An Introduction to D-Modules
Jean-Pierre Schneiders
Abstract
The purpose of these notes is to introduce the reader to the algebraic theory
of systems of partial differential equations on a complex analytic manifold. We
start by explaining how to switch from the classical point of view to the point
of view of algebraic analysis. Then, we perform a detailed study of the ring
of differential operators and its modules. In particular, we prove its coherence
and compute its homological dimension. After introducing the characteristic
variety of a system, we prove Kashiwara’s version of the Cauchy-Kowalewsky
theorem. Next, we consider the behavior of systems under proper direct images.
We conclude with an appendix giving the complements of algebra required to
fully understand the exposition.
Contents
1 Introduction
2
2 Motivations
4
3 Differential operators on manifolds
3.1 The ring DX and its modules . . . . . . . . . . . . . .
3.2 The order filtration of DX . . . . . . . . . . . . . . . .
3.3 Principal symbols and the symplectic structure of T ∗X
3.4 Local finiteness properties of DX . . . . . . . . . . . .
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4 Internal operations on D-modules
4.1 Internal products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Side changing functors and adjoint operators . . . . . . . . . . . . . . .
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5 The
5.1
5.2
5.3
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de Rham system
Koszul complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The universal Spencer and de Rham complexes . . . . . . . . . . . . .
The de Rham complex of a DX -module . . . . . . . . . . . . . . . . . .
1991 Mathematics Subject Classification. 32C38, 35A27.
Key words and phrases. D-modules, algebraic analysis, partial differential equations on manifolds.
1
2
Jean-Pierre Schneiders
6 The
6.1
6.2
6.3
6.4
characteristic variety
Definition and basic properties .
Additivity . . . . . . . . . . . .
Involutivity and consequences .
Definition of holonomic systems
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7 Inverse Images of D-modules
7.1 Inverse image functors and transfer modules . . . . . . . .
7.2 Non-characteristic maps . . . . . . . . . . . . . . . . . . .
7.3 Non-characteristic inverse images . . . . . . . . . . . . . .
7.4 Holomorphic solutions of non-characteristic inverse images
8 Direct images of D-modules
8.1 Direct image functors . . . . . . . . . . . . . .
8.2 Proper direct images . . . . . . . . . . . . . .
8.3 The D-linear integration morphism . . . . . .
8.4 Holomorphic solutions of proper direct images
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9 Non-singular D-modules
9.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Linear holomorphic connections . . . . . . . . . . . . . . .
9.3 Non-singular D-modules and flat holomorphic connections
9.4 Local systems . . . . . . . . . . . . . . . . . . . . . . . . .
9.5 Non-singular D-modules and local systems . . . . . . . . .
10 Appendix
10.1 Linear algebra . . . . .
10.2 Graded linear algebra .
10.3 Filtered linear algebra
10.4 Transition to sheaves .
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67
Introduction
The application of the powerful methods of homological algebra and sheaf theory to
the study of analytic systems of partial differential equations started in the late ’60s
under the impulse of M. Sato and later M. Kashiwara. Many essential results were
obtained both in the complex and real domain, and soon a whole theory was built.
This theory, which is now often referred as “Algebraic Analysis”, has grown steadily
during the past years and important applications to other fields of mathematics and
physics (e.g. singularities theory, group representations, Feynman integrals,. . . ) were
developed.
Many research areas are still open. However, the task facing a student who would
like to work in these directions is quite impressive: the research literature is huge and
few textbooks are available.
An Introduction to D-Modules
3
In these notes, we develop the elements of the algebraic theory of systems of partial
differential equations in the complex domain. All the results are well-known and our
contribution is only at the level of the presentation.
Of course, it is impossible to give a complete picture of the theory in these few
pages, but we hope to give the reader the basic knowledge he needs to understand the
main research papers on the subject.
To get a better perspective on algebraic analysis, the reader should have a look
at the works [15, 2, 9, 16, 4, 12, 3] and their bibliographies. A natural extension
of this course would be the study of the theory of holonomic systems as developed
in [6, 7, 8, 11, 10].
Throughout the text, we assume the reader has a basic knowledge of linear algebra,
homological algebra, sheaf theory and analytic geometry.
In the first section, as a motivation for the theory, we explain how to make the
switch from the usual point of view of systems of partial differential equations to the
point of view of D-modules used in algebraic analysis.
In the second section, we study the ring D of differential operators on a complex
manifold and its local finiteness properties (coherence, dimension, syzygies).
The third section is devoted to internal operations of the category of D-modules on
a complex manifold. After introducing the internal product, we explain how to switch
between left and right D-modules.
In the next section, as an example, we make a detailed study of the D-module
structure of the sheaf of holomorphic functions O.
After a section on the characteristic variety, we study non-characteristic inverse
images and proper direct images. In particular, we prove Kashiwara’s version of the
Cauchy-Kowalewsky theorem for systems.
In the section on non-singular systems, we identify the D-modules corresponding
to flat holomorphic connections and classify them.
The appendix is devoted to an informal study of homological dimension, Noether
property and syzygies for rings, graded rings and filtered rings. Most of the results of
filtered algebra needed for the study of DX are proven there. Hence, the reader should
have a quick look at the appendix before reading the main text and refer afterwards
to the detailed proofs as the need arises.
These notes are based on lectures given at the “Journées d’Analyse Algébrique”
held at the University of Liège in March 1993. Let me take this opportunity to thank
the participants for their constructive comments.
Finally, I wish to thank M. Kashiwara and P. Schapira through the work of whom
I learned to appreciate the power and beauty of algebraic analysis.
4
2
Jean-Pierre Schneiders
Motivations
Let U be an open subset of C
I n . Recall that a (complex analytic linear partial) differential operator on U is an operator of the form
P (z, ∂z ) =
X
aα(z )∂zα
|α|≤p
where aα(z ) is a holomorphic function on U , α being a multi-index α1 , . . . , αn of length
|α| = α1 + · · · + αn and ∂zα = ∂zα11 . . . ∂zαnn . Such an operator acts C
I -linearly on holomorphic functions defined on U and is characterized by this action since
P ((z − z0 )α )|z=z0 = α! aα (z0).
One checks easily that differential operators form a subring of End CI (O (U )). The
restriction of a differential operator
P =
X
aα∂zα
|α|≤p
defined on U to an open subset V is the operator
P|V =
X
|α|≤p
aα|V ∂zα .
With these restriction morphisms, differential operators appear as a presheaf of rings
on C
I n . We denote by DCIn the associated sheaf of rings. A section P ∈ Γ(U ; DCIn ) may
be written in a unique way as
X
aα(z )∂zα
α
where, aα = 0 in a neighborhood of z0 for |α| 0. If there is an integer p such that
aα = 0 for |α| > p and aα 6= 0 for at least an α of length p we say that P has order p.
Any operator P ∈ Γ(U, DCI n ) has locally a finite order. Note that OCIn has a natural
structure of DCIn -module. For an open subset U of C
I n, we denote by DU the restriction
to U of the sheaf DCIn .
Our basic problem in these lectures is to study systems of equations of the form
n
X
Pij (z, ∂z )uj = vi
(i = 1, . . . , m)
(2.1)
j =1
where Pij (z, ∂z ) is a differential operator of some finite order pij , the v1, . . . , vn being
given data and u1, . . . , um being indeterminates in some function space S (U ) on which
the ring Γ(U ; DCIn ) acts. In fact, since we want to be able to use the powerful methods
of sheaf theory to glue together local results, we will assume that S (U ) is the space of
sections on U of a sheaf S which is a DU -module. For example, we may take S = OU
the sheaf of holomorphic functions on U or S = AU the sheaf or real analytic functions
or S = BU the sheaf of hyperfunctions. Other possible examples for S are FU the
An Introduction to D-Modules
5
sheaf of infinitely differentiable functions on U or DbU the sheaf of distributions on U
or even the sheaf DU itself.
Our first step in studying systems like (2.1) will be to try to understand when two
of them should be considered as equivalent.
Let us first consider the case of homogeneous equations
n
X
Pij (z, ∂z )uj = 0
(i = 1, . . . , m).
j =1
Intuitively, equivalent systems are systems which have the same solutions in any solution space S . To translate this intuition into a mathematical notion, we will proceed
as follows. Denote by P the matrix (Pij ) and by S P the solution sheaf of the system
associated to P in the DU -module S . We have
Γ(U ; S P ) = {u ∈ Γ(U ; S )n : P · u = 0}.
The law S 7→ S P defines a functor from the category of DU -modules to the category
of sheaves of C
I -vector spaces on U and two systems P , Q should be considered as
equivalent when the associated functors S 7→ S P and S 7→ S Q are naturally equivalent.
