Constructible Derived Category
Dongkwan Kim
September 29, 2015
1
Category of Sheaves
In this talk we mainly deal with sheaves of C-vector spaces. For a topological space X, we denote by Sh(X) the abelian category of sheaves of C-vector
spaces on X. We assume that readers are familiar with basic sheaf theory. (i.e.
sheafification, Sh(X) is abelian, etc.)
1.1
Operations on Sheaves
Here we recapitulate some operations (i.e. functors) on sheaves. First we define
internal Hom and tensor product as follows.
(1) Internal Hom. For F, G ∈ Sh(X), we define Hom(F, G) = HomSh(X) (F, G) ∈
Sh(X) as the sheaf defined by
U 7→ Hom(F|U , G|U ) = HomSh(X) (F|U , G|U ).
(2) Tensor product. For F, G ∈ Sh(X), we define F ⊗ G = F ⊗C G ∈ Sh(X)
as the sheaf associated to the following presheaf
U 7→ F(U ) ⊗ G(U ).
Note that the natural map (Hom(F, G))x → Hom(Fx , Gx ) is in general neither
injective nor surjective. However, we have (F ⊗ G)x ' Fx ⊗ Gx . Now for a
continuous map f : X → Y , we define operations of sheaves which corresponds
to f as follows.
(3) Push-forward or direct image. For F ∈ Sh(X), we define f∗ F ∈ Sh(Y )
such that for open V ⊂ Y we have
f∗ F : V 7→ F(f −1 (V )).
In particular, if Y is a point, then we denote Γ(X, G) := f∗ G as we identify
Sh(Y ) with the category of C-vector spaces and call it the set of global
sections of F on X.
1
(4) Pull-back or inverse image. For G ∈ Sh(Y ), we define f ∗ G ∈ Sh(X) to
be a sheaf associated to the presheaf
U 7→
lim G(V ).
−→
f (U )⊂V
Note that the natural map (f∗ F)f (x) → Fx is not an isomorphism, however
(f ∗ G)x ' Gf (x) .
For a subspace f : Z ,→ X, using the functors above we have a way to
”restrict” a sheaf F ∈ Sh(X) to Z still considered as an element in Sh(X) by
taking f∗ f ∗ F. We will define two other analogous restriction functors when
the subspace is locally closed, which would coincide with f∗ f ∗ in some special
situations.
(5) For a closed subset f : Z ,→ X and for F ∈ Sh(X), we define FZ := f∗ f ∗ F.
For an open subset U , we set FU := ker(F → FX\U ). Finally, if A is
locally closed, say A = U ∩ Z for U open and Z closed, then we define
FA := (FU )Z .1
We briefly mention one of its properties, which would become obvious as we
discuss further.
Proposition 1. Suppose f : A ,→ X is locally closed and g : B = X \ A ,→ X.
Then for any F ∈ Sh(X),
f ∗ (FA ) = f ∗ F,
g ∗ (FA ) = 0.
(6) The sheaf of sections supported by a subspace. For a closed subset
f : Z ,→ X and F ∈ Sh(Z), we let ΓZ (F) be the sheaf defined by
V 7→ ker(F(V ) → F(V \ (V ∩ Z))).
For A = U ∩ Z locally closed with U open and Z closed, define ΓA (F) by
ΓA (F)(V ) := ΓA (F|U )(U ∩ V ).
It is straightforward to check that for an open embedding f : U ,→ X,
ΓU (F) = f∗ f ∗ F.
So far we defined Hom, ⊗, f∗ , f ∗ , •A , ΓA . If topological spaces are locally
compact2 , there is one more functor which is of our interest.
(7) Proper push-forward or direct image with proper support. Assume
that X, Y are locally compact topological spaces and given a continuous map
f : X → Y and a sheaf F ∈ Sh(X). We denote by f! F the sheaf defined by
U 7→ {s ∈ F(f −1 U ) | f |supp(s) : supp(s) → U is proper3 }.
1 This
is well-defined, that is this definition does not depend on the choice of U or Z.
this talk locally compact spaces are always Hausdorff.
3 Here f being proper means that every inverse image of compact set is again compact.
2 In
2
In particular, if Y is a point then we denote Γc (X, F) := f! (F) as we identify
Sh(Y ) with the cateogory of C-vector spaces and call it the set of compactly
(or properly) supported global sections of F on X.
