2.1

In this section we introduce a generalization of the holonomic condition on a $D$-module and show that it is preserved under $D$-module operations.

The method is standard. The main point is Lemma 2.7.1, which is a generalization of the fact that pushforward along an open embedding preserves holonomic objects, which is essentially equivalent to the usual $b$-function lemma. The main difference is that we cannot use finite-length methods. See also Remark 2.7.2 for indications on a different approach.

The presentation is based on [Reference KashiwaraKas76, Reference GinzburgGin86].

2.2 Gabber–Kashiwara–Sato (GKS) filtration

We begin by reviewing some material from [Reference GinzburgGin86, § 1].

Definition 2.2.1 Let $X$ be a variety and let ${\cal F} \in D(X)^{\heartsuit }$ be a coherent $D$-module. For an integer $i$, we let

(2.2.1)\begin{equation} F_i^{{\rm GKS}} {\cal F} := \operatorname{Image}(H^0({\mathbb{D}} \tau^{{\geq}{-}i} {\mathbb{D}} {\cal F}) \to {\cal F}). \end{equation}

We extend this definition to general ${\cal F}$ by ind-extension. That is, if we write ${\cal F}$ as a filtered colimit $\operatorname {colim}_j {\cal F}_j$ with each ${\cal F}_j$ coherent (as can always be done), we take $F_i^{{\rm GKS}} {\cal F} := \operatorname {colim}_j F_i^{{\rm GKS}} {\cal F}_j$.

Note that $F_{\bullet }^{{\rm GKS}}$ is an increasing filtration on ${\cal F}$. Because ${\mathbb {D}} {\cal F}$ is in cohomological degrees $[-\dim X,0]$, we have $F_i^{{\rm GKS}} {\cal F} = 0$ for $i<0$, and $F_i^{{\rm GKS}} {\cal F} = {\cal F}$ for $i \geq \dim X$. Formation of the GKS filtration is functorial for $D$-module morphisms, that is, a map ${\cal F}_1 \to {\cal F}_2 \in D(X)^{\heartsuit }$ sends $F_i^{{\rm GKS}} {\cal F}_1$ to $F_i^{{\rm GKS}} {\cal F}_2$.

Note that if ${\cal F} = \operatorname {colim}_j {\cal F}_j$ is a filtered colimit in $D(X)^{\heartsuit }$, then $F_i^{{\rm GKS}} {\cal F} = \operatorname {colim}_j F_i^{{\rm GKS}} {\cal F}_j$.

Therefore, many results about this filtration reduce to the case of smooth $X$ by taking Zariski local closed embeddings into affine space. The key property in the smooth case is given in the following theorem.

See [Reference GinzburgGin86, Proposition V.14]. Note that it is equivalent to say that $F_i^{{\rm GKS}} {\cal F}$ is the maximal submodule of ${\cal F}$ with singular support of dimension $\leq \dim X+i$.

2.3 Holonomic defect

For $\delta \in {\mathbb {Z}}^{\geq 0}$, we say that ${\cal F} \in D(X)^{\heartsuit }$ has holonomic defect $\leq \delta$ if $F_{\delta }^{{\rm GKS}} {\cal F} = {\cal F}$.

2.4

More generally, for ${\cal F} \in D(X)$ a complex of $D$-modules, we say that ${\cal F}$ has holonomic defect $\leq \delta$ if all of its cohomology groups do. By Lemmas 2.3.4 and 2.3.5, this defines a DG subcategory of $D(X)$ closed under (homotopy) colimits (equivalently, under cones, shifts, and arbitrary direct sums).

2.5

The following theorem is the main result of this section.

This theorem generalizes the preservation of holonomic objects under $D$-module operations, so the proof must follow similar lines. It is given below.

2.6 Verdier duality

The compatibility with Verdier duality in Theorem 2.5.1 is well known. Indeed, the result immediately reduces to $X$ being smooth, and then we have the following proposition.

See, for example, [Reference KashiwaraKas76, Theorem 2.3].

2.7 Affine open embeddings

The main case of Theorem 2.5.1 is pushforward along an open embedding.

Proof of Lemma 2.7.1 Step 1.  We may obviously assume $X$ is connected and affine and that $f$ is non-constant.