Now consider the DU -linear morphism
·P
DUm −→ DUn
R
7→
R·P
and denote by MP its cokernel. From the exact sequence
·P
DUm −→ DUn −→ MP −→ 0
we get the exact sequence
P·
0 −→ Hom DU (MP , S ) −→ S n −→ S m
u 7→ P · u
and hence a natural equivalence of functors
Hom DU (MP , S ) ' S P .
This shows that the functor S 7→ S P is representable by the DU -module MP . The homogeneous systems associated to P and Q are thus equivalent if and only if MP ' MQ
as DU -modules. So it appears that the natural object which represents a homogeneous
system of P.D.E. is the DU -module MP not the matrix of operators P .
What is the situation for inhomogeneous systems?
Assume that u1 , . . . , um ; v1, . . . , vm ∈ Γ(U, S ) and that
n
X
j =1
Pij uj = vi
(i = 1, . . . , m)
(2.2)
6
Jean-Pierre Schneiders
Then, for any differential operators Q1, . . . , Qm on U , we have
n
m
X
X
j =1
and it follows that
m
X
!
QiPij uj =
i=1
m
X
Qi v i ,
i=1
Qivi = 0 if
i=1
m
X
QiPij = 0.
i=1
Such a vector of m differential operators defines what we will call an algebraic compatibility condition for the inhomogeneous system (2.2). By definition, it is clear that
algebraic compatibility condition are the sections of the kernel NP of the map
·P
DUm −→ DUn .
Denote by IP the image of this same map. We have the exact sequences
α
0 −→ NP −→ DUm −→
0 −→ IP
β
−→
DUn
IP
−→ 0
−→ MP −→ 0
where β ◦ α = ·P . Hence, we get the exact sequences
0 −→ Hom DU (MP , S ) −→ S n −→ Hom DU (IP , S ) −→ E xt1DU (MP , S ) −→ 0
0 −→ Hom DU (IP , S ) −→ S m −→ Hom DU (NP , S ) −→ E xt1DU (IP , S ) −→ 0
The second one shows that
Hom DU (IP , S ) = {v ∈ S m : Q · v = 0, ∀Q ∈ NP }
so IP is a DU -module which represents the system of algebraic compatibility conditions
of the system P . Using the first exact sequence we see, at the level of germs, that the
elements of E xt1DU (MP , S )x are the classes of vectors vx of Sxm satisfying the algebraic compatibility conditions modulo those for which the system is truly compatible.
Moreover, for k ≥ 1,
E xtkDU (IP , S ) ' E xtkD+1 (MP , S ).
U
E xt kDU (MP , S )
Thus, all the
give meaningful information about the system P .
We hope that the preceding discussion has convinced the reader that the study of
the system
n
X
Pij uj = vi
(i = 1, . . . , m)
j =1
is equivalent to the study of the DU -module MP defined by the exact sequence
·P
DUm −→ DUn −→ MP −→ 0
and of its full solution complex
RHom DU (MP , S ).
An Introduction to D-Modules
7
In fact, restricting our interest to system which like MP have a global finite free
1-presentation is not a very good point of view because the system IP of algebraic
compatibility conditions of P admits in general no finite free 1-presentation. However,
we will show that it has locally such a presentation (DU is a coherent sheaf of rings). So
coherent DU -modules appear to be the natural algebraic objects representing systems of
linear partial differential equations. We will see in the sequel that the study of coherent
D-modules also allows us to understand in a clear and intrinsic manner systems of linear
partial differential equations on manifolds.
3
Differential operators on manifolds
Let X be a complex analytic manifold of dimension n. Let us recall that OX denotes
the sheaf of holomorphic functions on X and that ΘX denotes the sheaf of holomorphic
vector fields on X . Of course, OX is a sheaf of C
I -algebras. We know that ΘX is the
sheaf of holomorphic sections of the tangent bundle T X and that it is a locally free
OX -module of rank n. Through Lie derivative, ΘX acts C
I -linearly on OX . In fact, ΘX
may be identified to the sheaf of C
I -linear derivations of the C
I X -algebra OX . The Lie
bracket turns ΘX into a sheaf of Lie algebras over C
I . As usual, one denotes by Lθ the
Lie derivative along a vector field θ ∈ ΘX .
3.1
The ring DX and its modules
Definition 3.1.1 The ring of linear partial differential operators with analytic coefficients on X is the subsheaf of rings of E nd CIX (OX ) generated by Lie derivatives along
holomorphic vector fields and multiplications by holomorphic functions. We denote it
by DX . Locally, a section P ∈ Γ(U ; DX ) may be written in a unique way in the form
P =
X
aα(z )∂zα.
|α|≤p
where (z : U −→ C
I n ) is a local coordinate system on X .
Proposition 3.1.2 The ring DX is a simple sheaf of non-commutative C
I -algebras with
center C
IX.
Proof: It is sufficient to work at the levels of germs and we may thus assume X = C
In
and show that the center of (DCIn )0 is C
I.
For any central P ∈ (DCIn )0 we have [P, z i ] = 0 and [P, ∂zi ] = 0. Let z 0 =
(z 1, . . . , z n−1 ) and z 00 = z n . We may write P as
P (z, ∂z ) =
p
X
Pk (z 0 , z 00 , ∂z0 )∂zk00
k=0
where Pp (z 0, z 00 , ∂z0 ) 6= 0 and p is the order of P in ∂z00 . We have
00
0 = [P, z ] =
p
X
k=1
Pk (z 0, z 00 , ∂z0 )k∂zk00−1
8
Jean-Pierre Schneiders
hence P = P0 (z 0, z 00 , ∂z0 ). Iterating this process, we see that P is a germ a(z ) ∈ (OCIn )0 .
Since
0 = [ ∂z i , a ] = ∂z i a
the germ a is constant and the conclusion follows.
Now, let I be a two sided ideal of (DCIn )0. Assuming I 6= 0, we will show that 1 ∈ I .
With the same notations as above, we may find
p
X
P (z, ∂z ) =
Pk (z 0 , z 00 , ∂z0 )∂zk00 ∈ I
k=1
6 0. Since the order in ∂z00 of [P, ∂z00 ] is p − 1 we may even assume
where Pp (z 0, z 00 , ∂z0 ) =
P is of order 0 in ∂z00 . Iterating this procedure, we find a non zero germ of holomorphic
function a(z ) ∈ I . Up to a change of coordinates, we may assume a(z ) is of Weierstrass
type with respect to z 00 at 0. This implies that
a (z ) =
p
X
!
bk (z 0)(z 00 )k u(z )
k=0
where u(z ) is an invertible germ of holomorphic function on z and b0 (z 0), . . . , bp(z 0 ) are
germs of holomorphic function in z 0 with b0 (0) = · · · = bp−1 (0) = 0 and bp (z 0 ) = 1. It
follows that
p
b (z ) =
X
bk (z 0)(z 00 )k ∈ I
k=0
and since
[∂z00 , b(z )] =
p
X
bk (z 0 )k (z 00 )k−1
k=0
0
we find by iteration that p! bp (z ) ∈ I . Hence 1 ∈ I and I = (DCIn )0 , and the proof is
2
complete.
op
The category Mod(DX ) (resp. Mod(DX
)) of left (resp. right) DX -modules is of
course an Abelian category with enough injective objects as is any category of modules
over a sheaf of rings.
The construction of left or right DX -modules is simplified by the following proposition.
Proposition 3.1.3 Assume M is an OX -module and denote by
µ : OX −→ E nd CIX (M)
the associated multiplication. Assume
α : ΘX −→ E nd CIX (M)
is a C
I X linear action of holomorphic vector fields on M such that
(a) [α(θ), µ(h)] = µ(Lθ h) (resp. −µ(Lθ h)),
An Introduction to D-Modules
9
(b) [α(θ), α(ψ )] = α([θ, ψ ]) (resp. −α([θ, ψ ])),
(c) µ(h) ◦ α(θ) = α(hθ) (resp. α(θ) ◦ µ(h) = α(hθ))
for any θ, ψ ∈ ΘX and any h ∈ OX . Then there is a unique structure of left (resp.
right) DX -module on M extending the given actions of OX and ΘX .
Proof: The uniqueness of the extending structure is trivial since DX is generated by
OX and ΘX . The existence needs thus only to be proven locally. There, the structure
of a differential operator dictates the definition of its action and one checks easily that
this action is a left (resp. right) DX -module structure extending the given actions of
2
OX and ΘX .