Note that we have a natural embedding f! F ,→ f∗ F, which is clearly an isomorphism if f is a priori proper. Also we have the following property which would
be apparent by the argument later on. It is not in general true for the usual
push-forward.
Proposition 2. [[KS, Proposition 2.5.2]] If f : X → Y is a continuous map
between locally compact spaces, then for y ∈ Y and F ∈ Sh(X) the canonical
morphism
(f! F)y → Γc (f −1 (y), F|f −1 (y) )
is an isomorphism.
4
If f : X → Y is proper then f∗ f ∗ = f! f ∗ , which is true especially when f
is a closed embedding. One might already expect that we have the following
property.
Proposition 3. If X is locally compact and f : Z ,→ X is a closed embedding,
thus in particular Z is also locally compact, then for F ∈ Sh(X) we have
FZ = f! f ∗ F.
1.2
Exactness, Adjunction Formula, and Base Change
We state, but omit proofs of, the exactness properties of the operations defined
above.
Proposition 4. Let f : X → Y be a continuous map between topological spaces.
Then,
(1) Hom : Sh(X)op × Sh(X) → Sh(X) is left exact.
(2) ⊗ : Sh(X) × Sh(X) → Sh(X) is exact.5
(3) f∗ : Sh(X) → Sh(Y ) is left exact. Thus in particular, Γ is left exact.
(3’) f∗ : Sh(X) → Sh(Y ) is exact if f is a closed embedding.
(4) f ∗ : Sh(Y ) → Sh(X) is exact.
(5) F 7→ FX : Sh(Y ) → Sh(Y ) is exact if f is a locally closed embedding.
(6) ΓX : Sh(Y ) → Sh(Y ) is left exact if f is a locally closed embedding.
4 This also has an étale analogue which is difficult to prove. See the comments on [M, p.117]
after Theorem 17.7 and 17.10 for more infomation.
5 In general if we consider tensor product over a sheaf of rings then it is right exact. Here
we only consider sheaves of C-vector spaces, thus every sheaf is flat.
3
(7) f! : Sh(X) → Sh(Y ) is left exact if X and Y are locally compact.
(7’) f! : Sh(X) → Sh(Y ) is exact if Y is locally compact and f is a locally closed
embedding.
Also we have the following adjunction formulae.
Proposition 5. Let f : X → Y be a continuous map between topological spaces
and let F, F 0 , F 00 ∈ Sh(X), G, G 0 ∈ Sh(Y ). Then
(a) Hom(F ⊗ F 0 , F 00 ) ' Hom(F, Hom(F 0 , F 00 ))
(b) Hom(G, f∗ F) ' f∗ Hom(f ∗ G, F)
(c) Hom(GX , G 0 ) ' Hom(G, ΓX (G 0 )) if f is a locally closed embedding.
If we take the set of global sections on each formula, we get the ”usual”
adjunctions.
Meanwhile, suppose we are given the following diagram of locally compact
spaces.
A
f˜
/B
g
g̃
f
/D
C
Then for any F ∈ Sh(C) we have a well-defined morphism
f! F → f! g̃∗ g̃ ∗ F
by adjunction. Now for U ⊂ D, we have
f! g̃∗ F(U ) = {s ∈ F(g̃ −1 f −1 U ) | f : supp(g̃∗ s) → U is proper}
g∗ f˜! F(U ) = {s ∈ F(f˜−1 g −1 U ) | f˜ : supp(s) → g −1 U is proper}
In other words, if s ∈ f! g̃∗ F(U ) then s ∈ F(g̃ −1 f −1 U ) = F(f˜−1 g −1 U ) and
there exists Z ⊂ C such that supp(s) ⊂ g̃ −1 Z and f |Z : Z → U is proper.
Now suppose further that the diagram above is cartesian. Then f˜ : g̃ −1 Z →
−1
g U is also proper, thus so is supp(s) → g −1 U . It defines a natural morphism
f! g̃∗ → g∗ f˜! , and
f! F → f! g̃∗ g̃ ∗ F → g∗ f˜! g̃ ∗ F.