We let $D_X$ and $D_U$ denote the respective rings of global differential operators (as opposed to the sheaves of differential operators). Because $X$ and $U$ are affine, we may consider $D$-modules on either of these schemes as modules over these rings.

Let ${\cal F} \in D_U - \mathsf {mod}^{\heartsuit } \simeq D(U)^{\heartsuit }$. Because we are working with modules rather than sheaves, considering ${\cal F}$ as a $D_X$-module by restriction is the same as considering the sheaf $j_{*,{\rm dR}}({\cal F}) \in D(X)^{\heartsuit }$.

For $s \in {\cal F}$, we write $\operatorname {SS}_U(s) \subseteq T^* U$ for the singular support of $D_U \cdot s$ and $\operatorname {SS}_X(s) \subseteq T^* X$ for the singular support of $D_X \cdot s$. Note that $\operatorname {SS}_X(s)|_{T^* U} = \operatorname {SS}_U(s)$. We always understand singular support as a reduced subscheme.

We want to show that if every section $s \in {\cal F}$ has $\dim \operatorname {SS}_U(s) \leq \dim U + \delta = \dim X + \delta$, then the same is true of $\dim SS_X(s)$.

Step 2.  First, we observe that there is a $D_X$-submodule ${\cal G} \subseteq {\cal F}$ such that every section of ${\cal G}$ has singular support with dimension $\leq \dim X + \delta$, and which is a lattice, that is, ${\cal G} \otimes _{{\cal O}_X} {\cal O}_U \xrightarrow {\simeq } {\cal F}$.

Indeed, we can take ${\cal G} = F_{\delta }^{{\rm GKS}} {\cal F}$, where the GKS filtration is with ${\cal F}$ considered as a $D_X$-module. Because the GKS filtration commutes with open restriction, we must have ${\cal G}|_U = {\cal F}$.

(Note that by Theorem 2.2.4, we are trying to show that ${\cal G} = {\cal F}$.)

Step 3.  We begin with some preliminary constructions.

Let $\lambda$ be an indeterminate. We write ${\mathbb {A}}_{\lambda }^1$ for $\operatorname {Spec}(k[\lambda ])$. We let $k^{\prime }$ denote the fraction field $k(\lambda )$ of $k[\lambda ]$. We use similar notation for a base-change to $k^{\prime }$; for example, $X^{\prime }$ or ${\cal F}^{\prime }$, etc. We always consider $X^{\prime }$ and $U^{\prime }$ as schemes over $k^{\prime }$, so, for example, their cotangent bundles are understood relative to $k^{\prime }$, and $D_X^{\prime } = D_{X^{\prime }}$.

We now recall that there is certain standard object ${\cal O}_{U^{\prime }}f^{\lambda } \in D(U^{\prime })^{\heartsuit }$ which is $({\cal O}_{U^{\prime }},\nabla = d+\lambda (df\!{/}\!f))$. We denote the basis vector of $\Gamma (U^{\prime },{\cal O}_{U^{\prime }}f^{\lambda }) \in D_U^{\prime } - \mathsf {mod}^{\heartsuit }$ also by $f^{\lambda }$.

We similarly let ${\cal F}^{\prime }f^{\lambda }$ denote the tensor product of ${\cal F}^{\prime }$ with ${\cal O}_{U^{\prime }}f^{\lambda }$ over ${\cal O}_{U^{\prime }}$, equipped with its standard $D_{U^{\prime }}$-module structure. Explicitly, this $D_U^{\prime }$-module has the same underlying $\Gamma (U^{\prime },{\cal O}_{U^{\prime }})$-module as ${\cal F}^{\prime }$, except that we write this identification as

\[ (s \in {\cal F}^{\prime})) \mapsto (s f^{\lambda} \in {\cal F}^{\prime} f^{\lambda}). \]

Then the action of vector fields is modified: for $\xi$ a vector field and $s\in {\cal F}^{\prime }$, we have

\[ \xi \cdot (s f^{\lambda}) = (\xi s + \lambda\xi(f)f^{{-}1}s) f^{\lambda}. \]

Step 4.  We first show that the result is true for ${\cal F}^{\prime } f^{\lambda }$, that is, that every section has $SS_{X^{\prime }}$ with dimension $\leq \dim X + \delta$.

First, note that the singular support in $U^{\prime }$ of any section has dimension $\leq \dim X + \delta$: this follows because ${\cal O}_{U^{\prime }}f^{\lambda }$ is lisse on $U^{\prime }$ (that is, a vector bundle when considered as a mere quasi-coherent sheaf).

