathscr{B}$ , the diagram (2.2) rewrites as follows.

(2.3)

The difference between this diagram and (2.2) is that here, the ${\mathbf{G}_m}$ action on $\mathscr{A}^\Rightarrow \otimes \mathrm{QCoh}({\mathbf{G}_m})$ is the natural one coming from the pre-existing action on $\mathscr{A}^\Rightarrow$ , while in (2.2) the ${\mathbf{G}_m}$ action on $\mathscr{B}$ is trivial.

2.2.6.

Note that shearing induces a commutative diagram in which the horizontal arrows are equivalences.

Concatenating this with (2.3), we obtain the following commutative diagram where the lower horizontal equivalence is given by

\begin{align*} \mathscr{M} \mapsto \mathscr{M} \otimes \mathrm{QCoh}({\mathbf{G}_m})^\Leftarrow. \end{align*}

2.2.7.

This is our answer to the question posed in § 2.2.2: giving an action of $\mathscr{A}^\Rightarrow$ on $\mathscr{M}$ is equivalent to giving a ${\mathbf{G}_m}$ -equivariant action of $\mathscr{A}$ on

\begin{align*} \mathscr{M} \otimes \mathrm{QCoh}({\mathbf{G}_m})^\Leftarrow, \end{align*}

commuting with the natural ${\mathbf{G}_m}$ -equivariant action of $\mathrm{QCoh}({\mathbf{G}_m})^\Leftarrow$ on the right factor.

2.2.8 Remark

Since

\begin{align*}\mathrm{QCoh}({\mathbf{G}_m})^\Leftarrow \simeq k[\beta^\pm]{\operatorname{-mod}}\end{align*}

where $|\beta| = 2$ , the category $\mathscr{M} \otimes\mathrm{QCoh}({\mathbf{G}_m})^\Leftarrow$ is a 2-periodization of $\mathscr{M}$ . One may interpret the result of this subsection as a deperiodization statement, showing what additional data is needed to reconstruct an action of $\mathscr{A}^\Rightarrow$ on $\mathscr{M}$ from an action of $\mathscr{A}$ on the 2-periodization of $\mathscr{M}$ .

2.3.1.

For the rest this section, we will let Z denote a fixed choice of quasi-smooth scheme. Instead of performing Construction 2.1 as is, we will consider its reformulation in the terms provided by § 2.2.

Construction 2.2. There is a natural lift of

\begin{align*} \mathrm{IndCoh}(Z) \otimes \mathrm{QCoh}({\mathbf{G}_m})^\Leftarrow \in \mathrm{QCoh}({\mathbf{G}_m})^\Leftarrow\textbf{-mod}({\mathbf{G}_m}\textbf{-mod}) \end{align*}

to an object of

\begin{align*} ( \mathrm{D}(\mathrm{Sing}(Z)) \otimes \mathrm{QCoh}({\mathbf{G}_m})^\Leftarrow ) \textbf{-mod} ({\mathbf{G}_m} \textbf{-mod}). \end{align*}

2.3.2.

In fact, we will not perform Construction 2.2 directly either. Instead, we give a geometric incarnation of $\mathrm{QCoh}({\mathbf{G}_m})^\Leftarrow$ , viewed as a symmetric monoidal category with its dilation action of ${\mathbf{G}_m}$ . The resulting reformulation of Construction 2.2 will appear below as Construction 2.3.

Regard $\mathrm{IndCoh}(\Omega \mathbf{A}^1)$ as a symmetric monoidal category under convolution. Under the Koszul duality equivalence (1.4) for $V = \mathbf{A}^1$ , the monoidal localization

\begin{align*} (j^*)^\Rightarrow : \mathrm{QCoh}(\mathbf{A}^1)^\Rightarrow \to \mathrm{QCoh}({\mathbf{G}_m})^\Rightarrow \end{align*}

goes over the monoidal localization

\begin{align*} \mathrm{IndCoh}(\Omega \mathbf{A}^1) \to \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1). \end{align*}

Therefore, we obtain a ${\mathbf{G}_m}$ -equivariant equivalence of symmetric monoidal categories

\begin{align*} \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1) \simeq \mathrm{QCoh}({\mathbf{G}_m})^\Rightarrow \end{align*}

intertwining the inverse dilation action on the left-hand side with the dilation action on the right-hand side. After inverting the ${\mathbf{G}_m}$ action on the right-hand side, we obtain a symmetric monoidal equivalence

\begin{align*} \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1) \simeq \mathrm{QCoh}({\mathbf{G}_m})^\Leftarrow \end{align*}

which is equivariant for the dilation actions on both sides.

It follows that Construction 2.2 admits the following reformulation.

Construction 2.3. There is a natural lift of

(2.4) \begin{equation} \mathrm{IndCoh}(Z) \otimes \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1) \in \overset{\circ}{\mathrm{IndCoh}}(\Omega\mathbf{A}^1)\textbf{-mod}({\mathbf{G}_m}\textbf{-mod})\end{equation}

to an object of

\begin{align*} ( \mathrm{D}(\mathrm{Sing}(Z)) \otimes \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1)) \textbf{-mod} ({\mathbf{G}_m} \textbf{-mod}). \end{align*}

2.3.3.

We now carry out Construction 2.3 by studying the product

\begin{align*} Z \times \Omega \mathbf{A}^1, \end{align*}

equipped with the action of ${\mathbf{G}_m}$ by dilation on the second factor.

To start off, $\mathbf{P}\mathrm{Sing}(Z \times \Omega \mathbf{A}^1)$ inherits a ${\mathbf{G}_m}$ action by the naturality of $\mathrm{Sing}(-)$ , and we claim that

\begin{align*} \overset{\circ}{\mathrm{IndCoh}}(Z \times \Omega \mathbf{A}^1) \in {\mathbf{G}_m} \textbf{-mod} \end{align*}

naturally lifts to an object

(2.5) \begin{equation} \overset{\circ}{\mathrm{IndCoh}}(Z \times \Omega \mathbf{A}^1) \in \mathrm{D}(\mathbf{P}\mathrm{Sing}(Z \times \Omega \mathbf{A}^1))\textbf{-mod}({\mathbf{G}_m}\textbf{-mod}).\end{equation}

In other words, we are saying that

(2.6) \begin{equation} \overset{\circ}{\mathrm{IndCoh}}((Z \times \Omega\mathbf{A}^1)/{\mathbf{G}_m}) \simeq \overset{\circ}{\mathrm{IndCoh}}(Z \times \Omega\mathbf{A}^1)^{\mathbf{G}_m} \in \mathrm{Rep}({\mathbf{G}_m})\textbf{-mod}\end{equation}

admits a natural lift to

\begin{equation*}\begin{split} & \operatorname{ShCat}(\mathbf{P}\mathrm{Sing}(Z \times \Omega \mathbf{A}^1)_\mathrm{dR}/{\mathbf{G}_m}) \\ & \quad\simeq \mathrm{D}(\mathbf{P}\mathrm{Sing}(Z \times \Omega \mathbf{A}^1))^{\mathbf{G}_m}\textbf{-mod} (\mathrm{Rep}({\mathbf{G}_m})\textbf{-mod}).\end{split}\end{equation*}

Indeed, applying [Reference Arinkin and GaitsgoryAG18, Proposition-Construction 1.5.2] to the quotient stack

\begin{align*} (Z \times \Omega \mathbf{A}^1)/{\mathbf{G}_m} \end{align*}

produces a lift of (2.6) to a sheaf of categories over

\begin{align*} \mathbf{P}\mathrm{Sing}((Z \times \Omega \mathbf{A}^1)/{\mathbf{G}_m})_\mathrm{dR} \underset{((Z \times \Omega \mathbf{A}^1)/{\mathbf{G}_m})_\mathrm{dR}}{\times} (Z \times \Omega \mathbf{A}^1)/{\mathbf{G}_m}, \end{align*}

and this last stack maps to

\begin{equation*}\begin{split} & \mathbf{P}\mathrm{Sing}((Z \times \Omega \mathbf{A}^1)/{\mathbf{G}_m})_\mathrm{dR} \underset{((Z \times \Omega \mathbf{A}^1)/{\mathbf{G}_m})_\mathrm{dR}}{\times} (Z \times \Omega \mathbf{A}^1)_\mathrm{dR}/{\mathbf{G}_m} \\ & \quad \simeq \mathbf{P}\mathrm{Sing}(Z \times \Omega \mathbf{A}^1)_\mathrm{dR}/{\mathbf{G}_m}.\end{split}\end{equation*}

2.3.4.

One has

\begin{align*} \mathrm{Sing}(Z \times \Omega \mathbf{A}^1)_\mathrm{dR} \simeq (\mathrm{Sing}(Z) \oplus \mathbf{1})_\mathrm{dR}, \end{align*}

where the latter expression denotes the de Rham prestack of $\mathrm{Sing}(Z) \underset{Z}{\times} \mathbf{A}^1_Z$ .

In terms of this identification, the action of ${\mathbf{G}_m}$ on $\mathrm{Sing}(Z \times \Omega \mathbf{A}^1)$ obtained from the dilation action of ${\mathbf{G}_m}$ on $\Omega \mathbf{A}^1$ by the naturality of the $\mathrm{Sing}(-)$ construction matches with inverse dilation on $\mathbf{A}^1_Z$ .

Therefore, the open embedding

(2.7) \begin{equation} \mathrm{Sing}(Z) \hookrightarrow \mathbf{P}(\mathrm{Sing}(Z) \oplus \mathbf{1}) \simeq \mathbf{P}\mathrm{Sing}(Z \times \Omega \mathbf{A}^1) \quad \xi \mapsto [\xi : 1]\end{equation}

intertwines the action of ${\mathbf{G}_m}$ on its source by dilation with the preceding action of ${\mathbf{G}_m}$ on its target, and restricting (2.5) along this map defines a lift of

\begin{align*}& \mathrm{IndCoh}(Z) \otimes \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1) \\ & \quad \simeq \mathrm{D}(\mathrm{Sing}(Z)) \underset{\mathrm{D}(\mathbf{P}(\mathrm{Sing}(Z) \oplus \mathbf{1}))}{\otimes} \overset{\circ}{\mathrm{IndCoh}}(Z \times \Omega \mathbf{A}^1) \in {\mathbf{G}_m} \textbf{-mod}\end{align*}

to an object

(2.8) \begin{equation} \mathrm{IndCoh}(Z) \otimes \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1) \in \mathrm{D}(\mathrm{Sing}(Z))\textbf{-mod}({\mathbf{G}_m} \textbf{-mod}).\end{equation}

We claim the following.

Proposition 2.1. The ${\mathbf{G}_m}$ -equivariant action of $\mathrm{D}(\mathbf{P} \mathrm{Sing}(Z))$ on

\begin{align*} \mathrm{IndCoh}(Z) \otimes \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1) \in {\mathbf{G}_m} \textbf{-}{\kern.6pt}\mathbf{mod} \end{align*}

encoded by (2.8) commutes with the ${\mathbf{G}_m}$ -equivariant action of $\overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1)$ encoded by (2.4).

2.3.5.

Observe that Construction 2.3 follows immediately from Proposition 2.1. However, it is not clear a priori that Proposition 2.1 is actually a proposition (as opposed to a construction). This is shown in § 2.4, and the proof of Proposition 2.1 appears in § 2.5.

2.4.1.

In this subsection, we prove a general assertion about relative crystals of categories. A consequence of this result will be that the commutativity of the two actions appearing in Proposition 2.1 is merely a property and not additional data.

Proposition 2.2. Let $\mathscr{A} \to \mathscr{B}$ be a map of monoidal categories. If the multiplication map

\begin{align*} \mu: \mathscr{B} \underset{\mathscr{A}}{\otimes} \mathscr{B} \to \mathscr{B} \end{align*}

admits a $(\mathscr{B}, \mathscr{B})$ -bilinear fully faithful left adjoint, then the functor

\begin{align*} \mathbf{res}: \mathscr{B} \textbf{-}{\kern.6pt}\mathbf{mod} \to \mathscr{A} \textbf{-}{\kern.6pt}\mathbf{mod} \end{align*}

is locally fully faithful.Footnote ⁵

Proof of Proposition 2.2. A stronger assertion holds true: for every pair of $\mathscr{B}$ -modules $\mathscr{M}$ and $\mathscr{M}'$ , the functor

\begin{align*} \operatorname{Fun}_{\mathscr{B}\textbf{-mod}}(\mathscr{M}, \mathscr{M}') & \to \operatorname{Fun}_{\mathscr{A}\textbf{-mod}}(\mathbf{res}(\mathscr{M}), \mathbf{res}(\mathscr{M}')) \\ & \simeq \operatorname{Fun}_{\mathscr{B}\textbf{-mod}}(\mathbf{ind} \circ \mathbf{res}(\mathscr{M}), \mathscr{M}')\end{align*}

is a colocalization. Indeed, the counit

\begin{align*} \mathbf{ind} \circ \mathbf{res} \to \mathbf{1} \end{align*}

identifies with the image of $\mu$ under the functor of 2-categories

\begin{align*} (\mathscr{B}, \mathscr{B})\textbf{-bimod} \to \operatorname{End}(\mathscr{B}\textbf{-mod}) \quad \mathscr{N} \mapsto \mathscr{N} \otimes_{\mathscr{B}} - \end{align*}

and $\mu$ is assumed to admit a fully faithful left adjoint.

2.4.2.

Let $\mathscr{Z} \to \mathscr{Y}$ be a schematic morphism of prestacks, and write

\begin{align*} p : \mathscr{Z}_{\mathrm{dR}/\mathscr{Y}} = \mathscr{Z}_\mathrm{dR} \underset{\mathscr{Y}_\mathrm{dR}}{\times} \mathscr{Y} \to \mathscr{Y} \end{align*}

for its formal completion.

Proposition 2.3. The functor

\begin{align*} \mathbf{coind}_p : \operatorname{ShCat}(\mathscr{Z}_{\mathrm{dR}/\mathscr{Y}}) \to \operatorname{ShCat}(\mathscr{Y}) \end{align*}

is locally fully faithful.

Proof of Proposition 2.3. Note that it suffices to consider the case where $\mathscr{Y}$ is an affine scheme. Indeed, under the identifications

\begin{align*} \operatorname{ShCat}(\mathscr{Z}_{\mathrm{dR}/\mathscr{Y}}) \simeq \lim_S \operatorname{ShCat}((\mathscr{Z}_S)_{\mathrm{dR}/S}){,} \quad \operatorname{ShCat}(\mathscr{Y}) = \lim_S \operatorname{ShCat}(S), \end{align*}

where the limits are taken over all test affine schemes mapping to $\mathscr{Y}$ , the functor $\mathbf{coind}_p$ is the one induced by the functors

\begin{align*} \mathbf{coind}_{p_S} : \operatorname{ShCat}((\mathscr{Z}_S)_{\mathrm{dR}/S}) \to \operatorname{ShCat}(S). \end{align*}

Therefore, we may replace $\mathscr{Z} \to \mathscr{Y}$ with a map of schemes $Z \to Y$ , with Y affine.