Again by adjunction we have g ∗ f! F → f˜! g̃ ∗ F. Now for x ∈ B, we have
(g ∗ f! F)x = (f! F)g(x) = Γc (f −1 (g(x)), F|f −1 (g(x)) )
= Γc (f˜−1 (x), (g̃ ∗ F)| ˜−1 ) = (f˜! g ∗ F)x .
f
(x)
Thus we proved the following theorem
Theorem 6 (Proper Base Change). If the diagram above is cartesian, we have
a natural isomorphism
g ∗ f! ' f˜! g̃ ∗ .
Now we see that Proposition 1 and 2 are special cases of this theorem together with Proposition 3.
4
1.3
Constructible Sheaves
The main object with which we are concerned is (cohomologically) constructible
sheaves. We briefly discuss the corresponding definitions from now on. Let X
be a complex analytic space and P = (Xi )i∈I be a partition of X which satisfies
the following properties.
• Each Xi is nonempty, locally closed and connected.
• P is locally finite.
• Each Xi \ Xi is a union of elements in P.
• For each Xi ∈ P, Xi and Xi \ Xi are closed complex analytic subspaces in X.
Then we say F ∈ Sh(X) is constructible if there exists a partition P which
satisfies conditions above such that for each Xi ∈ P, F|Xi is a local system, i.e.
a locally constant sheaf of C-vector spaces of finite dimension. In this case we
also say that F is constructible with respect to P. We denote by Shc (X) the
full subcategory of constructible sheaves on Sh(X) and by Shc (X, P) its full
subcategory of constructible sheaves with respect to a fixed P. Then it is easy
to show that Shc (X) is an abelian category.
Also, it is easy to check that internal Hom and tensor product preserve constructibility. For an analytic morphism f : X → Y between complex analytic varieties, one can also easily show that f ∗ preserves constructibility. Push-forward
is a little tricky and does not preserve the property in general. However, if f is
indeed algebraic, then f∗ preserves it using Chevalley constructibility theorem.
Similarly, for F ∈ Shc (X) if f is proper on supp F, then f∗ F ∈ Shc (Y ). We
postpone going into details about it until we introduce a constructible derived
category.
Example 7. Suppose X = C and i : Z = {0} ,→ X, j : U = X \ Z ,→ X. We
let F ∈ Sh(U ) be the local system with monodromy matrix A ∈ GLn (C) which
has Jordan blocks A1 , · · · , Ar with eigenvalues λ1 , · · · , λr such that λ1 = · · · =
λs = 1. Then one can easily see that i∗ j∗ F ' Cs , i∗ j! F = 0.
1.4
Some Questions
So far we discussed definitions and some properties of operation on sheaves.
Now these questions arise naturally.
• Is there any description for ΓZ similar to FZ ' f! f ∗ F?
• Is there any adjunction formula for f! ?
• For a local system, we can easily take its dual. Can we expect such a thing
in general?
To answer these questions properly, we shall introduce the notion of derived
category.
5
2
2.1
Derived Category of Sheaves
Derived Category of Abelian Categories
We assume that we are already familiar with the language of a derived category,
but we briefly mention some properties. For an abelian category A, we denote by
K(A) the homotopy category of chain complexes in A. The derived category of
A, usually denoted by D(A), is the category obtained from K(A) by inverting
quasi-isomorphisms, which is a morphism of chain complexes which induces
isomorphisms on their cohomology. This process is called localization.
One of its major properties is that it is a triangulated category. Even though
D(A) is no longer abelian, it has an analogue of an exact sequence in an abelian
category, called a distinguished triangle. Then a functor between triangulated
category is called exact if it is additive, commutes with translations, and sends
distinguished triangles to distinguished ones. Note that K(A) is also triangulated and the localization functor K(A) → D(A) is exact.
Also we may consider at first the category of chain complexes of bounded cohomology, and we let Db (A) be its derived category. Likewise, D+ (A), D− (A),
respectively, is the derived category of bounded below complexes, bounded
above complexes, respectively. Sometimes Db (A) is the same as the category of
bounded chain complexes if we have some finiteness condition on A.
2.2
Derived Category of Sheaves
For a topological space X we may define D(Sh(X)), the derived category of
sheaves on X. We assume we already know that Sh(X) has enough injectives.
Therefore, it is possible to derive functors which are left exact on Sh(X). Let
f : X → Y be a continuous map between topological spaces. Then we have
• RHom : D− (Sh(X))op × D+ (Sh(X)) → D+ (Sh(X))
• Rf∗ : D+ (Sh(X)) → D+ (Sh(Y ))
• RΓX : D+ (Sh(Y )) → D+ (Sh(Y )) if f is a locally closed embedding.