We have a canonical field automorphism $\gamma \in \operatorname {Aut}(k^{\prime }/k)$ sending $\lambda \mapsto \lambda +1$. Of course, anything obtained by extension of scalars from $k$ to $k^{\prime }$ also carries such an automorphism $\gamma$; in particular, $D_X^{\prime }$ does (it sends a differential operator $P(\lambda )$ to $P(\lambda +1)$).

Similarly, ${\cal F}^{\prime }$ has such an automorphism: note that this is not an automorphism as a $D_X^{\prime }$-module, but rather intertwines the standard action with the one obtained by twisting by the automorphism $\gamma$ of $D_X^{\prime }$. That is, $\gamma (P \cdot s) = \gamma (P) \cdot \gamma (s)$ for $P \in D_X^{\prime }$ and $s \in {\cal F}^{\prime }$.

Define $\gamma$ on the $D_X^{\prime }$-module ${\cal O}_{U^{\prime }}f^{\lambda }$ by setting

\[ \gamma(g f^{\lambda}) =\gamma(g) f^{\lambda+1} := (\gamma(g)\cdot f) f^{\lambda} \]

for $g$ a function on $U^{\prime }$. Again, this morphism intertwines the actions of $D_X^{\prime }$ up to the automorphism $\gamma$ of $D_X^{\prime }$.

Tensoring, we obtain an automorphism $\gamma$ of ${\cal F}^{\prime } f^{\lambda }$ with similar semi-linearity.

By the semi-linearity, we have

\[ \operatorname{SS}_{X^{\prime}}(s f^{\lambda}) =\gamma \cdot \operatorname{SS}_{X^{\prime}}(\gamma(s f^{\lambda})), \]

where we are using $\gamma$ to indicate the induced automorphism of $T^* X^{\prime }$. In particular, we find that

\[ \dim \operatorname{SS}_{X^{\prime}} (s f^{\lambda}) = \dim \operatorname{SS}_{X^{\prime}} (\gamma(s f^{\lambda})). \]

Now let ${\cal G} = F_{\delta }^{{\rm GKS}} ({\cal F}^{\prime } f^{\lambda })$,Footnote ⁹ where the GKS filtration is taken with ${\cal F}^{\prime } f^{\lambda }$ considered as a $D_X^{\prime }$-module. By the above and Theorem 2.2.4, $s f^{\lambda } \in {\cal G}$ if and only if $\gamma (s f^{\lambda }) \in {\cal G}$.

For any $s \in {\cal F}$ (as opposed to ${\cal F}^{\prime }$), we clearly have $\gamma (s f^{\lambda }) = s f^{\lambda +1}$. Since ${\cal G}$ is a lattice (by Step 2), $\gamma ^N(s) = s f^{\lambda +N} \in {\cal G}$ for $N \gg 0$. But by the above, this means that $s \in {\cal G}$. Since ${\cal F}^{\prime } f^{\lambda }$ is $k^{\prime }$-spanned by such vectors, this means that ${\cal G} = {\cal F}^{\prime } f^{\lambda }$, as desired.

Step 5.  We now show that the result is true for our original ${\cal F}$. Let $s \in {\cal F}$; we want to show $\dim \operatorname {SS}_X(s) \leq \dim X+\delta$.

We now write ${\cal F} f^{\lambda }$ for the corresponding $D_X[\lambda ]$-module (as opposed to the fiber over the generic point in ${\mathbb {A}}_{\lambda }^1$, which is what we called by this name previously). Note that ${\cal F} f^{\lambda } = {\cal F} \otimes _k k[\lambda ]$ as a ${\cal O}_X[\lambda ]$-module.

Let ${\cal F}_0$ be the $D_X[\lambda ]$ submodule generated by $s f^{\lambda }$. Equip ${\cal F}_0$ with the filtration $F_i {\cal F}_0 = D_X^{\leq i}[\lambda ]s f^{\lambda }$, where $D_X^{\leq i}$ are differential operators of order $\leq i$.