Corollary 3.1.4 (a) There is a unique left DX -module structure on OX extending
its structure of OX -module in such a way that
θ · h = Lθ h
for any θ ∈ ΘX , h ∈ OX .
(b) There is a unique right DX -module structure on ΩX extending its structure of
OX -module in such a way that
ω · θ = −Lθ ω
for any θ ∈ ΘX , ω ∈ ΩX .
Proof: (a) is obvious by the definition of DX .
(b) It is well known that ΩX is an OX -module. We will show that the action of ΘX
on ΩX defined by setting
ω · θ = −Lθ ω
for ω ∈ (ΩX )x , θ ∈ (ΘX )x satisfies the conditions of Proposition 3.1.3. The easiest way
is to look what happens in a local coordinate system z : U −→ C
I n.
Recall that if
n
θ=
X
θ i ∂z i
i=1
is a holomorphic vector field on U generating the complex flow ϕτ and if ω = a(z )dz 1 ∧
. . . ∧ dz n is a holomorphic n-form on U by definition
Lθ ω = ∂τ (ϕ∗τ ω)|τ =0
= ∂τ (a(ϕτ ) dϕ1τ ∧ . . . ∧ dϕnτ)|τ =0
= Lθ adz 1 ∧ . . . ∧ dz n +
"
=
=
Lθ a +
" n
X
i=1
n
X
i=1
!
∂z i θ
i
n
X
a(z )dz 1 ∧ . . . dθi ∧ . . . ∧ dz n
i=1
#
a(z ) dz 1 ∧ . . . ∧ dz n
#
∂zi (θi a) dz 1 ∧ . . . ∧ dz n
10
Jean-Pierre Schneiders
Hence
ω.θ =
n
X
−∂zi (θi a)dz 1 ∧ . . . ∧ dz n
i=1
and the conditions of Proposition 3.1.3 are checked by easy direct computations.
3.2
2
The order filtration of DX
Definition 3.2.1 We define inductively the subsheaves Fk DX (k ∈ IN) of DX by
setting:
F0DX = OX ,
Fk+1 DX = Fk DX + ΘX · Fk DX
(k ≥ 0),
and extend the preceding sequence to negative integers by setting:
Fk DX = 0 (k < 0).
By definition, Fk DX ⊂ Fk+1 DX for k ∈ ZZ, and
DX =
[
Fk DX .
k∈ZZ
Hence, the sequence (Fk DX )k∈ZZ defines a filtration on the sheaf DX which is easily
seen to be compatible with its ring structure. The order of a section P of DX at x ∈ X
is the smallest integer k such that Px ∈ (Fk DX )x ; we denote it by ordx (P ). We also
set
ord(P ) = sup ordx (P ).
x∈ X
Obviously, a section P ∈ Γ(U, DX ) is a section of Fk DX if and only if ordx (P ) ≤ k
for each x ∈ U . Therefore, we call the sequence (Fk DX )k∈ZZ the order filtration of DX .
We denote by F DX the corresponding filtered ring and by GDX the associated graded
ring.
Locally, a section P ∈ Γ(U ; Fp DX ) may be written in a unique way in the form
P =
X
aα (z )∂zα.
|α|≤p
where (z : U −→ C
I n ) is a local coordinate system on X .
3.3
Principal symbols and the symplectic structure of T ∗X
Recall that to any local coordinate system (z : U −→ C
I n ) on X is associated a local
coordinate system ((z ; ξ ) : T ∗U −→ C
In × C
I n ) of T ∗X . The coordinates ξ (ω ) ∈ C
I n of
a form ω in Tu∗U are characterized by the formula
ω=
n
X
j =1
ξj (ω )dz j .
An Introduction to D-Modules
11
Lemma 3.3.1 Let (z : U −→ C
I n ) and (z 0 : U −→ C
I n ) be two coordinate systems on
an open subset U of X . Assume P ∈ Γ(U ; Fp DX ) may be written as
X
P =
aα(z )∂zα;
P =
|α|≤p
Then
X
X
aα0 (z 0)∂zα0 .
|α|≤p
X
aα (z )ξ α =
|α|=p
aα0 (z 0)(ξ 0 )α .
|α|=p
Proof: Denoting by J the Jacobian matrix
!
∂z
,
∂z 0
we have
∂z 0 = J ∗ ∂z .
Therefore,
X
aα0 (z 0 )∂zα0 =
|α|≤p
X
aα0 (z 0 ) (J ∗∂z )α .
|α|≤p
and, retaining only the terms in ∂zα with |α| = p, we get
X
|α|=p
Since
X
aα (z )ξ α =
aα0 (z 0)(J ∗ξ )α .
|α|=p
n
X
n
X
ξj dz j =
ξj0 dz 0 ,
j
j =1
j =1
we also get
ξ 0 = J ∗ξ,
2
and the conclusion follows easily.
Definition 3.3.2 Let P be a section of FpDX on X . In a local coordinate system
(z : U −→ C
I n ) of X , P may be written in a unique way in the form
X
P =
aα(z )∂zα.
|α|≤p
The preceding lemma shows that
σU =
X
aα (z )ξ α
|α|=p
is a holomorphic function on T ∗U which depends only on P . Moreover, for another
coordinate system (z : V −→ C
I n ) on X , we have
(σU )|U ∩V = (σV )|U ∩V .
Hence, there is a unique holomorphic function on T ∗X gluing all the σU together; we
call it the symbol of order p of P and denote it by σp (P ). The principal symbol of P is
its symbol of order ord(P ); we denote it by σ (P ).
12
Jean-Pierre Schneiders
Recall that a section of OT ∗ X on an open subset W is homogeneous of degree k ∈ ZZ
if for any (x; ω ) ∈ U there is an open neighborhood D of 0 in C
I such that
h(x; tω ) = tk h(x; ω )
for any t ∈ D. Obviously, these sections form a subsheaf of OT ∗X . We denote it by
OT ∗X (m). The reader will check easily that
OT ∗X (m)OT ∗X (n) ⊂ OT ∗X (m + n).
Hence,
⊕ π∗OT ∗X (k )
k∈IN
is canonically a sheaf of graded rings on X . In a local coordinate system (z : U −→ C
I n ),
one checks easily that
⊕ π∗OT ∗ U (k ) ' OU [ξ1, . . . , ξn ].
k∈IN
Proposition 3.3.3 The symbol maps induce a canonical isomorphism of graded rings
∼
GDX −→
⊕ π∗ OT ∗X (k ).
k∈IN
Proof: By definition, the symbol σk (P ) of an operator P ∈ Γ(X ; Fk DX ) belongs to
Γ(X ; OT ∗X (k )). Hence, for each k ∈ ZZ, we get an intrinsic morphism
σk : Fk DX −→ π∗ OT ∗X (k ).
Since σk is zero on Fk−1 DX , it induces a morphism
σk : Gk DX −→ π∗ OT ∗X (k ).
Hence, we get a canonical morphism
GDX −→ ⊕ π∗ OT ∗X (k ).
k∈IN
which is easily checked to be a morphism of graded rings. In a local coordinate system,
2
this morphism is clearly bijective and the conclusion follows.
The manifold T ∗X is endowed with a canonical 1-form α defined by
α(x,ω)(θ) = hω, π 0θi
for any θ ∈ T(x,ω)T ∗X , π : T ∗X −→ X being the canonical bundle map and π 0 denoting
its tangent map. If (z : U −→ V ) is a local coordinate system on U and
(z, ξ ) : T ∗U −→ V × C
In
is the associated trivialization of T ∗U , we have
α| T ∗ U =
n
X
k=1
ξk dz k .
An Introduction to D-Modules
13
We turn T ∗X into a symplectic manifold by endowing it with the non degenerate
skew symmetric form σ = dα. With the same notations as above, we have
σ|T ∗ U =
n
X
dξk ∧ dz k .
k=1
As on any symplectic manifold, we define the Hamiltonian isomorphism
H : T ∗T ∗X −→ T T ∗X
through the formula σp(θ, H (ω )) = hω, θi valid at any point p ∈ T ∗X and for any
ω ∈ Tp∗T ∗X , θ ∈ TpT ∗X .
Locally, we have
H (dz k ) = −∂ξk
H (dξk ) = ∂zk .
Hence, for a holomorphic function f defined on an open subset W of T ∗U we have
H (df ) =
n
X
∂ξ k f ∂z k − ∂z k f ∂ξ k
k=1
on W which is the usual expression for the Hamiltonian field Hf of the function f in
classical mechanics (see [1]).
Using Hamiltonian fields, we may introduce the classical notion of Poisson bracket
of two differentiable functions f , g defined on an open subset of T ∗X by the formula
{f, g } = Hf (g ),
and one checks easily that H{f,g} = [Hf , Hg ].