• Rf! : D+ (Sh(X)) → D+ (Sh(Y )) if X, Y are locally compact.
Also, since f ∗ , ⊗, •Z (where Z is a locally closed subset) are already exact, they
automatically define functors on the derived category of sheaves.
Meanwhile, in general Sh(X) does not have enough projectives as the next
example shows.
Example 8. Suppose X = C with the usual topology. Then we claim that the
skyscraper sheaf C0 supported by the origin does not admit an epimorphism
from a projective sheaf. Thus suppose otherwise and let P → C0 be such a
morphism. We will show that P(U ) → C0 (U ) is a zero map for all neighborhood
U of the origin. Choose ε > 0 such that D2ε ⊂ U , where D2ε = {x ∈ C | |x| <
2ε} is a disk of radius 2ε. Since we have a surjection CDε → C0 , we should be
6
able to lift P → C0 to obtain P → CDε . (Note that CDε is a constant sheaf on
Dε extended by zero on C \ Dε .) However, then CDε (U ) = 0 since U is strictly
bigger than Dε , which means P(U ) → C0 (U ) is a zero map. In other words,
P0 → (C0 )0 is zero, which contradicts the assumption.
2.3
Adjunction and Base Change
From adjunction of operations on sheaves, we easily deduce the following formulae. For the sake of simplicity, we restrict our situation to the bounded derived
category of sheaves.
Proposition 9. Let f : X → Y be a continuous map between topological spaces
and let F, F 0 , F 00 ∈ Db (Sh(X)), G, G 0 ∈ Db (Sh(Y )). Then
(a) RHom(F ⊗ F 0 , F 00 ) ' RHom(F, RHom(F 0 , F 00 ))
(b) RHom(G, Rf∗ F) ' Rf∗ RHom(f ∗ G, F)
(c) RHom(GX , G 0 ) ' RHom(G, RΓX (G 0 )) if f is a locally closed embedding.
One advantage to consider derived categories is that we may also find adjunction for f! .
Theorem 10. Suppose f : X → Y is a continuous map between locally compact
spaces and f! has finite cohomological dimension. Then we have a well-defined
functor f ! : Db (Sh(Y )) → Db (Sh(X)) such that for F ∈ Db (Sh(X)) and G ∈
Db (Sh(Y )) the following adjunction formula holds.
RHom(Rf! F, G) ' Rf∗ RHom(F, f ! G)
Note that f ! is unique up to isomorphism by Yoneda lemma. Also in general
f is not a derived functor of a functor on sheaves and only well-defined on
the derived category.6 Now it is a direct consequence from adjunction that if
f : X → Y is a locally closed embedding then
!
RΓX ' Rf∗ f ! .
For more information one may refer to [KS, Section 3.1].
Meanwhile, we have a derived version of proper base change. Suppose the
following diagram is cartesian as before.
A
f˜
g
g̃
C
/B
f
/D
Then we proved that f˜! g̃ ∗ ' g ∗ f! under suitable conditions. Since pull-back is
exact, if we derive both sides we get Rf˜! ◦ g̃ ∗ ' g ∗ ◦ Rf! , say as a morphism
form Db (Sh(C)) to Db (Sh(B)). (Here we need to verify the fact that g̃ ∗ sends
injective objects to f˜! -acyclic objects.) Also by adjunction, it also implies that
Rf˜∗ ◦ g̃ ! ' g ! ◦ Rf∗ .
6 If
f! were a left adjoint on the category of sheaves, then it should be right exact.
7
2.4
Constructible Derived Category
In this section we assume X is a complex analytic space. Then we may also
consider Db (Shc (X)), the bounded derived category of constructible sheaves.
However, since we mainly deal with the cohomology of a chain complex on
the derived category rather than each term on the complex, it is reasonable to
consider Dcb (Sh(X)), the bounded derived category of sheaves of constructible
cohomology, or the constructible derived category on X. A striking fact is that
when X is a complex algebraic variety, then they are the same7 , which is proved
in [N].8 From now on we only deal with Dcb (Sh(X)).
The operations we introduced so far preserve cohomological constructibility.