Then $\operatorname {gr}_{\bullet }({\cal F}_0)$ is the structure sheaf of some closed subscheme $Z \subseteq T^*X \times {\mathbb {A}}_{\lambda }^1$. We have seen that the base-change of $Z$ to the generic point of ${\mathbb {A}}_{\lambda }^1$ has dimension $\leq \dim X + \delta$, so the same is true for its fibers at closed points with only finitely many possible exceptions.

Choose a negative integer $-N$ not among this finite number of exceptions. Then the coherent $D_X$-module ${\cal F}_0/(\lambda +N)$ has singular support contained in $Z \times _{{\mathbb {A}}_{\lambda }^1} \{-N\}$, so has dimension $\leq \dim X + \delta$. We have the obvious morphism of $D_U$-modules (in particular, of $D_X$-modules):

\[ \begin{gathered} ({\cal F} f^{\lambda})/(\lambda+N) \to {\cal F} \\ \sum_{i = 0}^r \sigma_i \lambda^i f^{\lambda} \mapsto \sum_{i = 0}^r f^{{-}N} \cdot \sigma_i \cdot ({-}N)^i,\quad \sigma_i \in {\cal F} , \end{gathered} \]

inducing a map ${\cal F}_0/(\lambda +N)$ to ${\cal F}$ sending the generator to $f^{-N} s$. By functoriality of the GKS filtration (or standard singular support analysis), this means that $f^{-N} s \in F_{\delta }^{{\rm GKS}} {\cal F}$, and since $F_{\delta }^{{\rm GKS}} {\cal F}$ is a $D_X$-module, this means that $s \in F_{\delta }^{{\rm GKS}} {\cal F}$ as well.

2.8 Preservation of holonomic defect

We now proceed to prove Theorem 2.5.1. The argument is straightforward at this point, and we proceed in cases.

2.9

First, we treat pushforwards along an open embedding $j:U \to X$.

Suppose $X$ is smooth and affine. We proceed by induction on the number $n$ of basic affine opens required to cover $U$. For $n = 1$, the result is treated by Lemma 2.7.1.

The inductive step is a standard Čech argument.

For $n>1$, $U$ admits a cover by some $U_1$ and $U_2$ where $U_1$ is a basic affine open and $U_2$ admits a cover by $n-1$ basic affine opens. Let $j_i:U_i \to U$ denote the embeddings and let $j_{12}:U_1 \cap U_2 \to U$.

Fix ${\cal F} \in D(U)$ with holonomic defect $\leq \delta$. Note that

\[ {\cal F} = j_{1,*,{\rm dR}}j_1^!({\cal F}) \underset{ j_{12,*,{\rm dR}}j_{12}^!({\cal F})}{\times} j_{2,*,{\rm dR}}j_2^!({\cal F}) \]

(where this fiber product is a homotopyFootnote ¹⁰ fiber product). By induction, $j_{?,*,{\rm dR}}j_?^!({\cal F})$ has holonomic defect $\leq \delta$. Now recall from § 2.4 that $D$-modules with holonomic defect $\leq \delta$ are a DG subcategory and therefore closed under finite limits.

Finally, for general $X$, note that the problem is Zariski local, so we may assume $X$ is affine. Take a closed embedding $X \subseteq {\mathbb {A}}^N$. If $U = X \setminus Z$, then we have $U \hookrightarrow {\mathbb {A}}^N\setminus Z \hookrightarrow {\mathbb {A}}^N$, with the first map being closed and the second being open between smooth affine schemes. Therefore, this pushforward preserves having holonomic defect $\leq \delta$. Clearly this implies the result for the pushforward along $U \hookrightarrow X$.

2.10

Next, we treat restrictions to closed subschemes.

Let $i:Z \hookrightarrow X$ be closed and let $j:U = X\setminus Z \hookrightarrow X$. Then we have an exact triangle:

\[ i_{*,{\rm dR}} i^!({\cal F}) \to {\cal F} \to j_{*,{\rm dR}} j^!({\cal F}) \xrightarrow{+1}. \]

If ${\cal F}$ has holonomic defect $\leq \delta$, we have shown the same for $j_{*,{\rm dR}}j^!({\cal F})$, so $i_{*,{\rm dR}}i^!({\cal F})$ has holonomic defect $\leq \delta$, which is equivalent to $i^!({\cal F})$ having holonomic defect $\leq \delta$.

2.11

We can now show the result for restrictions in general.