Proposition 3.3.4 Let P ∈ Γ(X ; Fk DX ), Q ∈ Γ(X ; F` DX ) be two differential operators on X . Then [P, Q] ∈ Γ(X ; Fk+`−1 DX ) and
σk+`−1 ([P, Q]) = {σk (P ), σ` (Q)}.
Proof: Since the problem is local, we may assume that X is an open subset of C
I n . In
2
this case, the equality is easily checked by a direct computation.
Remark 3.3.5 In the preceding proposition, note that the order of [P, Q] may well
be strictly lower than k + ` − 1. Note also that the formula
σ ([P, Q]) = {σ (P ), σ (Q)}.
is false in general.
14
3.4
Jean-Pierre Schneiders
Local finiteness properties of DX
Proposition 3.4.1 The ring DX is Noetherian, syzygic and has finite homological
dimension. In particular, there is an integer p such that any 1-presentation
`1
`0
DX
−→ DX
−→ M −→ 0
may be extended locally into a free resolution of length p
`
`
0 −→ DVp −→ DVp−1 −→ · · · −→ DV`0 −→ M|V −→ 0.
Proof: Since the problem is local, we may assume that X is an open subset U of
C
I n . We know from classical results of analytic geometry that OU is a Noetherian
ring on U . Moreover, for any u ∈ U , the fiber (OU )u is a regular Noetherian local
ring of dimension n. Hence, by Corollary 10.1.7, OU is syzygic and its homological
dimension is lower than n. Therefore Proposition 10.1.1 combined with Corollary 10.2.5
and Proposition 10.4.10 shows that OU [ξ1, . . . , ξn ] is a syzygic Noetherian ring with
homological dimension lower than 2n. Hence, so is GDX , and Proposition 10.4.8 shows
that F DX is Noetherian and that DX is Noetherian. Moreover, Propositions 10.4.5
2
and 10.4.7 show that DX is syzygic with homological dimension glhd(DX ) ≤ 2n.
Remark 3.4.2 We will see in Corollary 6.3.2 that in fact glhd(DX ) is the complex
dimension of X .
4
Internal operations on D -modules
In this section, we will define an internal tensor product and compute its right adjoint.
We will also show that there is a canonical equivalence between the category of left
DX -modules and that of right DX -modules.
4.1
Internal products
The proofs of the following results are easy verifications based on Proposition 3.1.3.
We leave them to the reader.
Proposition 4.1.1 Let M, N , P be three left DX -modules.
(a) The action of ΘX on the OX -module M ⊗OX N defined by setting
θ(m ⊗ n) = θm ⊗ n + m ⊗ θn
for any θ ∈ ΘX , m ∈ M, n ∈ N extends to a left DX -module structure on M ⊗OX N .
(b) The action of ΘX on the OX -module Hom OX (N , P ) defined by setting
(θf )(n) = θf (n) − f (θn)
for any θ ∈ ΘX , f ∈ Hom OX (N , P ), n ∈ N , extends to a left DX -module structure on
Hom OX (N , P ).
An Introduction to D-Modules
15
(c) We have the adjunction isomorphism
Hom DX (M ⊗OX N , P ) −→ Hom DX (M, Hom OX (N , P ))
f
7→
(m 7→ (n 7→ f (m ⊗ n)).
In the same way, we have
Proposition 4.1.2 Let M, P be two right DX -modules and let N be a left DX -module
then
(a) the action of ΘX on M ⊗OX N defined by
(m ⊗ n)θ = mθ ⊗ n − m ⊗ θn
extends to a right DX -module structure,
(b) the action of ΘX on Hom OX (N , P ) defined by
(fθ)(n) = f (θn) − f (n)θ
extends to a right DX -module structure,
(c) the action of ΘX on Hom OX (M, P ) defined by
(θf )(n) = f (nθ) − f (n)θ
extends to a left DX -module structure,
(d) we have canonical adjunction isomorphisms
Hom DX (M ⊗OX N , P ) ' Hom DX (M, Hom OX (N , P ))
Hom DX (M ⊗OX N , P ) ' Hom DX (N , Hom OX (M, P )).
Proposition 4.1.3 Assume M is a right DX -module and N , P are left DX -modules
then we have a natural isomorphism
∼ M⊗
(M ⊗OX N ) ⊗DX P −→
DX (N ⊗OX P ).
4.2
Side changing functors and adjoint operators
The preceding results give us a way to switch between left and right DX -modules.
op
Proposition 4.2.1 The functor from Mod(DX ) to Mod(DX
) defined by
M 7→ ΩX ⊗OX M
op
)
is an equivalence of categories. Its quasi-inverse is given by the functor from Mod(DX
to Mod(DX ) defined by
N 7→ Hom OX (ΩX , N ).
16
Jean-Pierre Schneiders
Proof: Note that, since ΩX is locally free of rank 1 over OX ,
M ' Hom OX (ΩX , ΩX ⊗OX M)
N ' ΩX ⊗OX Hom OX (ΩX , N )
as OX -modules, and one checks easily that these isomorphisms are compatible with
2
the canonical DX -module structure of their sources and targets.
To better understand the preceding equivalence, let us introduce the notion of
adjoint of an operator.
Proposition 4.2.2 Let V be an open subset of X and assume to be given a generator
ω of ΩX on V . For any P ∈ Γ(V ; DX ) there is a unique operator P ∼ω ∈ Γ(V ; DX ) (the
adjoint of P with respect to ω ) such that
(P ∼ω h)ω = (hω ).P
for any h ∈ Γ(W ; OX ) (W open subset of V ). Moreover, we have
(P ∼ω )∼ω = P,
and
(Q ◦ P )∼ω = P ∼ω ◦ Q∼ω
for any Q ∈ Γ(V ; DX ).
Proof: The problem is clearly of local nature. In a local coordinate system (z : U −→
C
I n ) we have
P =
X
aα (z )∂zα
|α|≤p
ω = a(z )dz 1 ∧ . . . ∧ dz n ,
where a(z ) is an invertible holomorphic function on U .
Hence, we get
(ha dz 1 ∧ . . . ∧ dz n ).P =
X (−∂z )α (aα (z )ah)
|α|≤p
a
a dz 1 ∧ . . . ∧ dz n
for any holomorphic function h on W ⊂ U ; hence the conclusion.
2
Proposition 4.2.3 Assume M is a left DX -module. Then the right DX -module structure on ΩX ⊗OX M is locally given by
(ω ⊗ m).P = ω ⊗ P ∼ω .m
ω being a local generator of ΩX .
An Introduction to D-Modules
17
Proof: It is sufficient to check the formula in a local coordinate system (z : U −→ C
I n)
when P = ∂zi and ω = a(z )dz 1 ∧ . . . ∧ dz n . In this case, we have
(ω ⊗ m).∂zi = ω∂zi ⊗ m − ω ⊗ ∂zi m
= (−∂zi a)dz 1 ∧ . . . ∧ dz n ⊗ m − adz 1 ∧ . . . ∧ dz n ⊗ ∂zi m
= dz 1 ∧ . . . ∧ dz n ⊗ −∂zi (a.m)
= ω ⊗ ∂z∼i ω m
2
and the proof is complete.
5
The de Rham system
Let X be a complex analytic manifold of complex dimension n. We will now study in
details the DX -module OX . In a local coordinate system (z : U −→ C
I n), we clearly
have the exact sequence
Q
7→ Q · 1
−→
DU
−→ OU −→ 0
Pn
(Q1, . . . , Qn) 7→
i=1 Qi ∂zi
DUn
which shows that OU is the DU -module associated with the system


∂ 1u = 0

 z



..
.
∂z n u = 0
In order to understand the full system of algebraic compatibility conditions of this
system, we need an important algebraic tool: the Koszul complex.
5.1
Koszul complexes
Let A be any ring with unit and let M be an A-module. Denote by e1, . . . , ep the
canonical basis of ZZp and set
M (k) = M ⊗ Λk ZZp .
Note that any element of M (k) can be written in a unique way as
X
mi1 ...ik ⊗ ei1 ...ik
1≤i1 <···<ik ≤n
where mi1...ik ∈ M and ei1 ...ik = ei1 ∧ . . . ∧ eik .