In other words, we have the following
Proposition 11. Let f : X → Y be a morphism between complex analytic
spaces and F, F 0 ∈ Dcb (Sh(X)), G ∈ Dcb (Sh(Y )). Then,
(a) f ∗ G, f ! G ∈ Dcb (Sh(X)).
(b) Rf∗ F, Rf! F ∈ Dcb (Sh(Y )) if f is algebraic or f |supp F is proper.
(c) RHom(F, F 0 ), F ⊗ F 0 ∈ Dcb (Sh(X)).
We omit the proof.
3
3.1
Poincare-Verdier Duality
Duality Functor
We will define a duality functor on Db (Sh(X)), which would be a generalization
of taking dual of a local system. Thus it would be a contravariant endofunctor
on Db (Sh(X)) and it would be nice if it is also an involution. To that end,
among the operations we have it is natural to choose RHom(−, F) for some
F ∈ Db (Sh(X)). Now we give the definition of Verdier duality as follows.
Definition 12. Let f : X → Y be a continuous map of finite cohomological dimension, so that f ! is well-defined. Then we call ωX/Y := f ! CY ∈ Dcb (Sh(X)) ⊂
Db (Sh(X)) the relative dualizing complex on X over Y . If Y is a single point,
then we set ωX := ωX/Y and call it the dualizing complex on X.
Definition 13. Keep assuming the situation above. We define a contravariant
endofunctor
D = DX : Db (Sh(X)) → Db (Sh(X)) : F 7→ RHom(F, ωX )
and call it the Verdier duality on Db (Sh(X)).
7 It is not in general true if we fix a partition P on X and consider Sh (X, P) instead, or
c
X is just a complex analytic space
8 The ”perverse” analogue is proved by Beilinson.
8
Thus in particular, ωX = DCX . We investigate some interesting properties
of Verdier duality. First, we need the following lemmas.
Lemma 14 (Projection Formula). For continuous f : X → Y where X, Y are
locally compact and F ∈ Db (Sh(X)), G ∈ Db (Sh(Y )), we have
Rf! F ⊗ G ' Rf! (F ⊗ f ∗ G).
Lemma 15. Keep the setting above and assume further that f has finite cohomological dimension. Then for G, G 0 ∈ Db (Sh(Y )) we have
f ! RHom(G, G 0 ) ' RHom(f ∗ G, f ! G 0 ).
Proof. For any F ∈ Db (Sh(X)) we have
Hom(F, f ! RHom(G, G 0 )) = Hom(Rf! F, RHom(G, G 0 ))
= Hom(Rf! F ⊗ G, G 0 ) = Hom(Rf! (F ⊗ f ∗ G), G 0 )
= Hom(F ⊗ f ∗ G, f ! G 0 )
= Hom(F, RHom(f ∗ G, f ! G 0 ))
from which the result follows.
Proposition 16. Keep the setting above, then we have f ! D = Df ∗ .
Proof. For any G ∈ Db (Sh(Y )), we have
f ! DG = f ! RHom(G, ωY ) = RHom(f ∗ G, f ! ωY ) = RHom(f ∗ G, ωX ) = Df ∗ G.
Thus the result follows.
Proposition 17. Keep the setting above, then we have DRf! = Rf∗ D.
Proof. It follows from
Hom(G, DRf! F) = Hom(Rf! F ⊗ G, ωY ) = Hom(Rf! (F ⊗ f ∗ G), ωY )
= Hom(F ⊗ f ∗ G, f ! ωY ) = Hom(F ⊗ f ∗ G, ωX )
= Hom(f ∗ G, DF) = Hom(G, Rf∗ DF)
and Yoneda lemma.
From now on we investigate properties of the dualizing complex. Suppose
f : X → Y is a morphism of finite cohomological dimension between locally
compact spaces, and G, G 0 ∈ Db (Sh(Y )). Then by adjunction we have a natural
morphism Rf! f ! G → G, from which we also have Rf! f ! G ⊗ G 0 → G ⊗ G 0 . By projection formula it is equivalent to Rf! (f ! G ⊗ f ∗ G 0 ) → G ⊗ G 0 , thus by adjunction
again we have f ! G ⊗ f ∗ G 0 → f ! (G ⊗ G 0 ).
If we set G = CY , then f ! G = ωX/Y , thus we have ωX/Y ⊗ f ∗ G 0 → f ! G 0 .