If $f:X \to Y$ is smooth of relative dimension $d$, then $f^{*,{\rm dR}}[d] = f^![-d]$ commutes with Verdier duality and is $t$-exact. Therefore, it commutes with formation of the GKS filtration, and therefore preserves the property of having holonomic defect $\leq \delta$.

The case of general $f:X \to Y$ is immediately reduced to the case of affine varieties (since holonomic defect is Zariski local). We can then find a commutative diagram:

(2.11.1)

with the horizontal arrows being closed embeddings. This reduces to the case where $X$ and $Y$ are smooth.

Then we can factor $f$ through the graph as $X \to X \times Y \xrightarrow {p_1} Y$. The former map is a closed embedding, and the latter is smooth because $X$ is. We have treated each of these cases, so we obtain the result.

2.12

Next, we treat pushforwards along a proper morphism $f:X \to Y$ between smooth varieties.

This case does not need the work we have done so far. Let ${\cal F} \in D(X)^{\heartsuit }$ with holonomic defect $\leq \delta$ be given. By Lemma 2.3.5, we may assume ${\cal F}$ is coherent, so the hypothesis is that ${\cal F}$ has singular support $\operatorname {SS}_X({\cal F})$ with dimension $\dim X + \delta$.

Recall that $\operatorname {SS}_Y(H^i(f_{*,{\rm dR}}({\cal F})))$ is bounded in terms of $\operatorname {SS}_X({\cal F})$. More precisely, if we take the diagram

then the singular support of these cohomologies is contained in $\alpha (\beta ^{-1}\operatorname {SS}_X({\cal F}))$ (see, for example, [Reference KashiwaraKas76, Theorem 4.2]).

Because $\operatorname {SS}_X({\cal F})$ is coisotropic by [Reference GabberGab81], we have

\[ \dim \alpha(\beta^{{-}1}\operatorname{SS}_X({\cal F})) \leq \dim(\operatorname{SS}_X({\cal F})) + \dim Y - \dim X \]

by usual symplectic geometry. This immediately gives the claim.

2.13

Now observe that preservation of holonomic defect under pushforward along a general morphism $f:X \to Y$ between smooth varieties follows: by Nagata's compactification theorem (see, for example, [Reference ConradCon07]), there exist a scheme $\overline {X}{}$ and a factorization

\[ X \xrightarrow{j} \overline{X}{} \xrightarrow{\overline{f}{}} Y \]

of $f$ with $\overline {f}{}$ proper and $j$ an open embedding. Moreover, by resolution of singularities [Reference HironakaHir64, Main Theorem I], we may take $\overline {X}{}$ to be smooth.Footnote ¹¹ Then the result follows in this case from our earlier work.

2.14

We can now treat a general pushforward along $f:X \to Y$ a morphism between possibly singular varieties.

The problem is Zariski local on $Y$, so we may assume $Y$ is affine.

Next, we reduce to the case where $X$ is affine. It suffices to observe that the subcategory of $D(X)$ of $D$-modules of holonomic defect $\leq \delta$ is generated under colimitsFootnote ¹² by objects of the form $j_{*,{\rm dR}}({\cal F})$ for $j:U \to X$ an affine open and ${\cal F}$ of holonomic defect $\leq \delta$. This observation follows by induction on the number of affines required to cover $X$ by a Čech argument exactly as in § 2.9, and the fact (shown above) that pushforward along an open embedding preserves holonomic defect.

When $X$ and $Y$ are affine, we can find a commutative diagram (2.11.1) as before. This reduces to the case with $X$ and $Y$ smooth, which we have already treated.

3.1

The main result of this section says that $f_!$ is left $t$-exact for an affine morphism $f$. We also show that for $i:X \to Y$ a closed embedding, $i^{*,{\rm dR}}$ has cohomological amplitude $\geq -\dim (Y)+\dim (X)$, that is, $i^{*,{\rm dR}}[-\dim (Y)+\dim (X)]$ is left $t$-exact. Since $f_!$ and $i^{*,{\rm dR}}$ are not necessarily defined on non-holonomic $D$-modules, we use the language of pro-categories to formulate this result.

3.2

As was remarked in the introduction, we use some language from higher category theory in order to discuss pro-categories.

We use the theory of $\infty$-categories developed in [Reference LurieLur09]. In particular, we refer to [Reference LurieLur09, § 5.3] for a discussion of ind-categories in this formalism and we refer to [Reference LurieLur12, § 1.4.4] for the theory in the stable setting. There is a dual notion of pro-category: see [Reference LurieLur09, Remark 7.1.6.2].