Definition 5.1.1 Assume ϕ1 , . . . , ϕp are p commuting endomorphisms of M . Let
dk : M (k) −→ M (k+1)
18
Jean-Pierre Schneiders
be the A-linear morphism defined by setting
p
X
dk (m ⊗ ei1 ...ik ) =
ϕj (m) ⊗ ej ∧ ei1 ...ik .
j =1
By the commutativity of the ϕj ’s, we have dk+1 ◦ dk = 0. We denote by K · (Φ, M ) the
complex
d0
d1
0 −→ M (0) −→ M (1) −→ · · · −→ M (p) −→ 0
where M (0) is in degree zero. This is the positive Koszul complex of Φ = (ϕ1, . . . , ϕp).
Proposition 5.1.2 Assume ϕ1 is surjective and ϕj induces a surjective endomorphism
of ker ϕ1 ∩ . . . ∩ ker ϕj −1 for j ∈ {2, . . . , p}. Then
(
k
·
H (K (Φ, M )) =
ker ϕ1 ∩ . . . ∩ ker ϕp if k = 0
0
otherwise.
Proof: Note that, for p = 1, the result is obvious. For p > 1, the complex
K · (ϕ1 , . . . , ϕp ; M )
is isomorphic to the mapping cone of the morphism
ϕp
K · (ϕ1 , . . . , ϕp−1 ; M ) −→ K · (ϕ1 , . . . , ϕp−1 ; M )
and the conclusion follows by induction on p.
2
Definition 5.1.3 Consider the differential
δk : M (k) −→ M (k−1)
defined by setting
δk (m ⊗ ei1 ...ik ) =
k
X
(−1)j −1 ϕij (m) ⊗ ei1 ...ibj ...ik .
j =1
The commutativity of the ϕj ’s assures that δk−1 ◦ δk = 0. We will denote by K· (Φ, M )
the complex
δp
δp−1
0 −→ M (p) −→ M (p−1) −→ · · · −→ M (0) −→ 0
where M (0) is in degree 0. This is the negative Koszul complex of Φ.
Proposition 5.1.4 Assume ϕ1 is injective and ϕj induces an injective endomorphism
of M/(im ϕ1 + · · · + im ϕj −1 ) for j ∈ {2, . . . , p}. Then
(
Hk (K· (ϕ, M )) =
M/(im ϕ1 + · · · + im ϕp ) if k = 0
0
otherwise
Proof: Work as in the proof of 5.1.2.
2
An Introduction to D-Modules
19
Note that, for any Abelian group G, Hom(M, G) is canonically an Aop -module and
the endomorphisms ϕ1 , . . . , ϕp induce the Aop -linear endomorphisms
ψ1 = Hom(ϕ1, idG ), . . . , ψp = Hom(ϕp , idG )
of Hom(M, G) such that
Hom· (K· (ϕ, M ), G) ' K · (ψ, Hom(M, G))
Hom· (K · (ϕ, M ), G) ' K· (ψ, Hom(M, G)).
Let us denote
∗k : M (k) −→ M (p−k)
the only A-linear map such that
∗k (m ⊗ ei1 ...,ik ) = sm ⊗ ej1 ...jp−k
where {i1, . . . , ik } ∪ {j1 , . . . , jp−k } = {1, . . . , p} and
s = sign
i1 . . . ik j1 . . . jp−k
...
p
1
!
.
A direct computation shows that
dp−k ◦ ∗k = (−1)k−1 ∗k−1 ◦δ k .
Hence, the maps
(−1)
k(k−1)
2
∗k : M (k) −→ M (p−k)
define an isomorphism
∼ K · (Φ, M )[p]
K· (Φ, M ) −→
which induces the isomorphism
Hk (Φ; M ) ' H p−k (Φ; M ).
5.2
The universal Spencer and de Rham complexes
Definition 5.2.1 Let ΘpX denote the sheaf of holomorphic p-vector fields on X . The
universal Spencer complex of X is the complex SP ·X defined by setting
p
SP X
p = DX ⊗OX ΘX ,
the differential
X
δp : SP X
p −→ SP p−1
being defined by the formula
δ p (Q ⊗ θ 1 ∧ . . . ∧ θ p ) =
p
X
i=1
+
(−1)i−1 Qθi ⊗ θ1 ∧ . . . ∧ θbi ∧ . . . ∧ θp
X
1≤i<j ≤p
(−1)i+j Q ⊗ [θi, θj ] ∧ θ1 ∧ . . . θbi . . . θbj . . . ∧ θp.
Note that δp is well defined since the right hand side of the preceding formula is an
alternate OX -multilinear form in θ1, . . . , θp. A simple computation will confirm that
δp−1 ◦ δp = 0.
20
Jean-Pierre Schneiders
Proposition 5.2.2 The DX -linear morphism
DX −→ OX
Q
7→
Q·1
induces a DX -linear quasi-isomorphism
SP X
· −→ OX .
I n ) we have
Proof: In a local coordinate system (z : U −→ C
δp (Q ⊗ ∂zi1 ∧ . . . ∧ ∂zip ) =
p
X
i
(−1)j −1 Q∂zij ⊗ ∂zi1 ∧ . . . ∧ ∂d
zij ∧ . . . ∧ ∂z p .
j =1
Hence, the universal Spencer complex is isomorphic to the negative Koszul complex
associated to the sequence (·∂z1 , . . . , ·∂zn ) of DU -linear endomorphisms of DU . Since
·∂z1 is injective and ·∂zk induces an injective endomorphism of the quotient DU /DU ∂z1 +
· · · + DU ∂zk−1 , Proposition 5.1.4 allows us to conclude.
2
Definition 5.2.3 Recall that ΩkX denotes the sheaf of holomorphic k -forms. Set
DRkX = ΩkX ⊗OX DX
and define
dk : ΩkX ⊗OX DX −→ ΩkX+1 ⊗OX DX
in a local coordinate system (z : U −→ C
I n ) by the formula
d (ω ⊗ m) = dω ⊗ m +
k
k
k
n
X
dz j ∧ ωk ⊗ ∂zj m.
j =1
One checks easily that this definition does not depend on the chosen coordinate system
·
and that dk+1 ◦ dk = 0. The complex DRX
is the universal de Rham complex of X .
op
-linear morphism
Proposition 5.2.4 The DX
ΩX ⊗OX DX −→ ΩX
ω⊗Q
7→
ωQ
op
induces a DX
-linear quasi-isomorphism
·
[n] −→ ΩX .
DRX
Proof: Let us denote by ε the action morphism
ΩX ⊗OX DX −→ ΩX .
A local computation shows that
ε ◦ dn−1 = 0
An Introduction to D-Modules
21
op
hence ε induces a DX
-linear morphism of complexes
DR·X [n] −→ ΩX .
In a local coordinate system (z : U −→ C
I n ), the complex DR·U is isomorphic to the
positive Koszul complex associated to the sequence (∂z1· , . . . , ∂zn· ) of right DU -linear
endomorphisms of DU . Since ∂z1· is injective and ∂zk· induces an injective endomorphism of DU /∂z1 DU + · · · + ∂zk−1 DU we know that H j (DR·U ) = 0 for j 6= n, H n (DR·U )
being isomorphic to DU /∂z1 DU + · · · + ∂zn DU .
To conclude, we note that for ω = dz 1 ∧ . . . ∧ dz n , the diagram of isomorphisms:
P
↓
P ∼ω
P
7→ P 1
DU /DU ∂z1 + · · · + DU ∂zn −→ OU h
↓
↓
↓
n
1
/∂
∂
hω
DU z DU + · · · + z DU −→ ΩU
P
7→ ω.P
2
is commutative.
op
Proposition 5.2.5 There is a canonical isomorphism of complexes of DX
-modules
∼ Hom· (SP X , D ).
DR·X −→
X
DX
·
Proof: At the level of components, we have
k
Hom DX (SP X
k , DX ) = Hom DX (DX ⊗OX ΘX , DX )
' ΩkX ⊗OX DX .
The isomorphism
X
∼ Hom
DRkX −→
DX (SP k , DX )
sends ω ⊗ P to the DX -linear morphism
DX ⊗OX ΘkX −→ DX
(Q ⊗ θ )
7→
Q(hω, θi P ).
In a local coordinate system, it is easy to check that this induces a morphism of
complexes
DR·X −→ Hom·DX (SP ·X , DX )
2
hence the conclusion.
5.3
The de Rham complex of a DX -module
Definition 5.3.1 The de Rham complex of a left DX -module M is the complex
Ω·X (M) = DR·X ⊗DX M.
22
Jean-Pierre Schneiders
Of course,
ΩkX (M) ' ΩkX ⊗OX M,
and, in a local coordinate system (z : U −→ C
I n ), we have
d (ω ⊗ m) = dω ⊗ m +
k
k
k
n
X
dz j ∧ ω k ⊗ ∂zj m.
j =1
Proposition 5.3.2 For any left DX -module M there is a canonical quasi-isomorphism
·
RHom DX (OX , M) ' ΩX
(M ).