Now we are ready to state the following theorem.
9
Theorem 18. Suppose f : X → Y is a topological submersion with fiber dimension d. Then,
(a) H k (ωX/Y ) = 0 if k 6= −d and H −d (ωX/Y ) is a local system of rank 1. When
Y is a point it is called the orientation sheaf of X.
(b) The natural morphism ωX/Y ⊗ f ∗ G 0 → f ! G 0 above is an isomorphism.
(c) If X and Y are orientable manifolds, then ωX/Y ' CX [d]. Thus in particular f ∗ [d] ' f ! .
Proof. To show (a) it suffices to consider f : Rn → {pt}. For (b) we need to
keep track of the construction of this morphism. (c) is more complicated, see
[KS, Lemma 3.3.7].
Example 19 (Poincaré duality). Suppose X is a topological manifold. Then
for f : X → {pt} we apply DRf! ' Rf∗ D to CX to get DRf! CX ' Rf∗ DCX , or
DRΓc (X, CX ) ' RΓ(X, ωX ).
Thus by taking (−i)-th cohomology, we have Hci (X, CX )∨ ' H −i (X, ωX ). If
furthermore X is orientable, then ωX ' CX [d] where d = dim X, thus we have
Hci (X, CX )∨ ' H d−i (X, CX )
which is the Poincaré duality of X.
Example 20 (Alexander duality). Suppose X is an orientable topological manifold and Z ⊂ X is a closed subset. Then if we denote the closed embedding of
Z to X by f : Z → X, we have
RΓZ D ' Rf∗ f ! D ' Rf∗ Df ∗ ' DRf! f ∗
Thus if we apply this to CX ' ωX [−d] then we have
RΓZ CX [d] ' DCZ .
By taking (−i)-th cohomology on both sides we have
HZd−i (X, CX ) ' Hci (Z, CZ )∨
which is the Alexander duality.
3.2
Verdier Duality on Constructible Derived Category
Unfortunately, D is not a duality on the whole Db (Sh(X)). However, we have
Theorem 21. Suppose X is a complex analytic space. Then the restriction of D
is a contravariant endofunctor on Dcb (Sh(X)). Furthermore, D is an involution
on Dcb (Sh(X)).
We omit the proof.
10
3.3
Formalism of Six Functors
Suppose f : X → Y is an analytic morphism between complex analytic spaces.
So far we defined the following functors.
(1) RHom : Dcb (Sh(X))op × Dcb (Sh(X)) → Dcb (Sh(X))
(2) ⊗ : Dcb (Sh(X)) × Dcb (Sh(X)) → Dcb (Sh(X))
(3) Rf∗ : Dcb (Sh(X)) → Dcb (Sh(Y ))
(4) Rf! : Dcb (Sh(X)) → Dcb (Sh(Y ))
(5) f ∗ : Dcb (Sh(Y )) → Dcb (Sh(X))
(6) f ! : Dcb (Sh(Y )) → Dcb (Sh(X)).
Also we have the duality functor D : Dcb (Sh(X)) → Dcb (Sh(X))op which is an
involution, i.e. DD = id. So far we showed that they satisfy the following
properties.
• f ∗ a Rf∗ , Rf! a f ! , and ”Tensor-Hom adjunction”
• The adjunctions above are functorial, thus each f 7→ Rf∗ , Rf! , f ∗ , f ! is a
functor.
• There exists a natural morphism Rf! → Rf∗ which is an isomorphism if f is
proper.
• If f is smooth of relative complex dimension d, then there exists a natural
isomorphism f ∗ → f ! [−2d]
• If the following diagram is cartesian,
A
f˜
g
g̃
C
/B
f
/D
then there exist natural isomorphisms Rf˜! g̃ ∗ ' g ∗ Rf! and Rf˜∗ g̃ ! ' g ! Rf∗ .
• For F ∈ Dcb (Sh(X)) and G ∈ Dcb (Sh(Y )), we have the projection formula
F ⊗ Rf! G ' Rf! (f ∗ F ⊗ G).
• DRf! ' Rf∗ D, DRf∗ ' Rf! D, Df ! ' f ∗ D, Df ∗ ' f ! D.
This is a typical example of so-called ”Grothendieck’s six operations.” For another example, which corresponds to D-module theory, we refer readers to [E].
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References
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[GW2]
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