By Proposition 2.2, it suffices to check that the morphism of $\mathrm{QCoh}(Z_{\mathrm{dR}/Y})$ -bimodules

(2.9) \begin{align} & \mathrm{QCoh}(Z_{\mathrm{dR}/Y}) \underset{\mathrm{QCoh}(Y)}{\otimes} \mathrm{QCoh}(Z_{\mathrm{dR}/Y})\nonumber \\ & \quad \simeq \mathrm{QCoh}((Z \times_Y Z)_{\mathrm{dR}/Y}) \overset{\Delta_{\mathrm{dR}/Y}^*}{\longrightarrow} \mathrm{QCoh}(Z_{\mathrm{dR}/Y})\end{align}

admits a fully faithful $\mathrm{QCoh}(Z_{\mathrm{dR}/Y})$ -bilinear left adjoint.

Now Kashiwara’s lemma implies that the functor

\begin{align*} \Delta_\mathrm{dR}^* : \mathrm{QCoh}((Z \times_Y Z)_\mathrm{dR}) \to \mathrm{QCoh}(Z_\mathrm{dR}) \end{align*}

admits a fully faithful $\mathrm{QCoh}((Z \times_Y Z)_\mathrm{dR})$ -linear left adjoint. Tensoring up along

\begin{align*} - \underset{\mathrm{QCoh}(Y_\mathrm{dR})}{\otimes} \mathrm{QCoh}(Y) \end{align*}

shows that (2.9) admits a left adjoint with the desired properties.

2.4.3 Porism

Let us record some consequences of Proposition 2.3 (or, more precisely, its proof).

Let $\mathscr{C}$ be a sheaf of categories over $\mathscr{Z}_{\mathrm{dR}/\mathscr{Y}}$ . Then the data of $\mathscr{C}$ is encoded by the data of $\mathbf{coind}_p(\mathscr{C})$ along with compatible choices of colocalizing subcategories

(2.10) \begin{equation} \mathscr{C}_S \hookrightarrow \mathbf{\Gamma}(S; \mathbf{coind}_p(\mathscr{C}))\end{equation}

for every test affine scheme S mapping to $\mathscr{Z}_{\mathrm{dR}/\mathscr{Y}}$ .

We note for future reference that the functor (2.10) and its right adjoint are $\mathrm{QCoh}(S)$ -linear.

Let $\mathscr{D}$ be another sheaf of categories over $\mathscr{Z}_{\mathrm{dR}/\mathscr{Y}}$ . Then a morphism $\mathscr{C}\to \mathscr{D}$ in $\operatorname{ShCat}(\mathscr{Z}_{\mathrm{dR}/\mathscr{Y}})$ amounts to a morphism

\begin{align*} F : \mathbf{coind}_p(\mathscr{C}) \to \mathbf{coind}_p(\mathscr{D}) \end{align*}

in $\operatorname{ShCat}(\mathscr{Y})$ with the property: for every test affine scheme S mapping to $\mathscr{Z}_{\mathrm{dR}/\mathscr{Y}}$ , the functor

\begin{align*} \mathbf{\Gamma}(S; F) : \mathbf{\Gamma}(S; \mathbf{coind}_p(\mathscr{C})) \to \mathbf{\Gamma}(S; \mathbf{coind}_p(\mathscr{D})) \end{align*}

takes $\mathscr{C}_S$ into $\mathscr{D}_S$ .

Let $\mathscr{A}$ be a sheaf of monoidal categories over $\mathscr{Y}$ . Then an action of $\mathbf{cores}_p(\mathscr{A})$ on $\mathscr{C}$ amounts to an action of $\mathscr{A}$ on $\mathbf{coind}_p(\mathscr{C})$ with the property: for every test affine scheme S mapping to $\mathscr{Z}_{\mathrm{dR}/\mathscr{Y}}$ , the action map

\begin{equation*}\begin{split} \mathbf{\Gamma}(S; \mathbf{coind}_p(\mathscr{C})) \underset{\mathrm{QCoh}(S)}{\otimes} \mathbf{\Gamma}(S; \mathscr{A}) & \simeq \mathbf{\Gamma}(S; \mathbf{coind}_p(\mathscr{C}) \underset{\mathscr{Y}}{\otimes} \mathscr{A} ) \\ & \quad \overset{\mathbf{\Gamma}(S; \mathrm{act})}{\longrightarrow} \mathbf{\Gamma}(S; \mathbf{coind}_p(\mathscr{C}))\end{split}\end{equation*}

restricts to a functor

\begin{align*} \mathscr{C}_S \underset{\mathrm{QCoh}(S)}{\otimes} \mathbf{\Gamma}(S; \mathscr{A}) \to \mathscr{C}_S. \end{align*}

2.5.1.

We apply the paradigm of § 2.4 in the case

\begin{align*} \mathscr{Z} = \mathrm{Sing}(Z)/{\mathbf{G}_m} \subset \mathbf{P}(\mathrm{Sing}(Z) \oplus \mathbf{1})/{\mathbf{G}_m}{,} \quad \mathscr{Y} = B {\mathbf{G}_m}. \end{align*}

We interpret (2.8) as making

\begin{align*} \mathscr{C}^{\mathbf{G}_m} = (\mathrm{IndCoh}(Z) \otimes \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1) )^{\mathbf{G}_m} \end{align*}

into a sheaf of categories over $\mathscr{Z}_{\mathrm{dR}/\mathscr{Y}}$ , and (2.4) as encoding an action of

\begin{align*} \mathscr{A}^{\mathbf{G}_m} = \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1)^{\mathbf{G}_m}, \end{align*}

viewed as a constant sheaf of symmetric monoidal categories over $\mathscr{Y}$ , on $\mathbf{coind}_p(\mathscr{C}^{\mathbf{G}_m})$ .

In these terms, the assertion of Proposition 2.1 says that the action of $\mathscr{A}^{\mathbf{G}_m}$ on $\mathbf{coind}_p(\mathscr{C}^{\mathbf{G}_m})$ lifts to an action of $\mathbf{cores}_p(\mathscr{A}^{\mathbf{G}_m})$ on $\mathscr{C}^{\mathbf{G}_m}$ . Since $\mathscr{Z}$ and $\mathscr{Y}$ are 1-affine, this amounts to the assertion: for every test affine scheme S mapping to $\mathscr{Z}_{\mathrm{dR}/\mathscr{Y}}$ , the full subcategory

(2.11) \begin{equation} \mathrm{QCoh}(S) \underset{\mathrm{QCoh}(\mathscr{Z}_{\mathrm{dR}/\mathscr{Y}})}{\otimes} \mathscr{C}^{\mathbf{G}_m} \subset \mathrm{QCoh}(S) \underset{\mathrm{QCoh}(\mathscr{Y})}{\otimes} \mathscr{C}^{\mathbf{G}_m}\end{equation}

is stable under the action of $\mathscr{A}^{\mathbf{G}_m}$ .

Observe that the inclusion (2.11) rewrites as

\begin{align*}(\mathrm{QCoh}(S') \underset{\mathrm{D}(\mathscr{Z}')}{\otimes}\mathscr{C})^{\mathbf{G}_m} \subset ( \mathrm{QCoh}(S') \otimes \mathscr{C} )^{\mathbf{G}_m}, \end{align*}

where

\begin{align*} \mathscr{Z}' = \mathscr{Z} \underset{B{\mathbf{G}_m}}{\times} \mathrm{pt} \quad S' = S \underset{B{\mathbf{G}_m}}{\times} \mathrm{pt}. \end{align*}

Therefore, it suffices to show: for every test affine scheme T mapping to $\mathscr{Z}'_\mathrm{dR}$ , the full subcategory

(2.12) \begin{equation} \mathrm{QCoh}(T) \underset{\mathrm{D}(\mathscr{Z}')}{\otimes} \mathscr{C} \subset \mathrm{QCoh}(T) \otimes \mathscr{C}\end{equation}

is stable under the action of $\mathscr{A}$ .

2.5.2.

To proceed further, we need to recall the description of (2.12) from [Reference Arinkin and GaitsgoryAG18, § 2.4]. Write

\begin{align*} \Gamma_f \hookrightarrow T_\mathrm{red} \times \mathrm{Sing}(Z) \subset T_\mathrm{red} \times \mathbf{P}\mathrm{Sing}(Z \times \Omega \mathbf{A}^1) \end{align*}

for the graph of the associated morphism

\begin{align*} f : T_\mathrm{red} \to \mathrm{Sing}(Z) \subset \mathbf{P}(\mathrm{Sing}(Z) \oplus \mathbf{1}). \end{align*}

It corresponds to a closed conical subscheme

\begin{align*} \mathrm{Cone}(\Gamma_f) \hookrightarrow T_\mathrm{red} \times \mathrm{Sing}(Z \times \Omega \mathbf{A}^1). \end{align*}

By definition, (2.12) is the quotient of the full subcategory

(2.13) \begin{equation}\begin{split} & (\mathrm{QCoh}(T) \otimes \mathrm{IndCoh}(Z \times \Omega \mathbf{A}^1))_{\mathrm{Cone}(\Gamma_f)} \\ & \quad \subset \mathrm{QCoh}(T) \otimes \mathrm{IndCoh}(Z \times \Omega \mathbf{A}^1)\end{split}\end{equation}

by the full subcategory

\begin{align*} \mathrm{QCoh}(T) \otimes \mathrm{QCoh}(Z \times \Omega \mathbf{A}^1). \end{align*}

It remains for us to check that (2.13) is stable under the action of $\mathrm{IndCoh}(\Omega \mathbf{A}^1)$ .

2.5.3.

Let us make the identification

(2.14) \begin{equation}\begin{split} & (\mathrm{QCoh}(T) \otimes \mathrm{IndCoh}(Z \times \Omega \mathbf{A}^1))_{\mathrm{Cone}(\Gamma_f)} \otimes \mathrm{IndCoh}(\Omega \mathbf{A}^1) \\ & \quad \simeq(\mathrm{QCoh}(T) \otimes \mathrm{IndCoh}(Z \times \Omega\mathbf{A}^1 \times \Omega \mathbf{A}^1))_{\mathrm{Cone}(\Gamma_f) \oplus \mathbf{1}}\end{split}\end{equation}

as full subcategories of

(2.15) \begin{equation} \mathrm{QCoh}(T) \otimes \mathrm{IndCoh}(Z \times \Omega \mathbf{A}^1 \times \Omega \mathbf{A}^1).\end{equation}

In these terms, what remains to be seen is that the action map

\begin{equation*} \mathbf{1}_{\mathrm{QCoh}(T)} \otimes (\mathbf{1} \times \mathrm{add})_* : \mathrm{QCoh}(T) \otimes \mathrm{IndCoh}(Z \times \Omega \mathbf{A}^1 \times \Omega \mathbf{A}^1) \to \mathrm{QCoh}(T) \otimes \mathrm{IndCoh}(Z \times \Omega \mathbf{A}^1)\end{equation*}

takes (2.14) into (2.13).

2.5.4.

Consider the factorization of add into the composite

(2.16) \begin{equation} \Omega \mathbf{A}^1 \times \Omega \mathbf{A}^1 \overset{\sigma}{\longrightarrow} \Omega \mathbf{A}^1 \times \Omega \mathbf{A}^1 \overset{\mathrm{pr}_1}{\longrightarrow} \Omega \mathbf{A}^1\end{equation}

of the automorphism $\sigma$ defined by the formula

\begin{align*} (s, t) \mapsto (s + t, t) \end{align*}

and the coordinate projection $\mathrm{pr}_1$ .

The automorphism of

\begin{align*} S_\mathrm{red} \times \mathrm{Sing}(Z \times \Omega \mathbf{A}^1 \times \Omega \mathbf{A}^1) \end{align*}

induced by $\sigma$ restricts to an automorphism of $\mathrm{Cone}(\Gamma_f) \oplus \mathbf{1}$ , so

\begin{align*} \mathbf{1}_{\mathrm{QCoh}(T)} \otimes (\mathbf{1} \times \sigma)_* \end{align*}

defines an automorphism of (2.15) restricting to an automorphism of (2.14).Footnote ⁶

The functor

\begin{align*} \mathbf{1}_{\mathrm{QCoh}(T)} \otimes (\mathbf{1} \times \mathrm{pr}_1)_* \simeq \mathbf{1}_{\mathrm{QCoh}(T) \otimes \mathrm{IndCoh}(Z \times \Omega\mathbf{A}^1)} \otimes \Gamma_{\mathrm{IndCoh}} \end{align*}

takes (2.14) into (2.13).

3.1.1.

In this subsection, we collect some features of the inhomogeneous action which are more or less immediate from its construction.

Proposition 3.1. Let $\pi : \mathrm{Sing}(Z) \to Z$ denote the canonical projection. The restriction of the inhomogeneous action along the symmetric monoidal functor

\begin{align*} (\pi^!)^\Rightarrow : \mathrm{D}(Z) \to \mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow \end{align*}

recovers the usual action of $\mathrm{D}(Z)$ on $\mathrm{IndCoh}(Z)$ .

3.1.2 Proof

This follows from [Reference Arinkin and GaitsgoryAG18, Corollary 1.4.5(b)].

3.1.3.

Now consider the zero section $Z \to \mathrm{Sing}(Z)$ and view $\mathrm{D}(Z)$ as tensored over $\mathrm{D}(\mathrm{Sing}(Z))$ using this map. We claim the following.

Proposition 3.2. The full subcategory

\begin{align*} \mathrm{D}(Z) \underset{\mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow}{\otimes} \mathrm{IndCoh}(Z) \subset \mathrm{IndCoh}(Z) \end{align*}

is equal to

\begin{align*} \mathrm{QCoh}(Z) \subset \mathrm{IndCoh}(Z). \end{align*}

Proof of Proposition 3.2. Recall the action of $\mathrm{D}(\mathrm{Sing}(Z))$ on

\begin{align*} \mathrm{IndCoh}(Z) \otimes \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1) \end{align*}

from (2.8). We need to show that the full subcategory

\begin{align*} &\mathrm{D}(Z) \underset{\mathrm{D}(\mathrm{Sing}(Z))}{\otimes} ( \mathrm{IndCoh}(Z) \otimes \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1)) \\ & \quad \subset \mathrm{IndCoh}(Z) \otimes \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1)\end{align*}

is equal to

\begin{align*} \mathrm{QCoh}(Z) \otimes \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1) \subset \mathrm{IndCoh}(Z) \otimes \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1). \end{align*}

This is a consequence of the following assertion, Proposition 3.3.

Proposition 3.3. Let $Z_1$ and $Z_2$ be quasi-smooth schemes, and consider the closed embedding

\begin{align*} \mathbf{P}\mathrm{Sing}(Z_1) \times Z_2 \hookrightarrow \mathbf{P}\mathrm{Sing}(Z_1 \times Z_2) \quad \xi \mapsto [\xi : 0]. \end{align*}

Then

\begin{align*} & \mathrm{D}(\mathbf{P}\mathrm{Sing}(Z_1) \times Z_2) \underset{\mathrm{D}(\mathbf{P}\mathrm{Sing}(Z_1 \times Z_2))}{\otimes} \overset{\circ}{\mathrm{IndCoh}}(Z_1 \times Z_2) \\ & \quad \simeq \overset{\circ}{\mathrm{IndCoh}}(Z_1) \otimes \mathrm{QCoh}(Z_2)\end{align*}

as full subcategories of

\begin{align*} \overset{\circ}{\mathrm{IndCoh}}(Z_1 \times Z_2). \end{align*}

3.2.1.