Proof: By Proposition 5.2.5, we have the isomorphism of complex
·
∼ Hom· (SP X , D ).
DRX
−→
X
DX
·
Tensoring with M, we get the isomorphism
·
∼ RHom· (SP X , M).
ΩX
(M) −→
DX
·
in Db(X ). By Proposition 5.2.2, we have
∼
SP X
· −→ OX ,
2
and the proof is complete.
Corollary 5.3.3 In Db (X ), we have the canonical isomorphisms
·
RHom DX (OX , OX ) ' ΩX
'C
I X.
op
Corollary 5.3.4 In Db (DX
), we have the canonical isomorphisms
·
RHom DX (OX , DX ) ' DRX
' ΩX [−n].
In particular, hdDX (OX ) = n.
6
6.1
The characteristic variety
Definition and basic properties
Definition 6.1.1 Through the isomorphism
∼
GDX −→
⊕ π∗OT ∗X (k )
k∈IN
we may identify ΣGDX to a subsheaf of π∗OT ∗X . Using this identification, we define
g of a GD -module GM by the formula:
the analytic localization GM
X
−1
g =O ∗ ⊗
GM
T X π−1 ΣGD X π ΣGM.
An Introduction to D-Modules
23
Proposition 6.1.2 The analytic localization functor
g
GM 7→ GM
is faithful and exact. Moreover, a GDX -module is coherent if and only if its analytic
localization is a coherent OT ∗X -module.
Proof: It is a direct consequence of well-known results of Serre [17].
2
g of a coherent GD -module GM is
Corollary 6.1.3 The annihilating ideal Ann GM
X
g is a closed conic analytic subset
generated by π −1 Ann GM. In particular, supp GM
of T ∗X .
Lemma 6.1.4 (Comparison of filtrations) Let M be a coherent DX -module. Assume that FM and FM0 are two good filtrations on M. Then for any x ∈ X , we may
find a neighborhood U of x and integers d, d0 such that
Fk M|U ⊂ Fk+d M0|U
Fk M0|U ⊂ Fk+d0 M|U
Proof: Since FM is a coherent F DX -module, we may find a neighborhood U of x
and generators m1, . . . , mp of order d1 , . . . , dp of FM on U . After shrinking U , we may
assume that m1, . . . , mp have finite orders d01 , . . . , dp0 with respect to FM0 . Hence, for
k ∈ ZZ, we have
Fk M|U ⊂ Fk+d M0|U
if d = sup(d01 − d1 , . . . , dp0 − dp ). The other part of the result is obtained by reversing
2
the roles of FM and FM0 .
Proposition 6.1.5 Let M be a coherent DX -module. Assume that FM and FM0 are
two good filtrations on M and denote by GM and GM0 the corresponding coherent
GDX -modules. Then
g0
g = supp GM
.
supp GM
Proof: Since the problem is local on X and since the localization functor is shift
invariant, the preceding lemma allows us to assume that there is an integer d ∈ IN0
such that
Fk M0 ⊂ Fk M ⊂ Fk+d M0
for every k ∈ ZZ. We will proceed by increasing induction on d.
For d = 1, we define the auxiliary GD X -modules GL and GN by the formulas
GL = ⊕ Fk M0 /Fk−1 M;
k∈ZZ
GN = ⊕ Fk M/Fk M0.
k∈ZZ
We get the exact sequences
0 −→ GL −→ GM −→ GN −→ 0
0 −→ GN (−1) −→ GM0 −→ GL −→ 0.
24
Jean-Pierre Schneiders
Since GM and GM0 are coherent, so are GL and GN . Applying the localization
functor, we get the exact sequences
g −→ GM
g −→ GN
g −→ 0
0 −→ GL
g0
g −→ GM
g −→ 0.
−→ GL
0 −→ GN
The requested equality is then given by the formula
g0
g = supp GL
g ∪ supp GN
g = supp GM
.
supp GM
For d > 1, we define the auxiliary filtration FM00 by the formula
Fk M00 = Fk M + Fk+1 M0.
This is obviously a good filtration. Moreover, for any k ∈ ZZ, we have
Fk M ⊂ Fk M00 ⊂ Fk+1 M
and
Fk M0 ⊂ Fk M00 ⊂ Fk+1+(d−1) M0 .
By the induction hypothesis, we get
g 00
g0
g = supp GM
.
supp GM
= supp GM
2
and the conclusion follows.
Definition 6.1.6 Since any coherent DX -module admits locally a good filtration, the
preceding proposition shows that there is a unique closed conic analytic subvariety V
of T ∗X such that
g
V ∩ T ∗U = supp GM
for any open subset U of X and any good filtration FM of M on U . We call V the
characteristic variety of M and denote it by charM. For a complex M· ∈ Dbcoh (DX ),
we set
[
charM· =
charHk (M· ).
k∈ZZ
Proposition 6.1.7 Let F M· be a complex of F DX -modules and denote by M· the
underlying complex of DX -modules. Assume that the components of F M· are F DX coherent. Then the components of M· are DX -coherent and
charHk (M·) ⊂ supp Hk (G˜rF M· )
for k ∈ ZZ. In particular,
charM· ⊂ supp G˜rF M·.
An Introduction to D-Modules
25
Proof: Let us denote by F Hk the k -th cohomology group of M· endowed with the
filtration induced by that of F Mk . By construction, we have the exact sequence
0 −→ im F dk−1 −→ ker F dk −→ F Hk −→ 0.
Since both ker F dk and im F dk−1 are F DX -coherent, so is F Hk . By Proposition 10.3.2,
we have the inclusions
im G˜rF dk−1 ⊂ G˜r im F dk−1
G˜r ker F dk ⊂ ker G˜rF dk .
Hence, in a neighborhood of a point of T ∗X ,
im G˜rF dk−1 = ker G˜rF dk
implies
G˜r im F dk−1 = G˜r ker F dk .
This shows that
supp G˜rF Hk ⊂ supp Hk (G˜rF M·)
2
and the conclusion follows easily.
Remark 6.1.8 In general, the inclusion of the preceding proposition is strict. For
example, let F M· be the complex
Fu
F DX (−1) −→ F DX
where F u is the morphism corresponding to the inclusions Fk−1 DX ⊂ Fk DX . Then,
char(M·) = ∅,
but
supp(G˜rF M·) = T ∗X.
Proposition 6.1.9 Assume M is a coherent DX -module. Then
char(RHom DX (M, DX )) = char(M).
Moreover,
codim charE xtjDX (M, DX ) ≥ j
and
E xtjDX (M, DX ) = 0
for j < codim charM.
26
Jean-Pierre Schneiders
Proof: We only need to work locally. Hence, we may assume that M is the underlying
DX -module of a coherent F DX -module FM and that we have a finite resolution
0 −→ FLp −→ · · · −→ FL0 −→ FM −→ 0
of FM by finite free F DX -modules. Forgetting the filtration, the complex
F Hom FDX (FL· , F DX )
is quasi-isomorphic to RHom DX (M, DX ). Moreover,
G rF Hom FDX (FL· , F DX ) ' GHom GDX (G rFL· , GDX ).
Hence, by Proposition 6.1.7, we get
charE xtjDX (M, DX ) ⊂ supp E xtjOT ∗ X (G˜rFM, OT ∗X )
and the conclusion follows by a well-known result of analytic geometry.
6.2
2
Additivity
Proposition 6.2.1 If
0 −→ M0 −→ M −→ M00 −→ 0
is an exact sequence of DX -modules then
charM = charM0 ∪ charM00 .
Proof: Since the problem is local on X , we may assume that M is endowed with a
good filtration FM. This filtration induces good filtrations FM0 and FM00 on M0 and
M00 respectively and the sequence
0 −→ FM0 −→ FM −→ FM00 −→ 0
is strictly exact. Hence, the sequence
0 −→ GM0 −→ GM −→ GM00 −→ 0
is also exact, and, applying the localization functor, we get the exact sequence
g0
g 00
g −→ GM
0 −→ GM
−→ GM
−→ 0.
2
The conclusion follows by taking the supports.
Corollary 6.2.2 Assume
M0 −→ M −→ M00 −→
+1
is a distinguished triangle in Dbcoh (DX ). Then
charM ⊂ charM0 ∪ charM00 .
Proof: This is a direct consequence of the snake’s lemma and the preceding proposition.