We apply this theory in the setting of DG categories. We follow [Reference Gaitsgory and RozenblyumGR17, § I.1.10.3] in our terminology here, and in particular in viewing DG categories as $\infty$-categories with additional structure.

In what follows, we omit the prefix $\infty$ to avoid clutter, so, for example, ‘category’ means ‘$\infty$-category’.

3.3 Pro-categories

We refer to [Reference LurieLur11, § 3.1] and [Reference Gaitsgory and RozenblyumGR17, § III.1.3.1] for the material that follows.

For ${\cal C}$ an accessibleFootnote ¹³ category, we have the corresponding pro-category $\mathsf {Pro}({\cal C})$. If ${\cal C}$ is a DG category, $\mathsf {Pro}({\cal C})$ is as well. If ${\cal C}$ admits small colimits, then so does $\mathsf {Pro}({\cal C})$. For $F:{\cal C} \to {\cal D}$, there is an induced functor $\mathsf {Pro}({\cal C}) \to \mathsf {Pro}({\cal D})$, which we denote again by $F$ where there is no risk for confusion.

For any functor $G:{\cal D} \to {\cal C}$ commuting with finite colimits (for example, a DG functor), the induced functor $\mathsf {Pro}({\cal D}) \to \mathsf {Pro}({\cal C})$ admits a left adjoint $F$. We say that $F$ is defined on an object ${\cal F} \in {\cal C}$ if $F({\cal F}) \in {\cal D} \subseteq \mathsf {Pro}({\cal D})$. (This coincides with the usual notion of a left adjoint being defined on some object.)

If ${\cal C}$ is a DG category equipped with a $t$-structure, then $\mathsf {Pro}({\cal C})$ inherits one as well. It is characterized by the equality $\mathsf {Pro}({\cal C})^{\leq 0} = \mathsf {Pro}({\cal C}^{\leq 0})$. Truncation functors are the pro-extensions of the truncation functors on ${\cal C}$. In particular, we find that ${\cal C}$ is closed under truncations and inherits its given $t$-structure. We also find that $\mathsf {Pro}({\cal C})^{\geq 0} = \mathsf {Pro}({\cal C}^{\geq 0})$: if ${\cal F} = \operatorname {lim}_i {\cal F}_i \in \mathsf {Pro}({\cal C})^{\geq 0}$, then ${\cal F} = \tau ^{\geq 0} {\cal F} = \operatorname {lim}_i \tau ^{\geq 0} {\cal F}_i$.

3.4 Affine morphisms

For $f:X \to Y$, we have the functor $f_!:\mathsf {Pro}(D(X)) \to \mathsf {Pro}(D(Y))$ left adjoint to $f^!$.

Remark 3.4.2 Before giving the argument, we spell out the meaning of this result without the language of pro-categories. Suppose ${\cal F} \in D(X)^{c,\geq 0}$ is coherent and in cohomological degrees $\geq 0$. Let ${\cal G} \in D(Y)$ be arbitrary, and suppose we are given a morphism $\alpha :{\cal F} \to f^!({\cal G})$. Then the theorem is equivalent to asserting that when $f$ is affine, there necessarily exist ${\cal G}_0 \in D(Y)^{\geq 0}$, a morphism $\beta :{\cal G}_0\to {\cal G} \in D(Y)$, and a factorization

\[ {\cal F} \to f^!({\cal G}_0) \xrightarrow{f^!(\beta)} f^!({\cal G}) \]

of $\alpha$. In particular, the assertion of Theorem 3.4.1 can be reformulated using only triangulated categories.

Proof of Theorem 3.4.1 The problem isFootnote ¹⁴ Zariski local on $Y$, so we may assume $X$ and $Y$ are affine.

Note that $D$-module pushforward along closed embeddings remains fully faithful on pro-categories: the identity $i^!i_{*,{\rm dR}} = \operatorname {id}$ induces the same for the pro-functors. Therefore, the same argument as in § 2.11 allows us to assume $X$ and $Y$ are smooth.