Let $f : Z \to Y$ be a morphism of quasi-smooth schemes. We are interested in the category

\begin{align*} \mathrm{IndCoh}(Y^\wedge_Z). \end{align*}

By [Reference Arinkin and GaitsgoryAG18, Proposition 3.1.2], the functor of IndCoh $!$ -pullback

\begin{align*} f^! : \mathrm{IndCoh}(Y) \to \mathrm{IndCoh}(Y^\wedge_Z) \end{align*}

induces an equivalence of categories

(3.1) \begin{equation} \mathrm{D}(Z) \underset{\mathrm{D}(Y)}{\otimes} \mathrm{IndCoh}(Y) \overset{\sim}{\rightarrow} \mathrm{IndCoh}(Y^\wedge_Z).\end{equation}

In view of Proposition 3.1 and the relative Kunneth formula for D-modules on schemes (which follows from [Reference GaitsgoryGai15, Proposition 3.1.9 and Theorem 2.6.3], cf. [Reference ChenChe23, Lemma A.4.5]), we may regard (3.1) as defining an equivalence of categories

(3.2) \begin{equation} \mathrm{D}(\mathrm{Sing}(Y)_Z)^\Rightarrow \underset{\mathrm{D}(\mathrm{Sing}(Y))^\Rightarrow}{\otimes} \mathrm{IndCoh}(Y) \overset{\sim}{\rightarrow} \mathrm{IndCoh}(Y^\wedge_Z),\end{equation}

where

\begin{align*} \mathrm{Sing}(Y)_Z = Z \underset{Y}{\times} \mathrm{Sing}(Y). \end{align*}

In particular, this equips $\mathrm{IndCoh}(Y^\wedge_Z)$ an action of $\mathrm{D}(\mathrm{Sing}(Y)_Z)^\Rightarrow$ .

3.2.2.

Now consider the natural map

\begin{align*} \iota : Z \to Y^\wedge_Z \end{align*}

and the resulting adjunction

(3.3) \begin{equation} \iota_* : \mathrm{IndCoh}(Z) \rightleftharpoons \mathrm{IndCoh}(Y^\wedge_Z) : \iota^!.\end{equation}

Note that $\mathrm{IndCoh}(Y^\wedge_Z)$ is naturally tensored over $\mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow$ via the singular codifferential

\begin{align*} \mathrm{Sing}(f) : \mathrm{Sing}(Y)_Z \to \mathrm{Sing}(Z). \end{align*}

We prove the following assertion over the course of §§ 3.3 and 3.4.

3.2.3.

Recall that

\begin{align*} \mathrm{IndCoh}_0(Y^\wedge_Z) \subset \mathrm{IndCoh}(Y^\wedge_Z) \end{align*}

denotes the full subcategory of objects whose images under the functor $\iota^!$ of (3.3) belong to

\begin{align*} \mathrm{QCoh}(Z) \subset \mathrm{IndCoh}(Z). \end{align*}

Recall also the ‘relative’ scheme of singularities defined by the following fiber product.

One consequence of Theorem 3.1 is the following.

Corollary 3.1. The equivalence (3.2) restricts to an equivalence

\begin{align*} \mathrm{D}(\mathrm{Sing}(Z/Y))^\Rightarrow \underset{\mathrm{D}(\mathrm{Sing}(Y))^\Rightarrow}{\otimes} \mathrm{IndCoh}(Y) \overset{\sim}{\rightarrow} \mathrm{IndCoh}_0(Y^\wedge_Z). \end{align*}

3.3.1.

In this subsection, we will formulate Proposition 3.4, an analogue of Theorem 3.1 for the homogeneous action, and use it to deduce Theorem 3.1 itself. The proof of Proposition 3.4 will appear in § 3.4.

3.3.2.

Recall from [Reference Arinkin and GaitsgoryAG18, Proposition 3.1.8] that (3.1) restricts to an equivalence

(3.4) \begin{equation} \mathrm{D}(Z) \underset{\mathrm{D}(Y)}{\otimes} \overset{\circ}{\mathrm{IndCoh}}(Y) \overset{\sim}{\rightarrow} \overset{\circ}{\mathrm{IndCoh}}(Y^\wedge_Z)\end{equation}

onto the full subcategory

\begin{align*} \overset{\circ}{\mathrm{IndCoh}}(Y^\wedge_Z) \subset \mathrm{IndCoh}(Y^\wedge_Z) \end{align*}

which is right-orthogonal to $\mathrm{QCoh}(Y^\wedge_Z)$ .

Let us view (3.4) as tensored over $\mathrm{D}(\mathbf{P}\mathrm{Sing}(Y)_Z)$ using the homogeneous action of $\mathrm{D}(\mathbf{P}\mathrm{Sing}(Y))$ on $\overset{\circ}{\mathrm{IndCoh}}(Y)$ .

Now, the singular codifferential does not induce a morphism from the entirety of $\mathbf{P}\mathrm{Sing}(Y)_Z$ to $\mathbf{P}\mathrm{Sing}(Z)$ . Instead, we only have a morphism

(3.5) \begin{equation} \mathbf{P}\mathrm{Sing}(f) : \mathbf{P} \mathrm{Sing}(Y)_{\overset{\circ}{Z}} \to \mathbf{P} \mathrm{Sing}(Z)\end{equation}

out of the open subscheme

\begin{align*} \mathbf{P} \mathrm{Sing}(Y)_{\overset{\circ}{Z}} \hookrightarrow \mathbf{P} \mathrm{Sing}(Y) \end{align*}

defined via the following fiber product, where $\overset{\circ}{\mathrm{Sing}}(Z)$ is the complement of the zero section in $\mathrm{Sing}(Z)$ .

According to [Reference Arinkin and GaitsgoryAG18, Theorem 3.2.9], there is a natural identification

\begin{align*} \overset{\circ}{\mathrm{IndCoh}}_0(Y^\wedge_Z) = \mathrm{D}(\mathbf{P}(\mathrm{Sing}(f)^{-1}(0))) \underset{\mathrm{D}(\mathbf{P} \mathrm{Sing}(Y)_Z)}{\otimes} \overset{\circ}{\mathrm{IndCoh}}(Y^\wedge_Z) \end{align*}

as full subcategories of $\overset{\circ}{\mathrm{IndCoh}}(Y^\wedge_Z)$ . Combined with [Reference Arinkin and GaitsgoryAG18, Proposition 3.2.3], this shows that (3.3) restricts to an adjunction

(3.6) \begin{equation} \overset{\circ}{\mathrm{IndCoh}}(Z) \rightleftharpoons \mathrm{D}(\mathbf{P}\mathrm{Sing}(Y)_{\overset{\circ}{Z}}) \underset{\mathrm{D}(\mathbf{P}\mathrm{Sing}(Y)_Z)}{\otimes} \overset{\circ}{\mathrm{IndCoh}}(Y^\wedge_Z).\end{equation}

Both categories appearing in this adjunction are naturally tensored over $\mathrm{D}(\mathbf{P}\mathrm{Sing}(Z))$ ; the left-hand side via the homogeneous action and the right-hand side via the morphism (3.5). The analogue of Theorem 3.1 for the homogeneous action is then the following.

Proof of Theorem 3.1, assuming Proposition 3.4. Consider the adjunction

(3.7) \begin{align} & \iota_* \otimes \mathbf{1} : \mathrm{IndCoh}(Z) \otimes \mathrm{QCoh}({\mathbf{G}_m})^\Leftarrow \nonumber\\ & \quad \rightleftharpoons \mathrm{IndCoh}(Y^\wedge_Z) \otimes \mathrm{QCoh}({\mathbf{G}_m})^\Leftarrow : \iota^! \otimes \mathbf{1},\end{align}

taking place in

\begin{align*} \mathrm{QCoh}({\mathbf{G}_m})^\Leftarrow\textbf{-mod}({\mathbf{G}_m}\textbf{-mod}). \end{align*}

Both categories appearing in this adjunction are equipped with lifts to

\begin{align*} ( \mathrm{D}(\mathrm{Sing}(Z)) \otimes \mathrm{QCoh}({\mathbf{G}_m})^\Leftarrow )\textbf{-mod} ({\mathbf{G}_m} \textbf{-mod}), \end{align*}

and we are claiming that the adjunction itself lifts to this category. Note that Proposition 2.3 implies that this is a property and not additional structure, and that it suffices to check that (3.7) is $\mathrm{D}(\mathrm{Sing}(Z))$ -linear as an adjunction of plain dg categories.

Consider the adjunction (3.6) for the map

\begin{align*} Z \times \Omega \mathbf{A}^1 \to Y \times \Omega \mathbf{A}^1. \end{align*}

By Proposition 3.4, it respects the homogeneous action of

\begin{align*} \mathrm{D}(\mathbf{P}(\mathrm{Sing}(Z) \oplus \mathbf{1})), \end{align*}

so after restriction along the open embedding (2.7), we obtain that the adjunction

\begin{align*} \mathrm{IndCoh}(Z) \otimes \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1) \rightleftharpoons \mathrm{IndCoh}(Y^\wedge_Z) \otimes \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1) \end{align*}

is linear over $\mathrm{D}(\mathrm{Sing}(Z))$ .Footnote ⁷

3.4.1.

Let Y be a scheme and let $\mathscr{C}$ be a category tensored over $\mathrm{D}(Y)$ . So we are in the framework of § 2.4, but in the absolute case. In particular, for every test affine scheme S mapping to Y, the colocalization (2.10) becomes a $\mathrm{QCoh}(S)$ -linear colocalization

(3.8) \begin{equation} \mathscr{C}_S = \mathrm{QCoh}(S) \underset{\mathrm{D}(Y)}{\otimes} \mathscr{C} \rightleftharpoons \mathrm{QCoh}(S) \otimes \mathscr{C}.\end{equation}

1. (a) c belongs to $\mathscr{C}_S$ ;

2. (b) for every geometric point $i : s \to S$ , the functor

   \begin{align*} i^* \otimes \mathbf{1} : \mathrm{QCoh}(S) \otimes \mathscr{C} \to \mathrm{QCoh}(s) \otimes \mathscr{C} \end{align*}
   takes c into $\mathscr{C}_s$ .

Corollary 3.2. Let $\mathscr{C}$ and $\mathscr{D}$ be categories tensored over $\mathrm{D}(Y)$ , and let

\begin{align*} F : \mathscr{C} \to \mathscr{D} \end{align*}

be a functor of plain dg categories. The following are equivalent:

1. (a) F is $\mathrm{D}(Y)$ -linear;

2. (b) for every geometric point $\eta \to Y$ , the functor

   \begin{align*} \mathbf{1} \otimes F : \mathrm{QCoh}(\eta) \otimes \mathscr{C} \to \mathrm{QCoh}(\eta) \otimes \mathscr{D} \end{align*}
   takes $\mathscr{C}_\eta$ into $\mathscr{D}_\eta$ .

Proof of Proposition 3.4. Let $\eta$ be an arbitrary geometric point of $\mathbf{P}\mathrm{Sing}(Z)$ . By Corollary 3.2, it suffices to check that the adjunction

\begin{align*} \mathbf{1}_{\mathrm{QCoh}(\eta)} \otimes (\ref{eqn:rel_DMod_iota_adjunction_circ}) \end{align*}

restricts to an adjunction

\begin{equation*}\begin{split} & ( \mathrm{QCoh}(\eta) \otimes \overset{\circ}{\mathrm{IndCoh}}(Z))_\eta \\ & \quad\rightleftharpoons (\mathrm{QCoh}(\eta) \otimes \mathrm{D}(\mathbf{P}\mathrm{Sing}(Y)_{\overset{\circ}{Z}}) \underset{\mathrm{D}(\mathbf{P}\mathrm{Sing}(Y)_Z)}{\otimes} \overset{\circ}{\mathrm{IndCoh}}(Y^\wedge_Z))_\eta.\end{split}\end{equation*}

After extending the base field, we may assume that $\eta$ is a closed point. Then it remains to check that (3.6) restricts to an adjunction

\begin{equation*}\begin{split} & \mathrm{Vect} \underset{\mathrm{D}(\mathbf{P}\mathrm{Sing}(Z))}{\otimes} \overset{\circ}{\mathrm{IndCoh}}(Z) \\ & \quad \rightleftharpoons \mathrm{Vect} \underset{\mathrm{D}(\mathbf{P}\mathrm{Sing}(Z))}{\otimes} \mathrm{D}(\mathbf{P}\mathrm{Sing}(Y)_{\overset{\circ}{Z}}) \underset{\mathrm{D}(\mathbf{P}\mathrm{Sing}(Y)_Z)}{\otimes} \overset{\circ}{\mathrm{IndCoh}}(Y^\wedge_Z).\end{split}\end{equation*}

This follows from [Reference Arinkin and GaitsgoryAG18, Theorem 3.2.9] and [Reference Arinkin and GaitsgoryAG18, Proposition 3.2.3].

3.5.1.

To formulate the backwards compatibility between the inhomogeneous and homogeneous actions, consider the full subcategory

\begin{align*} \mathrm{D}(\overset{\circ}{\mathrm{Sing}}(Z))^\Rightarrow \underset{\mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow}{\otimes} \mathrm{IndCoh}(Z) \subset \mathrm{IndCoh}(Z). \end{align*}

By Proposition 3.2, it is equal to

\begin{align*} \overset{\circ}{\mathrm{IndCoh}}(Z) \subset \mathrm{IndCoh}(Z). \end{align*}

Therefore, we obtain an action of $\mathrm{D}(\overset{\circ}{\mathrm{Sing}}(Z))^\Rightarrow$ on $\overset{\circ}{\mathrm{IndCoh}}(Z)$ , and we may consider its restriction along the symmetric monoidal functorFootnote ⁸

\begin{align*} \mathrm{D}(\mathbf{P}\mathrm{Sing}(Z)) \to \mathrm{D}(\overset{\circ}{\mathrm{Sing}}(Z))^{\mathbf{G}_m} \simeq ( \mathrm{D}(\overset{\circ}{\mathrm{Sing}}(Z))^\Rightarrow )^{\mathbf{G}_m} \to \mathrm{D}(\overset{\circ}{\mathrm{Sing}}(Z))^\Rightarrow. \end{align*}

We claim the following.