2
An Introduction to D-Modules
6.3
27
Involutivity and consequences
For a linear subset L ⊂ Tp T ∗X , we denote by L⊥ the orthogonal of L with respect to
the symplectic form of T ∗X .
An analytic subvariety V of T ∗X is involutive (resp. Lagrangian; isotropic) if for
any smooth point p ∈ V , Tp V ⊥ ⊂ TpV (resp. TpV ⊥ = TpV ; Tp V ⊥ ⊃ TpV ). Note that
for a non empty involutive (resp. Lagrangian, isotropic) analytic subvariety V of T ∗X
we have dim V ≥ dim X (resp. dim V = dim X , dim V ≤ dim X ).
Since TpV ⊥ is generated by the values at p of the Hamiltonian fields of the holomorphic functions which are zero on V near p, we find that V is involutive if and only
if the germ {f, g } is zero on V near p for any germs of holomorphic functions f , g
which are zero on V near p. Hence, a necessary and sufficient condition for an analytic
subvariety of T ∗ X to be involutive is that
{IV , IV } ⊂ IV
where, as usual, IV denotes the defining ideal of V in OT ∗X .
This implies in particular that on the smooth part of an involutive analytic subvariety of T ∗ X , the sub-bundle T V ⊥ of T V satisfies the Frobenius integrability conditions.
The leaves of V are the corresponding maximal integral immersed sub-manifolds. Their
dimension is codimT ∗X V .
Theorem 6.3.1 The characteristic variety char(M) of a coherent DX -module M is a
closed conic involutive analytic subset of T ∗ X .
Proof: Locally, M is the underlying DX -module of a coherent F DX -module F M. By
Proposition 3.3.4, the Poisson bracket defined algebraically using the fact that GD X is
commutative is the same as the usual Poisson bracket of classical mechanics. Hence,
Theorem 10.3.7 shows that
√
√
√
{ Ann G rF M, Ann G rF M} ⊂ Ann G rF M.
√
Since Ann G rF M is the defining ideal of the characteristic variety of M, the proof
2
is complete.
Corollary 6.3.2 Assume M is a non zero coherent DX -module. Then,
(a) dim char(M) ≥ dim X ,
(b) hdDX (M) ≤ dim X ,
(c) glhd(DX ) = dim X .
Proof: Part (a) is an obvious consequence of the involutivity of char(M).
By Proposition 6.1.9, we know that
codim charE xtjDX (M, DX ) ≥ j
28
Jean-Pierre Schneiders
for any positive integer j . For j > dim X , we see that E xtjDX (M, DX ) = 0 since
otherwise we would get a contradiction with the involutivity of charE xtjDX (M, DX ).
Part (b) is then a consequence of Proposition 10.1.2.
From part (b) we already know that
glhd(DX ) ≤ dim X.
To conclude, we note that the equality holds thanks to Corollary 5.3.4.
2
Definition 6.3.3 We call the leaves of char(M) the bicharacteristic leaves of M.
6.4
Definition of holonomic systems
Definition 6.4.1 A DX -module is holonomic if it is coherent and has a Lagrangian
characteristic variety. Between non-zero DX -modules, the holonomic DX -modules are
the ones with a characteristic variety of the smallest possible dimension (i.e. dim X ).
Proposition 6.4.2 Assume M is a coherent DX -module. Then M is holonomic if
and only if
E xtjDX (M, DX ) = 0
6 dim X .
for j =
Proof: Set n = dim X .
Assume M is holonomic. Since
E xtjDX (M, DX ) = 0
for j < codim char(M) and for j > n and codim char(M) = n, the conclusion follows.
Assume now that
E xtjDX (M, DX ) = 0
for j 6= n. Since
codim char(E xtnDX (M, DX )) ≥ n,
op
n
the DX
-module E xtD
(M, DX ) is a holonomic. Moreover, since
X
M ' RHom Dop (E xtnDX (M, DX )[−n], DX ),
X
we see that
n
charM ⊂ charE xtD
(M, DX ),
X
and the proof is complete.
2
An Introduction to D-Modules
7
7.1
29
Inverse Images of D -modules
Inverse image functors and transfer modules
Definition 7.1.1 Let f : X −→ Y be a complex analytic map. We set
DX →Y = OX ⊗f −1OY f −1 DY .
Note that DX →Y is naturally both a left OX -module and a right f −1 DY -module and
these structure are clearly compatible. Let x0 be a point in X and set y0 = f (x0). For
θ ∈ ΘX,x0 , consider the map
θ· : ΘX,x0 ⊗OY,y DY,y0 −→ ΘX,x0 ⊗OY,y DY,y0
0
0
h⊗m
7→
θh ⊗ m +
p
X
hθ(y j ◦ f ) ⊗ ∂yj m
j =1
where (y 1, . . . , y p) is a coordinate system in a neighborhood of y0 in Y . Since this map
is easily checked to be independent of the chosen local coordinate system, we get a
canonical action morphism
ΘX ⊗ DX →Y −→ DX →Y .
A direct computation shows that it satisfies the conditions of Proposition 3.1.3; hence
DX →Y is canonically endowed with a structure of left DX -module compatible with
its structure of right f −1 DY -module. With this structure of bimodule, DX →Y is the
transfer module of f . It has a canonical section
1X →Y = 1X ⊗ f −1 1Y .
Definition 7.1.2 Let f : X −→ Y be a morphism of complex analytic manifolds. We
define the inverse image functor for D-modules
f ∗ : D− (DY ) −→ D− (DX )
by setting
f ∗ (M) = DX →Y ⊗fL−1 DY f −1 M
for any object M in D− (DY ). Note that, in D− (OX ), we have
f ∗ (M) ' OX ⊗fL−1 OY f −1 M.
Proposition 7.1.3 Assume f : X −→ Y , g : Y −→ Z are two morphisms of complex
analytic manifolds and set h = g ◦ f . Then we have a canonical isomorphism of
(DX ,DZop )-bimodules
∼ D
DX →
−→
h
Z
f
X →Y
g
.
⊗fL−1D f −1 DY →
Z
Y
It induces a canonical isomorphism
∼ f ∗ ◦ g∗ .
h∗ −→
30
Jean-Pierre Schneiders
Proof: Forgetting the DX -module structure, the isomorphism is a direct consequence
of the definition of the transfer module. To check that it is an isomorphism of (DX ,DZop )bimodules, we only need to prove its compatibility with the left action of ΘX . This
easily achieved in local coordinates.
2
By the preceding proposition, we may use the graph embedding to reduce the study
of inverse images to the case of a closed embedding or the case of the projection from
a product to one of its factors. Since the problem is local, the following proposition
will clarify the situation.
I n and C
I m respecProposition 7.1.4 Let U , V be two open neighborhoods of 0 in C
tively (m, n 6= 0). Then :
(a) The transfer module of the second projection
p : U ×V
−→ V
(u, v )
7→
v
is given by the formula
DU ×V →V ' {
X
aα(u, v )∂vα : aα (u, v ) ∈ OU ×V }.
α
As a left DU ×V -module it is isomorphic to
DU ×V /DU ×V ∂u1 + · · · + DU ×V ∂un .
It is a coherent module generated by 1U ×V →V . The Koszul complex
K· (DU ×V ; ·∂u1 , · · · , ·∂un )
is a free resolution of length n of this module.
As a right DV -module it is flat but not finitely generated.
b) The transfer module of the closed embedding
i : U −→ U × V
u
7→
(u, 0)
is given by the formula
DU →U ×V = {
X
aαβ (u)∂uα∂vβ : aαβ (u) ∈ OU }
α,β
As a right DU ×V -module, it is isomorphic to
DU ×V /v 1 DU ×V + · · · + v mDU ×V .
An Introduction to D-Modules
31
It is a coherent module generated by 1U →U ×V . The Koszul complex
K· (DU ×V ; v 1·, · · · , v m·)
is a free resolution of length m of this module.
As a left DU -module, it is flat but not finitely generated (it is a countable direct
sum of copies of DU ).
Corollary 7.1.5 For any morphism f : X −→ Y of complex analytic manifolds, the
functor f ∗ has bounded amplitude and induces functors
f ∗ : Db (DY ) −→ Db (DX )
f ∗ : D(DY ) −→ D(DX ).
In general, the inverse image of a coherent D-module is not coherent. It is however
the case for non-characteristic inverse images which we shall now investigate.
7.2
Non-characteristic maps
Consider a map f : X −→ Y between complex analytic manifolds. From a microlocal
point of view (i.e. at the level of cotangent bundles), we get the diagram
T ∗X
ρ
$
X ×Y T ∗ Y
u
πX
w
πY
π
u
X
u
∼
T ∗Y
u
X
u
f
w
Y
where vertical arrows are the canonical projections of the various bundles and
ρ(x; (f (x); ξ )) = (x; f ∗ ξ )
$(x; (f (x); ξ )) = (f (x); ξ ).