Recall that we have a Verdier duality equivalence ${\mathbb {D}}:D(X)^{{\rm op}} \xrightarrow {\simeq } \mathsf {Pro}(D(X)^c)$ induced by the usual Verdier duality equivalence ${\mathbb {D}}:D(X)^c \xrightarrow {\simeq } D(X)^{c,{\rm op}}$, and similarly for $Y$.

We then claim that

(3.4.1)\begin{equation} f_!({\cal F}) = {\mathbb{D}} f_{*,{\rm dR}} {\mathbb{D}} ({\cal F}). \end{equation}

This follows formally from the fact that $f_{*,{\rm dR}}$ and $f^!$ are dual functors in the sense of [Reference GaitsgoryGai12], but we give a direct proof below. Note that in this formula, $f_{*,{\rm dR}} {\mathbb {D}}({\cal F}) \in D(Y)$, and we are then using ${\mathbb {D}}$ to convert it to a pro-coherent object. Since this object is pro-coherent, it suffices to observe that for ${\cal G} \in D(Y)^c$, we have

\[ \begin{aligned} \underline{\operatorname{Hom}}_{\mathsf{Pro}(D(Y)^c)}({\mathbb{D}} f_{*,{\rm dR}} {\mathbb{D}} ({\cal F}), {\cal G}) &= \underline{\operatorname{Hom}}_{D(Y)}({\mathbb{D}} {\cal G}, f_{*,{\rm dR}} {\mathbb{D}} ({\cal F})) = \Gamma_{{\rm dR}}(Y, f_{*,{\rm dR}} {\mathbb{D}}({\cal F}) \overset{!}{\otimes} {\cal G}) \\ &=\Gamma_{{\rm dR}}(X, {\mathbb{D}}({\cal F}) \overset{!}{\otimes} f^!({\cal G})) = \underline{\operatorname{Hom}}_{D(X)}({\cal F},f^!({\cal G})). \end{aligned} \]

Here $\Gamma _{{\rm dR}}$ is the complex of de Rham cochains of a $D$-module, and we are repeatedly using the formulaFootnote ¹⁵ that if ${\cal F}_1$ is coherent, then

\[ \underline{\operatorname{Hom}}_{D(X)}({\cal F}_1,{\cal F}_2) = \Gamma_{{\rm dR}}(X, {\mathbb{D}}({\cal F}_1) \overset{!}{\otimes} {\cal F}_2). \]

Now suppose that ${\cal F} \in D(X)^{c,\heartsuit }$ is coherent and in cohomological degrees $\geq 0$. Note that ${\mathbb {D}} {\cal F}$ carries the canonical filtration with subquotients $(H^{-j}{\mathbb {D}}{\cal F})[j]$. By Proposition 2.6.1, $H^{-j} {\mathbb {D}} {\cal F}$ has holonomic defect $\leq j$. By Theorem 2.5.1, $f_{*,{\rm dR}} H^{-j} {\mathbb {D}} {\cal F}$ has holonomic defect $\leq j$ as well. Moreover, by affineness of $f$, the latter complex is in cohomological degrees $\leq 0$.

Note that by Proposition 2.6.1, if ${\cal G} \in D(Y)^{\heartsuit }$ has holonomic defect $\leq \delta$, then ${\mathbb {D}} {\cal G} \in \mathsf {Pro}(D(Y)^c)$ is in cohomological degrees $[-\delta ,0]$: indeed, this immediately reduces to the coherent case.

Therefore, ${\mathbb {D}} H^{-k} f_{*,{\rm dR}} H^{-j} {\mathbb {D}} {\cal F}$ is in cohomological degrees $[-j,0]$ for every $k$, which means ${\mathbb {D}} (H^{-k} (f_{*,{\rm dR}} H^{-j} {\mathbb {D}} {\cal F})[k])$ is in cohomological degrees $[-j+k,k]$. This complex vanishes unless $k\geq 0$, so ${\mathbb {D}} f_{*,{\rm dR}} H^{-j} {\mathbb {D}} {\cal F}$ is in cohomological degrees $\geq -j$. Finally, this means that ${\mathbb {D}} ((f_{*,{\rm dR}} H^{-j} {\mathbb {D}} {\cal F})[j])$ is in cohomological degrees $\geq 0$, so the same follows for ${\mathbb {D}} f_{*,{\rm dR}} {\mathbb {D}}({\cal F}) = f_!({\cal F})$.

3.5 Closed embeddings

Similarly, we have the following theorem.