Proof of Proposition 3.6. Consider the action of $\mathrm{D}(\overset{\circ}{\mathrm{Sing}}(Z))$ on the full subcategory

\begin{align*}& \mathrm{D}(\overset{\circ}{\mathrm{Sing}}(Z)) \underset{\mathrm{D}(\mathrm{Sing}(Z))}{\otimes} (\mathrm{IndCoh}(Z) \otimes \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1)) \\ & \quad \subset \mathrm{IndCoh}(Z) \otimes \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1).\end{align*}

After taking ${\mathbf{G}_m}$ -invariants, we obtain an action of $\mathrm{D}(\overset{\circ}{\mathrm{Sing}}(Z))^{\mathbf{G}_m}$ on

(3.9) \begin{equation}\begin{split} & ( \overset{\circ}{\mathrm{IndCoh}}(Z) \otimes \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1))^{\mathbf{G}_m} \\ & \quad \simeq (\overset{\circ}{\mathrm{IndCoh}}(Z) \otimes \mathrm{QCoh}({\mathbf{G}_m})^\Leftarrow)^{\mathbf{G}_m} \simeq \overset{\circ}{\mathrm{IndCoh}}(Z).\end{split}\end{equation}

This is the action obtained by restricting the action of $\mathrm{D}(\overset{\circ}{\mathrm{Sing}}(Z))^\Rightarrow$ on $\overset{\circ}{\mathrm{IndCoh}}(Z)$ along the functor

\begin{align*} \mathrm{D}(\overset{\circ}{\mathrm{Sing}}(Z))^{\mathbf{G}_m} \simeq ( \mathrm{D}(\overset{\circ}{\mathrm{Sing}}(Z))^\Rightarrow )^{\mathbf{G}_m} \to \mathrm{D}(\overset{\circ}{\mathrm{Sing}}(Z))^\Rightarrow. \end{align*}

Therefore, it suffices to show that the functor

(3.10) \begin{equation}\begin{split} & \overset{\circ}{\mathrm{IndCoh}}(Z) \otimes \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1) \\ & \quad \to( \overset{\circ}{\mathrm{IndCoh}}(Z) \otimes\overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1))_{\mathbf{G}_m} \\ & \quad \simeq (\overset{\circ}{\mathrm{IndCoh}}(Z) \otimes \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1))^{\mathbf{G}_m} \overset{(\small\ref{eqn:backwards_compat_eq1})}{\longrightarrow} \overset{\circ}{\mathrm{IndCoh}}(Z)\end{split}\end{equation}

intertwines the action of $\mathrm{D}(\overset{\circ}{\mathrm{Sing}}(Z))$ on its source with the homogeneous action of $\mathrm{D}(\mathbf{P}\mathrm{Sing}(Z))$ on its target, via the quotient map

(3.11) \begin{equation} \overset{\circ}{\mathrm{Sing}}(Z) \to \mathbf{P} \mathrm{Sing}(Z).\end{equation}

Note that the functor

\begin{align*} \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1) \to \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1)_{\mathbf{G}_m} \overset{\sim}{\rightarrow} \mathrm{Vect} \end{align*}

identifies with the composite

\begin{align*} \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1) \hookrightarrow \mathrm{IndCoh}(\Omega \mathbf{A}^1) \overset{i^!}{\longrightarrow} \mathrm{Vect}, \end{align*}

where $i: \mathrm{pt} \to \Omega \mathbf{A}^1$ is the inclusion of zero into the vector scheme $\Omega \mathbf{A}^1$ . Therefore, we may reinterpret (3.10) as the composite

\begin{align*} \overset{\circ}{\mathrm{IndCoh}}(Z) \otimes \overset{\circ}{\mathrm{IndCoh}}(\Omega \mathbf{A}^1) \hookrightarrow \overset{\circ}{\mathrm{IndCoh}}(Z \times \Omega \mathbf{A}^1) \overset{(\mathbf{1} \times i)^!}{\longrightarrow} \overset{\circ}{\mathrm{IndCoh}}(Z). \end{align*}

Now observe that i is a nil-isomorphism, so the natural map

\begin{align*} (Z \times \Omega \mathbf{A}^1) ^\wedge_Z \to Z \times \Omega \mathbf{A}^1 \end{align*}

is an isomorphism. Therefore, Proposition 3.4 implies that $(\mathbf{1} \times i)^!$ intertwines the action of $\mathrm{D}(\mathbf{P}\mathrm{Sing}(Z \times \Omega \mathbf{A}^1)_{\overset{\circ}{Z}})$ on $\overset{\circ}{\mathrm{IndCoh}}(Z \times \Omega \mathbf{A}^1)$ with the action of $\mathrm{D}(\mathbf{P}\mathrm{Sing}(Z))$ on $\overset{\circ}{\mathrm{IndCoh}}(Z)$ , via the map

\begin{align*} \mathbf{P}\mathrm{Sing}(\mathbf{1} \times i) : \mathbf{P}\mathrm{Sing}(Z \times \Omega \mathbf{A}^1)_{\overset{\circ}{Z}} \to \mathbf{P}\mathrm{Sing}(Z). \end{align*}

The result follows because the restriction of $\mathbf{P}\mathrm{Sing}(\mathbf{1} \times i)$ to the open subscheme

\begin{align*} \overset{\circ}{\mathrm{Sing}}(Z) \subset \mathbf{P}\mathrm{Sing}(Z \times \Omega \mathbf{A}^1)_{\overset{\circ}{Z}} \end{align*}

is the quotient map (3.11).

Proposition 3.7. For every closed conical subset $\mathscr{N} \subset \mathrm{Sing}(Z)$ , the full subcategory

\begin{align*} \mathrm{D}(\mathscr{N})^\Rightarrow \underset{\mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow}{\otimes} \mathrm{IndCoh}(Z) \subset \mathrm{IndCoh}(Z) \end{align*}

is equal to

\begin{align*} \mathrm{IndCoh}_{\mathscr{N}}(Z) \subset \mathrm{IndCoh}(Z). \end{align*}

Proof of Proposition 3.7. By Proposition 3.2, it suffices to show that the full subcategory

(3.12) \begin{equation} \mathrm{D}(\mathbf{P}\mathscr{N})^\Rightarrow \underset{\mathrm{D}(\mathbf{P}\mathrm{Sing}(Z))^\Rightarrow}{\otimes} \overset{\circ}{\mathrm{IndCoh}}(Z) \subset \overset{\circ}{\mathrm{IndCoh}}(Z)\end{equation}

is equal to

(3.13) \begin{equation} \overset{\circ}{\mathrm{IndCoh}}_{\mathscr{N}}(Z) \subset \overset{\circ}{\mathrm{IndCoh}}(Z).\end{equation}

Although the tensor product in (3.12) is formed using the inhomogeneous action, Proposition 3.6 implies that we may replace it with the homogeneous action. Now, the equality of (3.12) and (3.13) is [Reference Arinkin and GaitsgoryAG18, Theorem-Construction 1.4.2(a)].

3.6.1.

Suppose that Z has been presented as a global complete intersection; that is, it has been written as a fiber product where U is smooth and $\mathrm{pt} \to V$ is the inclusion of the zero vector into a vector space.

(3.14)

In this situation, it is possible to give a description of the action of $\mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow$ on $\mathrm{IndCoh}(Z)$ in the spirit of [Reference Arinkin and GaitsgoryAG15, § 5.4].

3.6.2.

Recall that we may regard $Z \times_U Z$ as a groupoid over Z, so that

\begin{align*} \mathrm{IndCoh}(Z \times_U Z) \end{align*}

acts on $\mathrm{IndCoh}(Z)$ by convolution.

If we let f denote the given map from U to V, then we obtain a Cartesian diagram where the upper horizontal arrow identifies with the given map $Z \to U$ .

We therefore obtain a map of groupoids

(3.15) \begin{equation}\epsilon: Z \times_U Z \to U \times_{(U \times V)} U \simeq U \times \Omega V.\end{equation}

To obtain Construction 3.1, we precompose the convolution action of $\mathrm{IndCoh}(Z \times_U Z)$ on $\mathrm{IndCoh}(Z)$ with the monoidal functor

(3.16) \begin{equation}\begin{split} & \mathrm{D}(U) \otimes \mathrm{D}(V^\vee)^\Rightarrow \overset{\mathrm{oblv} \otimes \mathrm{oblv}^\Rightarrow}{\longrightarrow} \mathrm{IndCoh}(U) \otimes \mathrm{QCoh}(V^\vee)^\Rightarrow \\ & \simeq \mathrm{IndCoh}(U) \otimes \mathrm{IndCoh}(\Omega V) \overset{\epsilon^!}{\to} \mathrm{IndCoh}(Z \times_U Z).\end{split}\end{equation}

3.6.3.

Since U is smooth, the morphism

\begin{align*} \mathbf{L}_Z \to \mathbf{L}_{Z/U} \simeq \mathscr{O}_Z \otimes V^\vee [1] \end{align*}

induces a closed embedding

(3.17) \begin{equation} \delta : \mathrm{Sing}(Z) \hookrightarrow Z \times V^\vee \hookrightarrow U \times V^\vee.\end{equation}

We claim the following.

Proposition 3.8. The action of Construction 3.1 agrees with composition of the inhomogeneous action with the monoidal colocalization

\begin{align*} (\delta^!)^\Rightarrow : \mathrm{D}(U \times V^\vee)^\Rightarrow \to \mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow. \end{align*}

3.6.4 A tautology

Let $\mathscr{A}$ be a monoidal category for which the multiplication map

\begin{align*} \mathrm{mult} : \mathscr{A} \otimes \mathscr{A} \to \mathscr{A} \end{align*}

admits a fully faithful $(\mathscr{A}, \mathscr{A})$ -bilinear left adjoint. According to Proposition 2.2, this implies that the forgetful functor

\begin{align*} \mathscr{A} \textbf{-mod} \to \textbf{dgCat} \end{align*}

is locally fully faithful.

Suppose now that a category $\mathscr{M}$ has been given two actions of $\mathscr{A}$ , to be denoted

\begin{align*} \mathrm{act}_1 : \mathscr{A} \otimes \mathscr{M} \to \mathscr{M} \quad \mathrm{act}_2 : \mathscr{A} \otimes \mathscr{M} \to \mathscr{M}. \end{align*}

Tensoring up the $(\mathrm{mult}^L, \mathrm{mult})$ adjunction shows that $\mathrm{act}_1$ and $\mathrm{act}_2$ admit fully faithful $\mathscr{A}$ -linear left adjoints realizing $\mathscr{M}$ as a full subcategory of $\mathscr{A} \otimes \mathscr{M}$ stable under the action of $\mathscr{A}$ . Therefore, the two actions of $\mathscr{A}$ on $\mathscr{M}$ coincide if and only if $\mathrm{act}_1$ and $\mathrm{act}_2$ are isomorphic as functors of plain dg categories.

We claim that $\mathrm{act}_1$ and $\mathrm{act}_2$ are isomorphic as functors once $\mathrm{act}_1$ is $\mathscr{A}$ -linear for the tautological action of $\mathscr{A}$ on $\mathscr{A} \otimes \mathscr{M}$ and the second action of $\mathscr{A}$ on $\mathscr{M}$ . Indeed, this follows from precomposing the following commutative diagram with the map

\begin{align*} \mathbf{1} \otimes \mathrm{unit} \otimes \mathbf{1} : \mathscr{A} \otimes \mathscr{M} \to \mathscr{A} \otimes \mathscr{A} \otimes \mathscr{M}. \end{align*}

Proof of Proposition 3.8. We apply the setup of § 3.6.4 to the data

\begin{align*} \mathscr{A} = \mathrm{D}(U \times V^\vee)^\Rightarrow \quad \mathscr{M} = \mathrm{IndCoh}(Z), \end{align*}

with $\mathrm{act}_1$ being the action of Construction 3.1 and $\mathrm{act}_2$ being the composite of the inhomogeneous action with $(\delta^!)^\Rightarrow$ .

To check that the action map

\begin{align*} \mathrm{act}_1: \mathscr{A} \otimes \mathscr{M} \to \mathscr{M} \end{align*}

is $D(U \times V^\vee)^\Rightarrow$ -linear, it suffices to check separately that it is linear for D(U) and for $D(V^\vee)^\Rightarrow$ .

Since the action of Construction 3.1 is $\mathrm{D}(U)$ -linear, the linearity of $\mathrm{act}_1$ for D(U) follows from Proposition 3.1. Therefore, it remains to show that $\mathrm{act}_1$ is $D(V^\vee)^\Rightarrow$ -linear.

Recall from [Reference Arinkin and GaitsgoryAG15, § 5.3.1] that the group scheme $\Omega V$ acts on Z, and there is an isomorphism $Z \times \Omega V \simeq Z \times_U Z$ making the following diagram commute.

In these terms, the action of $D(V^\vee)^\Rightarrow$ on $\mathrm{IndCoh}(Z)$ from Construction 3.1 identifies with the composite

\begin{align*}D(V^\vee)^\Rightarrow \otimes \mathrm{IndCoh}(Z) \xrightarrow{\mathrm{oblv} \otimes \mathbf{1}} \mathrm{IndCoh}(\Omega V) \otimes \mathrm{IndCoh}(Z) \xrightarrow{\mathrm{act}_*} \mathrm{IndCoh}(Z).\end{align*}

We claim that the functor

(3.18) \begin{equation} \mathrm{act}_*: \mathrm{IndCoh}(Z \times \Omega V) \to \mathrm{IndCoh}(Z)\end{equation}

is already $D(V^\vee)^\Rightarrow$ -linear in the appropriate sense. It is enough to show that (3.18) intertwines the action of $\mathrm{D}(V^\vee)^\Rightarrow \otimes \mathrm{D}(V^\vee)^\Rightarrow$ on its source with the action of $\mathrm{D}(V^\vee)^\Rightarrow$ on its target, via the map

\begin{align*} \Delta : V^\vee \to V^\vee \times V^\vee. \end{align*}

Indeed, this follows from Theorem 3.1 for the map act, together with the following commutative diagram.

\begin{align*}\vphantom{a}\\[-36pt]\end{align*}

4.1.1.

Let $\mathscr{M}$ be a module over

\begin{align*} \mathscr{A} = \mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow, \end{align*}

and suppose that $\mathscr{M}$ admits a dual $\mathscr{M}^\vee$ as a plain dg category. By [Reference ChenChe23, § A.3.9], the composite

\begin{align*} \mathrm{Vect} \to \mathscr{M} \otimes \mathscr{M}^\vee \to \mathscr{M} \otimes_{\mathscr{A}} \mathscr{M}^\vee \end{align*}

forms the unit of an $\mathscr{A}$ -module duality.Footnote ⁹

The counit of the duality

\begin{align*} \mathrm{ev} : \mathscr{M}^\vee \otimes \mathscr{M} \to \mathscr{A} \end{align*}

is given by the composite

\begin{align*} \mathscr{M}^\vee \otimes \mathscr{M} \overset{\mathbf{1} \otimes \mathrm{co}\mathrm{act}}{\longrightarrow} \mathscr{M}^\vee \otimes \mathscr{M} \otimes \mathscr{A} \to \mathscr{A}, \end{align*}

where $\mathrm{co}\mathrm{act}$ is the morphism

\begin{align*} \mathrm{co}\mathrm{act} : \mathscr{M} \to \mathscr{M} \otimes \mathscr{A}^\vee \overset{\sim}{\rightarrow} \mathscr{M} \otimes \mathscr{A} \end{align*}

obtained by transposing the action map and applying Verdier duality.

We observe, for future reference, that the functor

\begin{align*} \mathscr{A} \simeq \mathscr{A}^\vee \overset{\mathrm{ev}^\vee}{\longrightarrow} (\mathscr{M}^\vee \otimes \mathscr{M})^\vee \simeq \mathscr{M}^\vee \otimes \mathscr{M} \simeq \operatorname{End}(\mathscr{M}) \end{align*}

is the map of $(\mathscr{A}, \mathscr{A})$ -bimodules classifying the action of $\mathscr{A}$ on $\mathscr{M}$ .

4.1.2.

Now let us consider the module $\mathscr{M} = \mathrm{IndCoh}(Z)$ . In this case, Serre duality provides a canonical equivalence of dg categories

(4.1) \begin{equation} \mathrm{IndCoh}(Z)^\vee \simeq \mathrm{IndCoh}(Z).\end{equation}

An irksome feature of the theory is that this equivalence does not intertwine the $\mathscr{A}$ -module structure on the left-hand side from § 4.1.1 with the pre-existing $\mathscr{A}$ -module structure on the right-hand side.

To correct this, let

\begin{align*} \mathrm{IndCoh}(Z)^\tau \end{align*}

denote the category $\mathrm{IndCoh}(Z)$ equipped with the action of $\mathscr{A}$ obtained by precomposing the standard action with the involution of $\mathscr{A}$ induced by the negation map:

\begin{align*} \tau : \mathrm{Sing}(Z) \to \mathrm{Sing}(Z) \quad \xi \mapsto -\xi. \end{align*}

Proposition 4.1. The equivalence of plain dg categories (4.1) lifts to an equivalence of $\mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow$ -modules

\begin{align*} \mathrm{IndCoh}(Z)^\vee \simeq \mathrm{IndCoh}(Z)^\tau. \end{align*}

Proof of Proposition 4.1. We need to lift the map of left $\mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow$ -modules

(4.2) \begin{equation}\begin{split} \mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow & \overset{\mathrm{coev}^\vee}{\longrightarrow} \mathrm{IndCoh}(Z) \otimes \mathrm{IndCoh}(Z)^\vee \\ & \quad \simeq \mathrm{IndCoh}(Z) \otimes \mathrm{IndCoh}(Z)^\tau\end{split}\end{equation}

to a map of bimodules. It suffices to show that the functor (4.2) factors through the full subcategory

(4.3) \begin{align}& \mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow \underset{\mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow \otimes \mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow}{\otimes} ( \mathrm{IndCoh}(Z) \otimes \mathrm{IndCoh}(Z)^\tau ) \nonumber\\ & \quad \subset \mathrm{IndCoh}(Z) \otimes \mathrm{IndCoh}(Z)^\tau,\end{align}

where $\mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow \otimes \mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow$ acts on $\mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow$ via the diagonal embedding

\begin{align*} \mathrm{Sing}(Z) \hookrightarrow \mathrm{Sing}(Z) \times \mathrm{Sing}(Z). \end{align*}

Observe that (4.2) is given by acting on the object

\begin{align*} \Delta_*(\omega_Z) \in \mathrm{IndCoh}(Z \times Z), \end{align*}

and (4.3) is a submodule inclusion. Therefore, it suffices to show that $\Delta_*(\omega_Z)$ belongs to the submodule (4.3).

Let us rewrite the full subcategory (4.3) as

(4.4) \begin{align}& \mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow \underset{\mathrm{D}(\mathrm{Sing}(Z \times Z))^\Rightarrow}{\otimes} \mathrm{IndCoh}(Z \times Z) \nonumber\\ & \quad \subset \mathrm{IndCoh}(Z \times Z),\end{align}

where $\mathrm{D}(\mathrm{Sing}(Z \times Z))^\Rightarrow$ now acts on $\mathrm{IndCoh}(Z \times Z)$ in the usual way and on $\mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow$ via the closed embedding

\begin{align*} \mathbf{1} \times \tau : \mathrm{Sing}(Z) \to \mathrm{Sing}(Z) \times \mathrm{Sing}(Z) \simeq \mathrm{Sing}(Z \times Z). \end{align*}

This embedding identifies with the morphism

\begin{align*} \mathrm{Sing}(Z/Z \times Z) \to \mathrm{Sing}(Z \times Z), \end{align*}

so Corollary 3.1 identifies (4.4) with the full subcategory

\begin{align*} \mathrm{IndCoh}_0((Z \times Z)^\wedge_Z) \subset \mathrm{IndCoh}(Z \times Z) \simeq \mathrm{IndCoh}(Z) \otimes \mathrm{IndCoh}(Z)^\tau. \end{align*}

4.1.3 Porism

Since the inclusion (4.3) identifies with

\begin{align*} \operatorname{End}_{\mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow}(\mathrm{IndCoh}(Z)) \subset \operatorname{End}(\mathrm{IndCoh}(Z)), \end{align*}

the proof of Proposition 4.1 shows that

\begin{align*} \operatorname{End}_{\mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow}(\mathrm{IndCoh}(Z)) \simeq \mathrm{IndCoh}_0((Z \times Z)^\wedge_Z) \end{align*}

as full monoidal subcategories of

\begin{align*} \operatorname{End}(\mathrm{IndCoh}(Z)) \simeq \mathrm{IndCoh}(Z \times Z), \end{align*}

acting on the right on $\mathrm{IndCoh}(Z)$ .

By definition,

\begin{align*} \mathbf{H}(Z) = \mathrm{IndCoh}_0((Z \times Z)^\wedge_Z), \end{align*}

so we have upgraded $\mathrm{IndCoh}(Z)$ to a bimodule

\begin{align*} \mathrm{IndCoh}(Z) \in (\mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow, \mathbf{H}(Z)) \textbf{-bimod}. \end{align*}

Theorem 4.1. Let Z be a quasi-smooth scheme. The bimodule $\mathrm{IndCoh}(Z)$ defines a Morita equivalence

\begin{align*} \mathbf{H}(Z)\textbf{-}{\kern.6pt}\mathbf{mod} \overset{\sim}{\rightarrow} \mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow\textbf{-}{\kern.6pt}\mathbf{mod}. \end{align*}

4.1.4.

The symmetric monoidal structure on $\mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow$ gives a canonical identification of its categories of left and right modules. We use this identification freely.

The monoidal involution

\begin{align*} \sigma : \mathbf{H}(Z)^\mathrm{rev} \overset{\sim}{\rightarrow} \mathbf{H}(Z) \end{align*}

induced by the swap map on $Z \times Z$ gives a canonical identification between the categories of left and right modules over $\mathbf{H}(Z)$ . This identification exchanges $\mathrm{IndCoh}(Z)$ , viewed as a right $\mathbf{H}(Z)$ -module under convolution, with $\mathrm{IndCoh}(Z)$ , viewed as a left $\mathbf{H}(Z)$ -module under convolution. We will use this identification freely.

In particular, we may regard $\mathrm{IndCoh}(Z)^\tau$ as a bimodule

(4.5) \begin{equation} \mathrm{IndCoh}(Z)^\tau \in (\mathbf{H}(Z), \mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow)\textbf{-bimod}.\end{equation}

4.1.5.

By definition, the equivalence of Theorem 4.1 fits into the following commutative diagrams.

After passing to left (respectively right) adjoints, we obtain the following diagrams.

This shows, in particular, that the inverse to the Morita equivalence of Theorem 4.1 is given by the bimodule (4.5).

4.2.1.

The assertion of Theorem 4.1 takes place in the context of the Morita $(\infty, 2)$ -category of algebras and bimodules. For the convenience of the reader, we recall the basic features of this theory which are relevant to us.

4.2.2.

Let $\mathscr{A}$ and $\mathscr{B}$ be monoidal categories, and let $\mathscr{M}$ be a $(\mathscr{B}, \mathscr{A})$ -bimodule. Suppose that $\mathscr{M}$ admits a left dual $\mathscr{M}^\vee$ , i.e. an $(\mathscr{A}, \mathscr{B})$ -bimodule equipped with a pair of bimodule morphisms

(4.6) \begin{equation} \mathrm{coev} : \mathscr{B} \to \mathscr{M} \otimes_{\mathscr{A}} \mathscr{M}^\vee{,} \quad \mathrm{ev} : \mathscr{M}^\vee \otimes_{\mathscr{B}} \mathscr{M} \to \mathscr{A}\end{equation}

satisfying the triangle identities.

4.2.3.

Suppose that the morphisms appearing in the duality data (4.6) admit continuous right adjoints, which are moreover linear for the apparent bimodule structures

\begin{align*} \mathscr{B} \leftarrow \mathscr{M} \otimes_{\mathscr{A}} \mathscr{M}^\vee : \mathrm{coev}^R{,} \quad \mathscr{M}^\vee \otimes_{\mathscr{B}} \mathscr{M} \leftarrow \mathscr{A} : \mathrm{ev}^R. \end{align*}

Since passing to adjoints is functorial, these right adjoints identify $\mathscr{M}^\vee$ as a right dual of $\mathscr{M}$ .

Proof of Proposition 4.2. It suffices to show that the morphism

\begin{align*} \mathscr{M}^\vee \otimes_{\mathscr{B}} \mathscr{M} \overset{\mathrm{ev}}{\to} \mathscr{A} \overset{\mathrm{ev}^R}{\to} \mathscr{M}^\vee \otimes_{\mathscr{B}} \mathscr{M} \end{align*}

is isomorphic to the identityFootnote ¹⁰

\begin{align*} \mathscr{M}^\vee \otimes_{\mathscr{B}} \mathscr{M} \overset{\mathbf{1}}{\to} \mathscr{M}^\vee \otimes_{\mathscr{B}} \mathscr{M}. \end{align*}

After transposition, we see that it is equivalent to compare the morphisms

\begin{align*} \mathbf{1} \otimes \mathrm{ev}^R : \mathscr{M} \to \mathscr{M} \otimes_{\mathscr{A}} \mathscr{M}^\vee \otimes_{\mathscr{B}} \mathscr{M} \end{align*}

and

\begin{align*} \mathrm{coev} \otimes \mathbf{1} : \mathscr{M} \to \mathscr{M} \otimes_{\mathscr{A}} \mathscr{M}^\vee \otimes_{\mathscr{B}} \mathscr{M}. \end{align*}

Since coev is an equivalence, so is $\mathrm{coev}^R$ . Therefore, it suffices to see that both morphisms are the identity after composition with $\mathrm{coev}^R \otimes \mathbf{1}$ . For the first morphism, this is a triangular identity. For the second morphism, we use $\mathrm{coev}^R \simeq \mathrm{coev}^{-1}$ .

Note that Proposition 4.2 provides a more direct proof of [Reference Ben-Zvi, Gunningham and OremBGO20, Theorem 1.3].

4.3.1.

The goal of this subsection is to show that the hypothesis of Corollary 4.1 is applicable to the data

(4.7) \begin{equation} \mathscr{A} = \mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow{,} \quad \mathscr{B} = \mathbf{H}(Z){,} \quad \mathscr{M} = \mathrm{IndCoh}(Z).\end{equation}

That is:

• – $\mathscr{M}$ admits a left dual as a $(\mathscr{B}, \mathscr{A})$ -bimodule (Proposition 4.3);

• – the coevaluation map for this duality is an equivalence (see § 4.1.3); and

• – the evaluation map admits an $(\mathscr{A}, \mathscr{A})$ -bilinear right adjoint (Proposition 4.4).

Proof of Proposition 4.3. We need to show that the functor

\begin{align*} \mathscr{A} \textbf{-mod} \to \mathscr{B} \textbf{-mod} \end{align*}

admits a left adjoint. Indeed, its composite with the forgetful functor

\begin{align*} \mathrm{oblv} : \mathscr{B} \textbf{-mod} \to \operatorname{dgCat} \end{align*}

admits a left adjoint because $\mathscr{M}$ is dualizable as an $\mathscr{A}$ -module, and $\mathscr{B} \textbf{-mod}$ is generated under colimits by the image of $\operatorname{ind}$ .

4.3.2.

Note that the proof of Proposition 4.3 identifies the dg category underlying the left dual of $\mathscr{M}$ with its dual $\mathscr{M}^\vee$ as a plain dg category. In these terms, the resulting duality data

\begin{align*} \mathrm{coev} : \mathscr{B} \to \mathscr{M} \otimes_{\mathscr{A}} \mathscr{M}^\vee{,} \quad \mathrm{ev} : \mathscr{M}^\vee \otimes_{\mathscr{B}} \mathscr{M} \to \mathscr{A} \end{align*}

can be described as follows.

1. (i) The composite

   \begin{align*} \mathscr{B} \overset{\mathrm{coev}}{\longrightarrow} \mathscr{M} \otimes_{\mathscr{A}} \mathscr{M}^\vee \simeq \operatorname{End}_{\mathscr{A}}(\mathscr{M}) \end{align*}
   is the action map. In the case at hand, we saw in § 4.1.3 that this map in equivalence.

2. (ii) The composite

   (4.8) \begin{equation} \mathscr{M}^\vee \otimes \mathscr{M} \to \mathscr{M}^\vee \otimes_{\mathscr{B}} \mathscr{M} \overset{\mathrm{ev}}{\longrightarrow} \mathscr{A} \end{equation}
   is the counit of the $\mathscr{A}$ -module duality between $\mathscr{M}^\vee$ and $\mathscr{M}$ .

Proof of Proposition 4.4. Since $\mathscr{B}$ is rigid, the functor

\begin{align*} \mu : \mathscr{M}^\vee \otimes \mathscr{M} \to \mathscr{M}^\vee \otimes_{\mathscr{B}} \mathscr{M} \end{align*}

admits a continuous right adjoint given by the tensor product

\begin{align*} \mathrm{mult}^R \otimes \mathbf{1} : \mathscr{B} \underset{\mathscr{B} \otimes \mathscr{B}^\mathrm{rev}}{\otimes} (\mathscr{M}^\vee \otimes \mathscr{M}) \to (\mathscr{B} \otimes \mathscr{B}^\mathrm{rev}) \underset{\mathscr{B} \otimes \mathscr{B}^\mathrm{rev}}{\otimes} (\mathscr{M}^\vee \otimes \mathscr{M}). \end{align*}

Now $\mu^R$ is visibly $(\mathscr{A}, \mathscr{A})$ -bilinear, and it is conservative because the image of $\mu$ generates its target. Therefore, it suffices to check that the composite (4.8) admits a continuous $(\mathscr{A}, \mathscr{A})$ -bilinear right adjoint.

Recall from § 4.1.1 that the composite (4.8) identifies with the functor dual to the action map

(4.9) \begin{equation} \mathscr{A} \to \mathscr{M} \otimes \mathscr{M}^\vee.\end{equation}

Therefore, it suffices to show that (4.9) admits an $(\mathscr{A}, \mathscr{A})$ -bilinear left adjoint.

After Zariski localization, we may assume that Z is presented as a global complete intersection (3.14). Recall the closed immersion (3.17) and the morphism (3.15)

\begin{align*} \delta : \mathrm{Sing}(Z) \hookrightarrow Z \times V^\vee, \quad \epsilon : Z \times_U Z \to U \times \Omega V \end{align*}

from § 3.6.

Since the $(\delta_!, \delta^!)$ adjunction is a colocalization, it suffices to check that the composite of monoidal functors

(4.10) \begin{equation} \mathrm{D}(U \times V^\vee)^\Rightarrow \overset{(\delta^!)^\Rightarrow}{\longrightarrow} \mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow \overset{(\small\ref{eqn:A_M_act})}{\longrightarrow} \mathrm{IndCoh}(Z \times Z)\end{equation}

admits a $\mathrm{D}(U \times V^\vee)^\Rightarrow$ -bilinear left adjoint.

By Proposition 3.8, we may replace (4.10) with the sequence of monoidal functors

\begin{align*} & \mathrm{D}(U) \otimes \mathrm{D}(V^\vee)^\Rightarrow \overset{\mathrm{oblv} \otimes \mathrm{oblv}^\Rightarrow}{\longrightarrow} \mathrm{IndCoh}(U) \otimes \mathrm{QCoh}(V^\vee)^\Rightarrow \\ & \quad \overset{\sim}{\rightarrow} \mathrm{IndCoh}(U) \otimes \mathrm{IndCoh}(\Omega V) \overset{\epsilon^!}{\longrightarrow} \mathrm{IndCoh}(Z \times_U Z) \overset{(\Delta_{Z/U})_*}{\longrightarrow} \mathrm{IndCoh}(Z \times Z),\end{align*}

where the last functor is that of IndCoh $*$ -pushforward along the relative diagonal

\begin{align*} \Delta_{Z/U} : Z \times_U Z \to Z \times Z. \end{align*}

The left adjoint

\begin{align*} \operatorname{ind} \otimes \operatorname{ind}^\Rightarrow : \mathrm{IndCoh}(U) \otimes \mathrm{QCoh}(V^\vee)^\Rightarrow \to \mathrm{D}(U) \otimes \mathrm{D}(V^\vee)^\Rightarrow \end{align*}

of $\mathrm{oblv} \otimes \mathrm{oblv}^\Rightarrow$ is $\mathrm{D}(U) \otimes \mathrm{D}(V^\vee)^\Rightarrow$ -bilinear.

Since U is smooth, the monoidal category

(4.11) \begin{equation} \mathrm{IndCoh}(U) \otimes \mathrm{QCoh}(V^\vee)^\Rightarrow \simeq \mathrm{QCoh}(U) \otimes \mathrm{QCoh}(V^\vee)^\Rightarrow\end{equation}

is rigid. This implies that any left adjoint to the monoidal functor $(\Delta_{Z/U})_* \circ \epsilon^!$ is automatically bilinear for (4.11) and, hence, for $\mathrm{D}(U) \otimes \mathrm{D}(V^\vee)^\Rightarrow$ . Therefore, it remains to show that both $\epsilon^!$ and $(\Delta_{Z/U})_*$ are left adjointable.

The morphism $\epsilon$ is proper because it is a base change of the graph

\begin{align*} U \to U \times V, \end{align*}

so $\epsilon^!$ identifies with the right adjoint of $\epsilon_*$ .

The morphism $\Delta_{Z/U}$ is quasi-smooth because it is a base change of the diagonal

\begin{align*} U \to U \times U, \end{align*}

so it has finite tor amplitude and admits an IndCoh $*$ -pullback.

4.4.1.

By Corollary 4.1, it remains to check that the functor

\begin{align*} \mathrm{ev}^R : \mathscr{A} \to \mathscr{M}^\vee \otimes_{\mathscr{B}} \mathscr{M} \end{align*}

is conservative. It suffices to prove that the composite

\begin{align*} \mathscr{A} \overset{\mathrm{ev}^R}{\to} \mathscr{M}^\vee \otimes_{\mathscr{B}} \mathscr{M} \overset{\mu^R}{\to} \mathscr{M}^\vee \otimes \mathscr{M} \end{align*}

is conservative. As before, we may Zariski localize and assume that Z is presented as a global complete intersection.

4.4.2.

The description of

\begin{align*} (\mu \circ \mathrm{ev})^{R, \vee} \simeq (\mu \circ \mathrm{ev})^{\vee, L} \end{align*}

from the proof of Proposition 4.4 identifies $\mu^R \circ \mathrm{ev}^R$ with

\begin{align*}\mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow & \overset{(\delta^!)^{\Rightarrow, \vee}}{\longrightarrow} \mathrm{D}(U) \otimes \mathrm{D}(V^\vee)^\Rightarrow \overset{\operatorname{ind}^\vee \otimes (\operatorname{ind}^\Rightarrow)^\vee}{\longrightarrow} \mathrm{IndCoh}(U) \otimes \mathrm{QCoh}(V^\vee)^\Rightarrow \\ & \overset{\sim}{\rightarrow} \mathrm{IndCoh}(U) \otimes \mathrm{IndCoh}(\Omega V) \overset{(\epsilon_*)^\vee}{\longrightarrow} \mathrm{IndCoh}(Z \times_U Z) \overset{(\Delta_{Z/U}^*)^\vee}{\longrightarrow} \mathrm{IndCoh}(Z \times Z).\end{align*}

Now, $\Delta_{Z/U}^*$ differs from $\Delta_{Z/U}^!$ by a shifted line bundle because $\Delta_{Z/U}$ is quasi-smooth and hence Gorenstein. Therefore, it is equivalent to show that composition of the functor

(4.12) \begin{equation}\begin{split} \mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow & \overset{(\delta_*)^\Rightarrow}{\longrightarrow} \mathrm{D}(U) \otimes \mathrm{D}(V^\vee)^\Rightarrow \overset{\mathrm{oblv} \otimes \mathrm{oblv}^\Rightarrow}{\longrightarrow} \mathrm{IndCoh}(U) \otimes \mathrm{QCoh}(V^\vee)^\Rightarrow \\ & \overset{\sim}{\rightarrow} \mathrm{IndCoh}(U \times \Omega V) \overset{\epsilon^!}{\to} \mathrm{IndCoh}(Z \times_U Z)\end{split}\end{equation}

with the functor of IndCoh $*$ -pushforward along the map $\Delta_{Z/U}$ is conservative.

4.4.3.

Both of the functors $(\delta_*)^\Rightarrow$ and $\mathrm{oblv} \otimes \mathrm{oblv}^\Rightarrow$ are conservative, and their composition takes values in the full subcategory of $\mathrm{IndCoh}(U \times \Omega V)$ consisting of objects whose singular support is contained in the image of $\delta$ . Since the functor $\epsilon^!$ is conservative when restricted to objects that are set-theoretically supported on the image of $\epsilon$ , it follows that the functor (4.12) is conservative. Furthermore, [Reference Arinkin and GaitsgoryAG15, Proposition 7.1.3(a)] implies that the image of (4.12) is contained in the full subcategory consisting of objects whose singular support is contained in the image of the map

(4.13) \begin{equation} \mathrm{Sing}(Z) \hookrightarrow Z \times V^\vee \hookrightarrow \mathrm{Sing}(Z \times_U Z),\end{equation}

where the first arrow is the one appearing in the definition (3.17) of $\delta$ and the second is the one induced by the singular codifferential of $\epsilon$ .

By [Reference Arinkin and GaitsgoryAG15, Proposition 7.3.10], the functor of IndCoh $*$ -pushforward along $\Delta_{Z/U}$ is conservative when restricted to those objects whose singular support is contained in the image of the singular codifferential

\begin{align*} \mathrm{Sing}(\Delta_{Z/U}) : \mathrm{Sing}(Z \times Z)_Z \hookrightarrow \mathrm{Sing}(Z \times_U Z). \end{align*}

We observe that this locus contains the image of (4.13).

4.5.1.

Let $f : Z \to Y$ be a morphism of quasi-smooth schemes. We may form the category

\begin{align*} \mathbf{H}_{Z \to Y} = \mathrm{IndCoh}_0((Z \times Y)^\wedge_Z) \end{align*}

associated to the graph of f by the procedure of § 3.2.3, regarded as an $(\mathbf{H}(Z), \mathbf{H}(Y))$ -bimodule under convolution. According to [Reference BeraldoBer19, Proposition 5.2.8], the bimodule $\mathbf{H}_{Z \to Y}$ admits a right dual (and also a left dual) canonically identified with

\begin{align*} \mathbf{H}_{Y \leftarrow Z} = \mathrm{IndCoh}_0((Y \times Z)^\wedge_Z), \end{align*}

viewed as an $(\mathbf{H}(Y), \mathbf{H}(Z))$ -bimodule.Footnote ¹¹

In particular, we obtain a pair of adjoint functors

\begin{align*} \mathbf{H}_{Z \to Y} : \mathbf{H}(Y) \textbf{-mod} \rightleftharpoons \mathbf{H}(Z) \textbf{-mod} : \mathbf{H}_{Y \leftarrow Z}. \end{align*}

The purpose of this subsection is to describe the adjunction

\begin{align*} \mathrm{D}(\mathrm{Sing}(Y))^\Rightarrow \rightleftharpoons \mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow \end{align*}

corresponding to this one under the Morita equivalence of Theorem 4.1.

4.5.2.

Consider the following correspondence induced by the singular codifferential of f.

(4.14)

It defines a bimodule

(4.15) \begin{equation} \mathrm{D}(\mathrm{Sing}(Y)_Z)^\Rightarrow \in (\mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow, \mathrm{D}(\mathrm{Sing}(Y))^\Rightarrow) \textbf{-bimod}.\end{equation}

We claim the following.

Construction 4.1. The following diagram commutes.

(4.16)

4.5.3.

Note that we may also view the correspondence (4.14) as defining a bimodule

(4.17) \begin{equation} \mathrm{D}(\mathrm{Sing}(Y)_Z)^\Rightarrow \in (\mathrm{D}(\mathrm{Sing}(Y))^\Rightarrow, \mathrm{D}(\mathrm{Sing}(Z))^\Rightarrow) \textbf{-bimod}.\end{equation}

This bimodule is canonically right dual (and also left dual) to (4.15), so after taking right adjoints of the vertical arrows, we obtain from (4.16) the following commutative diagram.

(4.18)

5.1.1.

Let $\mathscr{Y}$ be a quasi-smooth algebraic stack, and let $\operatorname{Sm}_{/\mathscr{Y}}$ denote the category whose objects are schemes S equipped with a smooth morphism $S \to \mathscr{Y}$ and whose morphisms are composites $T \to S \to \mathscr{Y}$ with $T \to S$ smooth.

5.1.2.

Recall that [Reference BeraldoBer19] assembles the assignment

(5.2) \begin{equation} S \rightsquigarrow \mathbf{H}(S)\textbf{-mod} \quad (T \to S) \rightsquigarrow \mathbf{H}_{T \to S} \otimes_{\mathbf{H}(S)} (-)\end{equation}

into a functor

\begin{align*} \mathbf{H}\textbf{-mod} : (\operatorname{Sm}_{/\mathscr{Y}})^\mathrm{op} \to \textbf{2-dgCat}. \end{align*}

Now let

\begin{align*} \operatorname{ShCat}_\mathbf{H}(\mathscr{Y}) = \lim_{S \to \mathscr{Y}} \mathbf{H}(S) \textbf{-mod} \end{align*}

denote the limit of the functor $\mathbf{H}\textbf{-mod}$ .Footnote ¹³ The $\mathbf{H}$ -affineness result of [Reference BeraldoBer19] exhibits an adjoint equivalence

\begin{align*} \mathbf{Loc}_\mathbf{H} : \mathbf{H}(\mathscr{Y})\textbf{-mod} \rightleftharpoons \operatorname{ShCat}_\mathbf{H}(\mathscr{Y}) : \mathbf{\Gamma}_\mathbf{H}. \end{align*}

In particular, viewing $\mathrm{IndCoh}(\mathscr{Y})$ a module over $\mathbf{H}(\mathscr{Y})$ , we obtain a canonical object

(5.3) \begin{equation} \mathbf{Loc}_\mathbf{H}(\mathrm{IndCoh}(\mathscr{Y})) \in \operatorname{ShCat}_\mathbf{H}(\mathscr{Y}).\end{equation}

5.1.3.

According to [Reference BeraldoBer19, Lemma 6.6.1], the value of (5.3) at an object $S \in \operatorname{Sm}_{/\mathscr{Y}}$ is

\begin{align*} \mathbf{Loc}_\mathbf{H}(\mathrm{IndCoh}(\mathscr{Y}))_S \simeq \mathrm{IndCoh}(S) \end{align*}

and its value at a morphism $T \to S$ is given by the convolution map

\begin{align*} \mathbf{H}_{T \to S} \underset{\mathbf{H}(S)}{\otimes} \mathrm{IndCoh}(S) \overset{\sim}{\rightarrow} \mathrm{IndCoh}(T). \end{align*}

5.1.4.

Now let us describe the counterparts of $\mathbf{H}\textbf{-mod}$ and $\mathrm{Sh}\mathrm{Cat}_\mathbf{H}(\mathscr{Y})$ that will appear on the other side of the Morita equivalence.

Let $f : T \to S$ be any morphism in $\operatorname{Sm}_{/\mathscr{Y}}$ . Since f is required to be smooth, its singular codifferential is an isomorphism

\begin{align*} \mathrm{Sing}(S)_T \overset{\sim}{\rightarrow} \mathrm{Sing}(T) \end{align*}

and the correspondence (4.14) induces a map

(5.4) \begin{equation} \mathrm{Sing}(T) \to \mathrm{Sing}(S).\end{equation}

In particular, we may consider the adjunction

\begin{align*} \mathbf{ind} : \mathrm{D}(\mathrm{Sing}(S))^\Rightarrow \textbf{-mod} \rightleftharpoons \mathrm{D}(\mathrm{Sing}(T))^\Rightarrow \textbf{-mod} : \mathbf{oblv}. \end{align*}

The left adjoints assemble into a functor

\begin{align*} \mathbf{S}^\Rightarrow\textbf{-mod} : (\operatorname{Sm}_{/\mathscr{Y}})^\mathrm{op} \to \textbf{2-dgCat}, \end{align*}

given at the level of objects and morphisms by the assignment

\begin{align*} (T \to S) \rightsquigarrow ( \mathrm{D}(\mathrm{Sing}(S))^\Rightarrow \textbf{-mod} \overset{\mathbf{ind}}{\longrightarrow} \mathrm{D}(\mathrm{Sing}(T))^\Rightarrow \textbf{-mod} ). \end{align*}

Let

(5.5) \begin{equation} \operatorname{ShCat}_{\mathbf{S}^\Rightarrow}(\mathscr{Y}) = \lim_{S \to \mathscr{Y}} \mathrm{D}(\mathrm{Sing}(S))^\Rightarrow \textbf{-mod}\end{equation}

denote the limit of ${\mathbf{S}^\Rightarrow}\textbf{-mod}$ , viewed as a functor. It is equipped with tautological evaluation functors

\begin{align*} \mathbf{\Gamma}(S; -) : \operatorname{ShCat}_{\mathbf{S}^\Rightarrow}(\mathscr{Y}) \to \mathrm{D}(\mathrm{Sing}(S))^\Rightarrow \textbf{-mod}. \end{align*}

For any stack $\mathscr{W}$ equipped with a ${\mathbf{G}_m}$ -equivariant morphism

\begin{align*} \mathscr{W} \to \mathrm{Sing}(\mathscr{Y}), \end{align*}

we obtain an object

\begin{align*} \underline{\mathrm{D}}(\mathscr{W})^\Rightarrow \in \operatorname{ShCat}_{\mathbf{S}^\Rightarrow}(\mathscr{Y}) \end{align*}

with

\begin{align*} \mathbf{\Gamma}(S; \underline{\mathrm{D}}(\mathscr{W})) \simeq \mathrm{D}(\mathscr{W}_S)^\Rightarrow \quad \mathscr{W}_S = S \times_{\mathscr{Y}} \mathscr{W}. \end{align*}

Construction 5.1. There is an equivalence of categories

(5.6) \begin{equation} \operatorname{ShCat}_\mathbf{H}(\mathscr{Y}) \simeq \operatorname{ShCat}_{\mathbf{S}^\Rightarrow}(\mathscr{Y})\end{equation}

exchanging the objects

\begin{align*} \mathbf{Loc}_\mathbf{H}(\mathrm{IndCoh}(\mathscr{Y})) \quad \text{and} \quad \underline{\mathrm{D}}(\mathrm{Sing}(\mathscr{Y}))^\Rightarrow. \end{align*}

5.1.5.

We will obtain Construction 5.1 from a slightly stronger result, Construction 5.2, whose statement we will now explain.

Let $\textbf{dgCat}$ denote the constant functor

(5.7) \begin{equation} (\operatorname{Sm}_{/\mathscr{Y}})^\mathrm{op} \to \textbf{2-dgCat}\end{equation}

with value $\textbf{dgCat}$ . Then, we may regard $\mathbf{Loc}_H(\mathrm{IndCoh}(\mathscr{Y}))$ as a natural transformation

(5.8) \begin{equation} \textbf{dgCat} \to \mathbf{H} \textbf{-mod}\end{equation}

of functors of the shape (5.7), whose value at an object $S \in \operatorname{Sm}_{/\mathscr{Y}}$ is

(5.9) \begin{equation} \mathrm{IndCoh}(S) : \textbf{dgCat} \to \mathbf{H}(S) \textbf{-mod}.\end{equation}

Similarly, we may regard $\underline{\mathrm{D}}(\mathrm{Sing}(\mathscr{Y}))^\Rightarrow$ as a natural transformation

(5.10) \begin{equation} \textbf{dgCat} \to {\mathbf{S}^\Rightarrow} \textbf{-mod},\end{equation}

whose value at an object $S \in \operatorname{Sm}_{/\mathscr{Y}}$ is

(5.11) \begin{equation} \mathbf{ind} : \textbf{dgCat} \to \mathrm{D}(\mathrm{Sing}(S))^\Rightarrow \textbf{-mod}.\end{equation}

To perform Construction 5.1, it suffices to perform the following and pass to the limit.

5.1.6.

The value of the putative natural isomorphism

\begin{align*} \mathbf{H} \textbf{-mod} \simeq {\mathbf{S}^\Rightarrow} \textbf{-mod} \end{align*}

at an individual morphism $T \to S$ is the following diagram provided by Construction 4.1.

(5.12)

Therefore, the content of Construction 5.2 is that the diagram (5.12) can be assembled into a natural transformation. This will be done in § 5.2.

5.2.1.

It turns out that Construction 5.2 as stated is difficult to approach directly, so we will have to provide a more tractable reformulation. This will appear below as Construction 5.3.

5.2.2.

Recall from § 4.5.1 that all of the bimodules $\mathbf{H}_{T \to S}$ admit right duals. Therefore, by passing to right adjoints in $\textbf{2-dgCat}$ , we obtain from $\mathbf{H}\textbf{-mod}$ a functor

\begin{align*} \mathbf{H} \textbf{-mod}^R : \operatorname{Sm}_{/\mathscr{Y}} \to \textbf{2-dgCat} \end{align*}

given at the level of objects and morphisms by the assignment

\begin{align*} S \rightsquigarrow \mathbf{H}(S)\textbf{-mod}{,} \quad (T \to S) \rightsquigarrow \mathbf{H}_{S \leftarrow T} \underset{\mathbf{H}(T)}{\otimes} (-). \end{align*}

Applying the same procedure to ${\mathbf{S}^\Rightarrow}\textbf{-mod}$ , we obtain a functor

\begin{align*} {\mathbf{S}^\Rightarrow} \textbf{-mod}^R : \operatorname{Sm}_{/\mathscr{Y}} \to \textbf{2-dgCat} \end{align*}

given at the level of objects and morphisms by the assignment

\begin{align*} (T \to S) \rightsquigarrow ( \mathrm{D}(\mathrm{Sing}(T))^\Rightarrow \textbf{-mod} \overset{\mathbf{oblv}}{\longrightarrow} \mathrm{D}(\mathrm{Sing}(S))^\Rightarrow \textbf{-mod} ). \end{align*}

Now let $\textbf{dgCat}^R$ denote the constant functor

(5.13) \begin{equation} \operatorname{Sm}_{/\mathscr{Y}} \to \textbf{2-dgCat}\end{equation}

with value $\textbf{dgCat}$ (alternatively, it is obtained from the functor $\textbf{dgCat}$ by passing to right adjoints). Since $\mathrm{IndCoh}(S)$ is self-dual as a module over $\mathbf{H}(S)$ , all of the functors (5.9) admit right adjoints in $\textbf{2-dgCat}$ . Therefore, by passing to right adjoints, we obtain from (5.8) a natural transformation

(5.14) \begin{equation} \mathbf{H}\textbf{-mod}^R \to \textbf{dgCat}^R\end{equation}

of functors of the shape (5.13). Similarly, by virtue of the $(\mathbf{ind}, \mathbf{oblv})$ adjunction, we obtain from (5.10) a natural transformation

(5.15) \begin{equation} {\mathbf{S}^\Rightarrow}\textbf{-mod}^R \to \textbf{dgCat}^R\end{equation}

whose value at $S \in \operatorname{Sm}_{/\mathscr{Y}}$ is the forgetful functor

(5.16) \begin{equation} \mathbf{oblv} : \mathrm{D}(\mathrm{Sing}(S))^\Rightarrow \textbf{-mod} \to \textbf{dgCat}.\end{equation}

Construction 5.2 is equivalent to the following.

5.2.3.

The feature of Construction 5.3 which makes it manageable is the fact that the forgetful functors (5.16) are locally fully faithful (this follows from Proposition 2.2). This means (see Proposition A.5) that, in order to perform Construction 5.3, it is equivalent to do the following.

1. (i) For every object $S \in \operatorname{Sm}_{/\mathscr{Y}}$ , specify a commutative diagram.

   

2. (ii) For every morphism $T \to S$ in $\operatorname{Sm}_{/\mathscr{Y}}$ , verify that the resulting diagram (in which the back face is empty) can be completed to a commutative prism.

   (5.17)

   

Note that the data required by (i) is provided by Theorem 4.1, so it remains to verify the assertion of (ii). This occupies the remainder of the subsection.

5.2.4.

The front face of (5.17) encodes the isomorphism of right $\mathbf{H}(T)$ -modules

(5.18) \begin{equation} \mathrm{IndCoh}(S) \underset{\mathbf{H}(S)}{\otimes} \mathbf{H}_{S \leftarrow T} \to \mathrm{IndCoh}(T)\end{equation}

given by convolution. We need to check that (5.18) is linear for the pre-existing left actions of $\mathrm{D}(\mathrm{Sing}(S))^\Rightarrow$ on its source and target.

Indeed, let us write (5.18) as the composite

(5.19) \begin{equation} \begin{split} & \mathrm{IndCoh}(S) \underset{\mathbf{H}(S)}{\otimes} \mathbf{H}_{S \leftarrow T} \overset{\sim}{\rightarrow} \mathrm{IndCoh}(S) \underset{\mathbf{H}(S)}{\otimes} \mathrm{IndCoh}(S)^\tau \underset{\mathrm{D}(\mathrm{Sing}(S))^\Rightarrow}{\otimes} \mathrm{IndCoh}(T) \\ & \quad \hookrightarrow \mathrm{IndCoh}(S) \underset{\mathbf{H}(S)}{\otimes} \mathrm{IndCoh}(S)^\tau \otimes \mathrm{IndCoh}(T) \overset{\mathrm{ev} \otimes \mathbf{1}}{\longrightarrow} \mathrm{IndCoh}(T). \end{split}\end{equation}

By § 4.6.4, we may further rewrite (5.19) as

\begin{equation*} \begin{split} & \mathrm{IndCoh}(S) \underset{\mathbf{H}(S)}{\otimes} \mathbf{H}_{S \leftarrow T} \overset{\sim}{\rightarrow} \mathrm{IndCoh}(S) \underset{\mathbf{H}(S)}{\otimes} \mathrm{IndCoh}(S)^\tau \underset{\mathrm{D}(\mathrm{Sing}(S))^\Rightarrow}{\otimes} \mathrm{IndCoh}(T) \\ & \quad \overset{\sim}{\rightarrow} \mathrm{D}(\mathrm{Sing}(S))^\Rightarrow \underset{\mathrm{D}(\mathrm{Sing}(S))^\Rightarrow}{\otimes} \mathrm{IndCoh}(T) \hookrightarrow \mathrm{D}(\mathrm{Sing}(S))^\Rightarrow \otimes \mathrm{IndCoh}(T) \\ & \quad \overset{(\Gamma_\mathrm{dR})^\Rightarrow \otimes \mathbf{1}}{\longrightarrow} \mathrm{IndCoh}(T). \end{split}\end{equation*}

This last expression identifies with

\begin{align*} & \mathrm{IndCoh}(S) \underset{\mathbf{H}(S)}{\otimes} \mathbf{H}_{S \leftarrow T} \overset{\sim}{\rightarrow} \mathrm{IndCoh}(S) \underset{\mathbf{H}(S)}{\otimes} \mathrm{IndCoh}(S)^\tau \underset{\mathrm{D}(\mathrm{Sing}(S))^\Rightarrow}{\otimes} \mathrm{IndCoh}(T) \\ & \quad \overset{\sim}{\rightarrow} \mathrm{D}(\mathrm{Sing}(S))^\Rightarrow \underset{\mathrm{D}(\mathrm{Sing}(S))^\Rightarrow}{\otimes} \mathrm{IndCoh}(T) \overset{\sim}{\rightarrow} \mathrm{IndCoh}(T). \end{align*}

5.3.1.

Let $\mathscr{Z} \to \mathscr{Y}$ be a schematic morphism of quasi-smooth stacks. Then $\mathrm{IndCoh}(\mathscr{Y}^\wedge_{\mathscr{Z}})$ is naturally acted on by $\mathbf{H}(\mathscr{Y})$ , and we may form the object

\begin{align*} \mathbf{Loc}(\mathrm{IndCoh}(\mathscr{Y}^\wedge_{\mathscr{Z}})) \in \operatorname{ShCat}_\mathbf{H}(\mathscr{Y}). \end{align*}

On the other hand, from the morphism

\begin{align*} \mathrm{Sing}(\mathscr{Y})_{\mathscr{Z}} \to \mathrm{Sing}(\mathscr{Y}), \end{align*}

we obtain an object

\begin{align*} \underline{\mathrm{D}}(\mathrm{Sing}(\mathscr{Y})_{\mathscr{Z}})^\Rightarrow \in \operatorname{ShCat}_{\mathbf{S}^\Rightarrow}(\mathscr{Y}). \end{align*}

We note for future reference that for $S \in \operatorname{Sm}_{/\mathscr{Y}}$ , we have

(5.20) \begin{equation} \mathrm{Sing}(S)_{\mathscr{Z}_S} \simeq (\mathrm{Sing}(\mathscr{Y})_{\mathscr{Z}})_S.\end{equation}

Construction 5.4. The Morita equivalence (5.6) exchanges the objects

\begin{align*} \underline{\mathrm{D}}(\mathrm{Sing}(\mathscr{Y})_{\mathscr{Z}})^\Rightarrow \quad \text{and} \quad \mathbf{Loc}(\mathrm{IndCoh}(\mathscr{Y}^\wedge_{\mathscr{Z}})). \end{align*}

5.3.2.

Let us explain why Construction 5.4 is non-trivial. Consider the object

\begin{align*} \mathscr{D} \in \operatorname{ShCat}_{\mathbf{S}^\Rightarrow}(\mathscr{Y}) \end{align*}

corresponding to $\mathbf{Loc}(\mathrm{IndCoh}(\mathscr{Y}^\wedge_{\mathscr{Z}}))$ under the Morita equivalence (5.6).

The value of $\mathbf{Loc}(\mathrm{IndCoh}(\mathscr{Y}^\wedge_{\mathscr{Z}}))$ at an object $S \in \operatorname{Sm}_{/\mathscr{Y}}$ identifies with

\begin{align*} \mathbf{Loc}(\mathrm{IndCoh}(\mathscr{Y}^\wedge_{\mathscr{Z}}))_S \simeq \mathbf{H}_{S \to \mathscr{Y}} \underset{\mathbf{H}(\mathscr{Y})}{\otimes} \mathrm{IndCoh}(\mathscr{Y}^\wedge_{\mathscr{Z}}) \simeq \mathrm{IndCoh}(S^\wedge_{\mathscr{Z}_S}), \end{align*}

with its natural action of $\mathbf{H}(S)$ . Therefore, the equivalence (3.2), combined with the Morita equivalence for schemes, produces an equivalence of categories

(5.21) \begin{equation} \mathrm{D}(\mathrm{Sing}(S)_{\mathscr{Z}_S})^\Rightarrow \simeq \mathscr{D}_S\end{equation}

linear over $\mathrm{D}(\mathrm{Sing}(S))^\Rightarrow$ . However, for the purposes of Construction 5.4, this observation by itself is insufficient. It is not obvious how to supply the functoriality required to promote the data of (5.21) to an equivalence in $\operatorname{ShCat}_{\mathbf{S}^\Rightarrow}(\mathscr{Y})$ .

So the challenge is to produce a morphism

(5.22) \begin{equation} \underline{\mathrm{D}}(\mathrm{Sing}(\mathscr{Y})_{\mathscr{Z}})^\Rightarrow \to \mathscr{D}\end{equation}

in $\operatorname{ShCat}_{\mathbf{S}^\Rightarrow}(\mathscr{Y})$ . Once this has been done, it will be straightforward to verify that the value of (5.22) at every object of $\operatorname{Sm}_{/\mathscr{Y}}$ agrees with the equivalence (5.21), and hence that (5.22) itself is an equivalence.

We shall delay the execution of Construction 5.4 to § 5.4, and use the remainder of this subsection to explain some of its consequences.

5.3.3.

The colocalization

\begin{align*} \mathrm{IndCoh}_0(\mathscr{Y}^\wedge_{\mathscr{Z}}) \rightleftharpoons \mathrm{IndCoh}(\mathscr{Y}^\wedge_{\mathscr{Z}}) \end{align*}

of $\mathbf{H}(\mathscr{Y})$ -modules induces a colocalization

(5.23) \begin{equation} \mathbf{Loc}(\mathrm{IndCoh}_0(\mathscr{Y}^\wedge_{\mathscr{Z}})) \rightleftharpoons \mathbf{Loc}(\mathrm{IndCoh}(\mathscr{Y}^\wedge_{\mathscr{Z}})).\end{equation}

On the other hand, observe that we may identify

\begin{align*} \mathrm{Sing}(\mathscr{Z}_S/S) \simeq S \underset{\mathscr{Y}}{\times} \mathrm{Sing}(\mathscr{Z}/\mathscr{Y}) \end{align*}

as closed subschemes of (5.20). In particular, we obtain a colocalization

\begin{align*} \underline{\mathrm{D}}(\mathrm{Sing}(\mathscr{Z}/\mathscr{Y}))^\Rightarrow \rightleftharpoons \underline{\mathrm{D}}(\mathrm{Sing}(\mathscr{Y})_{\mathscr{Z}})^\Rightarrow \end{align*}

in $\operatorname{ShCat}_{\mathbf{S}^\Rightarrow}(\mathscr{Y})$ whose value at an object $S \in \operatorname{Sm}_{/\mathscr{Y}}$ is given by the colocalization

\begin{align*} \mathrm{D}(\mathrm{Sing}(\mathscr{Z}_S/S))^\Rightarrow \rightleftharpoons \mathrm{D}(\mathrm{Sing}(S)_{\mathscr{Z}_S})^\Rightarrow. \end{align*}

Proposition 5.1. Construction 5.4 exchanges the colocalizations

\begin{align*} \underline{\mathrm{D}}(\mathrm{Sing}(\mathscr{Z}/\mathscr{Y}))^\Rightarrow \quad \text{and} \quad \mathbf{Loc}(\mathrm{IndCoh}_0(\mathscr{Y}^\wedge_{\mathscr{Z}})). \end{align*}

5.3.4.

Under our running assumptions on stacks (see § 1.5.6), for every schematic morphism $f : \mathscr{Z} \to \mathscr{Y}$ of quasi-smooth stacks, the graph $\Gamma_f: \mathscr{Z} \to \mathscr{Z} \times \mathscr{Y}$ is also schematic. Recall that one may consider the $(\mathbf{H}(\mathscr{Z}), \mathbf{H}(\mathscr{Y}))$ -bimodule

\begin{align*} \mathbf{H}_{\mathscr{Z} \to \mathscr{Y}} = \mathrm{IndCoh}_0((\mathscr{Z} \times \mathscr{Y})^\wedge_{\mathscr{Z}}). \end{align*}

Recall also the correspondence of stacks induced by the singular codifferential of f.

(5.24)

Let us form the object

\begin{align*} \underline{\mathrm{D}}(\mathrm{Sing}(\mathscr{Y})_{\mathscr{Z}})^\Rightarrow \in \operatorname{ShCat}_{\mathbf{S}^\Rightarrow}(\mathscr{Z} \times \mathscr{Y}) \end{align*}

obtained from the morphism

(5.25) \begin{equation} \mathrm{Sing}(\mathscr{Y})_{\mathscr{Z}} \overset{\text{(\ref{eqn:singular_codifferential_stack})}}{\longrightarrow} \mathrm{Sing}(\mathscr{Z}) \times \mathrm{Sing}(\mathscr{Y}) \overset{\tau \times \mathbf{1}}{\longrightarrow} \mathrm{Sing}(\mathscr{Z}) \times \mathrm{Sing}(\mathscr{Y}).\end{equation}

Since (5.25) identifies with the map

(5.26) \begin{equation} \mathrm{Sing}(\mathscr{Z}/\mathscr{Z} \times \mathscr{Y}) \to \mathrm{Sing}(\mathscr{Z} \times \mathscr{Y})_{\mathscr{Z}}\end{equation}

associated to the graph of f

\begin{align*} \mathscr{Z} \to \mathscr{Z} \times \mathscr{Y}, \end{align*}

we obtain the following consequence of Proposition 5.1.

Construction 5.5. The Morita equivalence (5.6) for $\mathscr{Z} \times \mathscr{Y}$ exchanges the objects

\begin{align*} \underline{\mathrm{D}}(\mathrm{Sing}(\mathscr{Y})_{\mathscr{Z}})^\Rightarrow \quad \text{and} \quad \mathbf{Loc}(\mathbf{H}_{\mathscr{Z} \to \mathscr{Y}}). \end{align*}

5.4.1.

As indicated in § 5.3.2, the task at hand is more or less to construct the morphism (5.22). This will be accomplished by subjecting the functor of IndCoh $!$ -pullback

(5.27) \begin{equation} \mathrm{IndCoh}(\mathscr{Y}) \to \mathrm{IndCoh}(\mathscr{Y}^\wedge_{\mathscr{Z}}),\end{equation}

viewed as a morphism of $\mathbf{H}(\mathscr{Y})$ -modules, to a series of transformations.

First we apply $\mathbf{Loc}$ to (5.27), and obtain a morphism

(5.28) \begin{equation} \mathbf{Loc}(\mathrm{IndCoh}(\mathscr{Y})) \to \mathbf{Loc}(\mathrm{IndCoh}(\mathscr{Y}^\wedge_{\mathscr{Z}}))\end{equation}

in $\operatorname{ShCat}_\mathbf{H}(\mathscr{Y})$ .

Then, (5.28) may be viewed as a morphism

(5.29) \begin{equation} \underline{\mathrm{D}}(\mathrm{Sing}(\mathscr{Y}))^\Rightarrow \to \mathscr{D}\end{equation}

in $\operatorname{ShCat}_{\mathbf{S}^\Rightarrow}(\mathscr{Y})$ .

5.4.2 Digression

To proceed further, we need to introduce the category

\begin{align*} \operatorname{ShCat}_{\mathbf{S}^\Rightarrow}(\mathscr{Y})_{\mathscr{Z}} = \lim_{S \to \mathscr{Y}} \mathrm{D}(\mathrm{Sing}(S)_{\mathscr{Z}_S})^\Rightarrow \textbf{-mod}, \end{align*}

where the limit is taken over $(\operatorname{Sm}_{/\mathscr{Y}})^\mathrm{op}$ with respect to the transition functors of induction:

\begin{equation*} \mathbf{ind} : \mathrm{D}(\mathrm{Sing}(S)_{\mathscr{Z}_S})^\Rightarrow \textbf{-mod} \to \mathrm{D}(\mathrm{Sing}(T)_{\mathscr{Z}_T})^\Rightarrow \textbf{-mod}.\end{equation*}

Consider the adjunctions

\begin{align*} \mathbf{ind}_S : \mathrm{D}(\mathrm{Sing}(S))^\Rightarrow \textbf{-mod} \rightleftharpoons \mathrm{D}(\mathrm{Sing}(S)_{\mathscr{Z}_S})^\Rightarrow \textbf{-mod} : \mathbf{oblv}_S. \end{align*}

Observe that the left adjoints $\mathbf{ind}_S$ assemble tautologically into a functor

(5.30) \begin{equation} \mathbf{ind} : \operatorname{ShCat}_{\mathbf{S}^\Rightarrow}(\mathscr{Y}) \to \operatorname{ShCat}_{\mathbf{S}^\Rightarrow}(\mathscr{Y})_{\mathscr{Z}}\end{equation}

making the following diagram commute.

Since the map $\mathscr{Z} \to \mathscr{Y}$ is schematic, the Beck–Chevalley transformations are isomorphisms.

(5.31)

Therefore, by passing to the limit, we obtain that (5.30) admits a right adjoint

\begin{align*} \mathbf{oblv} : \operatorname{ShCat}_{\mathbf{S}^\Rightarrow}(\mathscr{Y})_{\mathscr{Z}} \to \operatorname{ShCat}_{\mathbf{S}^\Rightarrow}(\mathscr{Y}) \end{align*}

making the following diagram commute.

Note that we may identify

(5.32) \begin{equation} \mathbf{oblv} \circ \mathbf{ind}(\underline{\mathrm{D}}(\mathrm{Sing}(\mathscr{Y}))^\Rightarrow) \simeq \underline{\mathrm{D}}(\mathrm{Sing}(\mathscr{Y})_{\mathscr{Z}})^\Rightarrow.\end{equation}

5.4.3.

Note that the value of (5.28) at an object $S \in \operatorname{Sm}_{/\mathscr{Y}}$ is given by the functor of IndCoh $!$ -pullback

(5.33) \begin{equation} \mathrm{IndCoh}(S) \to \mathrm{IndCoh}(S^\wedge_{\mathscr{Z}_S}),\end{equation}

regarded as a morphism of $\mathbf{H}(S)$ -modules.

We may view (5.33) as the functor

\begin{align*} \mathrm{IndCoh}(S) \to \mathrm{IndCoh}(S) \underset{\mathrm{D}(\mathrm{Sing}(S))^\Rightarrow}{\otimes} \mathrm{D}(\mathrm{Sing}(S)_{\mathscr{Z}_S})^\Rightarrow, \end{align*}

induced by the D-module $!$ -pullback

(5.34) \begin{equation} \mathrm{D}(\mathrm{Sing}(S))^\Rightarrow \to \mathrm{D}(\mathrm{Sing}(S)_{\mathscr{Z}_S})^\Rightarrow.\end{equation}

In other words, the value of (5.29) at S identifies with the morphism (5.34).

In particular, this lifts $\mathscr{D}_S$ to an object

\begin{align*} \tilde{\mathscr{D}}_S \in \mathrm{D}(\mathrm{Sing}(S)_{\mathscr{Z}_S})^\Rightarrow \textbf{-mod}. \end{align*}

We claim the following.

Proposition 5.2. The data of $\tilde{\mathscr{D}}_S$ for $S \in \operatorname{Sm}_{/\mathscr{Y}}$ determine a lift of $\mathscr{D}$ to an object

\begin{align*} \tilde{\mathscr{D}} \in \operatorname{ShCat}_{\mathbf{S}^\Rightarrow}(\mathscr{Y})_{\mathscr{Z}}. \end{align*}

5.4.4.

Note that this assertion is a proposition and not a construction, because Proposition 2.3 (which is applicable by the running assumption that $\mathscr{Z} \to \mathscr{Y}$ is schematic) implies that the functors $\mathbf{oblv}_S$ are locally fully faithful. Namely, it amounts to the assertion that for every arrow $T \to S$ in $\operatorname{Sm}_{/\mathscr{Y}}$ , the map of $\mathrm{D}(\mathrm{Sing}(T))^\Rightarrow$ -modules

(5.35) \begin{equation} \mathbf{oblv}_T \circ \mathbf{ind}_{T \to S} (\tilde{\mathscr{D}}_S) \simeq \mathbf{ind}_{T \to S} \circ \mathbf{oblv}_S (\tilde{\mathscr{D}}_S) \simeq \mathbf{ind}_{T \to S} (\mathscr{D}_S) \overset{\sim}{\rightarrow} \mathscr{D}_T \simeq \mathbf{oblv}_T(\tilde{\mathscr{D}}_T)\end{equation}

is actually linear over $\mathrm{D}(\mathrm{Sing}(T)_{\mathscr{Z}_T})^\Rightarrow$ .

5.4.5.

We now take Proposition 5.2 as given and resume the construction of (5.22). Its proofs appear at the end of this section, see § 5.4.6.

By Proposition 5.2, we may perceive (5.29) as a morphism

(5.36) \begin{equation} \underline{\mathrm{D}}(\mathrm{Sing}(\mathscr{Y}))^\Rightarrow \to \mathbf{oblv}(\tilde{\mathscr{D}})\end{equation}

in $\operatorname{ShCat}_{\mathbf{S}^\Rightarrow}(\mathscr{Y})$ .

The transpose of (5.36) under the $(\mathbf{ind}, \mathbf{oblv})$ adjunction is a morphism

(5.37) \begin{equation} \mathbf{ind}(\underline{\mathrm{D}}(\mathrm{Sing}(\mathscr{Y}))^\Rightarrow) \to \tilde{\mathscr{D}}\end{equation}

in $\operatorname{ShCat}_{\mathbf{S}^\Rightarrow}(\mathscr{Y})_{\mathscr{Z}}$ .

After applying $\mathbf{oblv}$ to (5.37), we obtain a morphism

(5.38) \begin{equation} \mathbf{oblv} \circ \mathbf{ind}(\underline{\mathrm{D}}(\mathrm{Sing}(\mathscr{Y}))^\Rightarrow) \to \mathscr{D}\end{equation}

in $\operatorname{ShCat}_{\mathbf{S}^\Rightarrow}(\mathscr{Y})$ .

Finally, after making the identification (5.32), we find that (5.38) defines the sought-after morphism

(5.39) \begin{equation} \underline{\mathrm{D}}(\mathrm{Sing}(\mathscr{Y})_{\mathscr{Z}})^\Rightarrow \to \mathscr{D}\end{equation}

in $\operatorname{ShCat}_{\mathbf{S}^\Rightarrow}(\mathscr{Y})$ .

5.4.6.

To finish Construction 5.4, it remains to verify that (5.39) is an isomorphism. Indeed, by the description of (5.29) given in § 5.4.3, the value of (5.39) at an object $S \in \operatorname{Sm}_{/\mathscr{Y}}$ is the canonical arrow

\begin{align*}\mathrm{D}(\mathscr{Z}_S) \underset{\mathrm{D}(S)}{\otimes} \mathrm{D}(\mathrm{Sing}(S))^\Rightarrow \overset{\sim}{\rightarrow} \mathrm{D}(\mathrm{Sing}(S)_{\mathscr{Z}_S})^\Rightarrow. \end{align*}

Proof of Proposition 5.2. Consider the following diagram, where the upper back-to-front maps are the composites.

\begin{align*}\mathbf{H}(\mathscr{Z}_S) \textbf{-mod} \xrightarrow{\sim} D(\mathrm{Sing}(\mathscr{Z}_S))^\Rightarrow \textbf{-mod} \xrightarrow{\mathbf{ind}} D(\mathrm{Sing}(S)_{\mathscr{Z}_S})^\Rightarrow \textbf{-mod}\end{align*}

The two back faces are commutative by the functoriality of $\mathbf{H}$ , the front face is commutative by the Beck–Chevalley transformation (5.31), the two lower faces are commutative by the construction of the Morita equivalence (5.6), and the two vertical faces are commutative by Construction 4.1.

At the moment, we have not filled in the upper face of the cube. Observe that the isomorphism (5.35) is encoded by the commutativity of the preceding diagram, when evaluated on the object

\begin{align*} \mathrm{IndCoh}(\mathscr{Z}) \in \mathbf{H}(\mathscr{Z}) \textbf{-mod}, \end{align*}

and that a completion of the right half of the diagram into a commutative cube would lift (5.35) to an isomorphism in $\mathrm{D}(\mathrm{Sing}(T)_{\mathscr{Z}_T})^\Rightarrow$ . Let us construct this completion.

All five faces of the diagram obtained by taking right adjoints of the horizontal arrows remain commutative, so it is equivalent to complete this new diagram into a commutative cube.

To do so, apply Construction 4.2 to the composites

\begin{align*} \mathscr{Z}_T \to \mathscr{Z}_S \to S{,} \quad \mathscr{Z}_T \to T \to S \end{align*}

and paste together the two resulting prisms.

5.4.7 Porism

Construction 5.4 comes with an additional compatibility: the Morita equivalence (5.6) exchanges the morphism

\begin{align*} \underline{\mathrm{D}}(\mathrm{Sing}(\mathscr{Y}))^\Rightarrow \to \underline{\mathrm{D}}(\mathrm{Sing}(\mathscr{Y})_{\mathscr{Z}})^\Rightarrow \end{align*}

induced by D-module $!$ -pullback with the morphism