The kernel of ρ will be denoted by TX∗ Y . In general, it is a conic complex analytic
subset of X ×Y T ∗Y . If f has locally constant rank, it becomes a holomorphic bundle.
Proposition 7.2.1 Assume that V is a closed conic complex analytic subset of T ∗Y .
Then the following conditions are equivalent :
(i) ρ is proper on $−1 (V ),
(ii) ρ is finite on $−1 (V ),
(iii) TX∗ Y ∩ $−1 (V ) is in the zero section of X ×Y T ∗ Y .
Proof: The equivalence of (i) and (ii) comes from the fact that a compact analytic subset of C
I n has a finite number of points. The equivalence of (i) and (iii) is a consequence
2
of Lemma 7.2.2 below.
32
Jean-Pierre Schneiders
Lemma 7.2.2 Let Γ1 , Γ2 be two closed cones in a real vector bundle p : E −→ X .
Then the map
+ : Γ1 ×X Γ2 −→ Γ1 + Γ2
is proper if and only if
Γ1 ∩ Γ2a ⊂ E0
where E0 is the zero section of E and (Γ2 )a is the antipodal of Γ2 .
Definition 7.2.3 Let f : X −→ Y be a morphism of complex analytic manifolds and
let V be a conic complex analytic subset of T ∗Y . We say that f is non-characteristic
for V if the equivalent conditions of Proposition 7.2.1 hold for f and V . If M is an
object of Dbcoh (DY ) and f is non-characteristic for char(M) then we say, for short, that
f is non characteristic for M.
The meaning of the non-characteristicity condition may be clarified by the following
simple remark.
Remark 7.2.4 (i) An analytic submersion is non-characteristic for any coherent
DY -module.
(ii) Assume Z is a closed analytic hypersurface of X . Then, the embedding i : Z −→
X is non-characteristic for the DX -module DX /DX P associated to an operator P
of DX if and only if σ (P )(z ; ξ ) 6= 0 for any non zero conormal covector ξ ∈ TZ∗ X .
7.3
Non-characteristic inverse images
Let us first investigate a very special but important case.
I n and C
I respectively
Proposition 7.3.1 Let U , V be open neighborhoods of 0 in C
and let P be an operator in Γ(U × V ; DU ×V ). Consider the closed embedding
i : U −→ U × V
u
7→
(u, 0)
and assume i is non-characteristic for DU ×V /DU ×V P . Then up to shrinking U and V ,
P = A0 (u, v )∂vd +
d
X
Ak (u, v, ∂u)∂vd−k
k=1
where A0 is an invertible holomorphic function and Ak is a differential operator of
degree at most k in ∂u . For such a P , the map
DUd −→ DU →U ×V
d−1
⊕ Qk (u, ∂u)
`=0
7→
d
−1
X
`=0
Qk (u, ∂u )∂vk
An Introduction to D-Modules
33
is left DU -linear and induces a quasi-isomorphism between DUd and the complex
·P
DU →U ×V −→ DU →U ×V .
In particular, we have an isomorphism
∼ f ∗ (D
DUd −→
U ×V /DU ×V P ).
Proof: Since i is non-characteristic for DU ×V /DU ×V P ,
σ (P )((u, 0); (0, τ )) 6= 0
for any u ∈ U and any τ ∈ C
I × . Hence, up to shrinking U and V , Weierstrass lemmas
show that
d
X
d
σ (P )((u, v ); (η, τ )) = a0 (u, v )τ +
ak (u, v, η )τ d−k
k=1
where a0 is an invertible holomorphic function and ak is a holomorphic function which
is homogeneous of degree k in η . The first part of the proposition follows easily. The
second part is obvious and so is the third one since it is clear that any operator S of
order s in Γ(U × V ; DU ×V ) may be written as
S =Q·P +R
2
where R is of degree strictly lower than d in ∂v .
In the general case, we have the following result.
Theorem 7.3.2 Let f : X −→ Y be a morphism of complex analytic manifolds and
let N be a coherent DY -module. Assume f is non-characteristic for N . Then
(i) f ∗ N is a coherent DX -module,
(ii) char(f ∗N ) ⊂ ρ$−1 (char(N )).
Proof: Using the factorization of f through its graph embedding, we may restrict
ourselves to the special cases where f is a closed embeddings or a projection from a
product to one of its factors. Since the problem is local on X , we may even assume f
is the embedding (case a)
i : U −→ U × V
u
7→
(u, 0)
or the projection (case b)
p:U ×V
(u, v )
−→ U
7→
u
34
Jean-Pierre Schneiders
where U , V are open neighborhoods of 0 in C
I n and C
I m respectively. Finally, by
factorizing i and p we may assume m = 1.
Let us prove part (i) of the theorem. In case (b), it is a direct consequence of
Proposition 7.1.4 so we only have to consider case (a).
Let s be a section of N . Since DU ×V s ⊂ N it is clear from our hypothesis that
TU∗ U × V ∩ char(DU ×V s) ⊂ TU∗ ×V U × V
Let us denote FI the annihilating ideal of s in DU ×V endowed with the order filtration.
Since N is coherent, we know that char(DU ×V s) is the zero variety of the OT ∗U ×V -ideal
generated by G rFI . Hence, up to shrinking U and V , we may assume that there is an
operator P in Γ(U × V ; DX ) such that
σ (P )((u, 0); (0, τ )) 6= 0
for some τ ∈ C
I × . For such a P , i is non-characteristic for DU ×V /DU ×V P .
We will now prove that i∗ N is concentrated in degree zero. By definition, as an
OU -modules
i∗ N = OU ⊗iL−1 OU ×V i−1N .
since the sequence
·v
0 −→ i−1 OU ×V −→ i−1 OU ×V −→ OU −→ 0
is exact, all we have to prove is that the kernel of the morphism
v·
N|U −→ N|U
is zero. If it is not the case, up to shrinking U and V , we would have a non zero section
s of N such that v · s = 0. Let P be the operator associated to s by the construction
of the preceding paragraph. Since v · s = 0 and P · s = 0 we get by induction
adnv (P )s = 0
for any n ∈ IN0. From Proposition 7.3.1 it follows that addv (P ) = d!A0 (u, v ). Hence s
is zero and we get a contradiction.
To prove that i∗ N is DU ×V -coherent, we will proceed as follows. Up to shrinking
U and V , we may assume N is generated by a finite number of sections s1 , . . . , sN .
Denote by P1 , . . . , PN the non-characteristic operators associated to these sections by
the above procedure. By construction, we have an epimorphism
N
⊕ DU ×V /DU ×V Pk −→ N −→ 0.
k=1
Let us denote by N 0 its kernel. Applying the inverse image functor, we get the exact
sequence
N
i∗N 0 −→ ⊕ i∗DU ×V /DU ×V Pk −→ i∗N −→ 0.
k=1
An Introduction to D-Modules
35
By construction i is non-characteristic for each Pk . Hence, by Proposition 7.3.1, the
middle term is a finite free DU -module and i∗N is of finite type. Since i is noncharacteristic for N 0 , i∗ N 0 is also of finite type and the conclusion follows.
Let us now prove part (ii) of the theorem.
Up to shrinking U nd V , we may assume N is endowed with a good filtration F N .
Let us denote by f ∗ F N the F DX -module obtained by filtering OX ⊗f −1 OY f −1 FN by
the images of the sheaves OX ⊗f −1 OY f −1 Fk N . From the exact sequence
0 −→ Fk−1 N −→ Fk N −→ G rk F N −→ 0
we get the exact sequence
OX ⊗f −1 OY f −1 Fk−1 N −→ OX ⊗f −1 OY f −1 Fk N −→ OX ⊗f −1 OY f −1 G rk F N −→ 0
Hence we have a canonical epimorphism of G rF DX -module
OX ⊗f −1 OY f −1 G rF N −→ G rf ∗ F N −→ 0.
Since ρ is finite on supp G˜rF N , we get a canonical epimorphism
ρ∗ $∗ G˜rF N −→ G˜rf ∗ F N −→ 0
2
and the conclusion follows.
Remark 7.3.3 In fact, the equality holds in part (ii) of the preceding proposition. To
prove this fact, we would need the theory of characteristic cycles which is not developed
here.
7.4
Holomorphic solutions of non-characteristic inverse images
In this section, following Kashiwara, we will obtain a wide generalization of the CauchyKowalewsky theorem in the form of a formula for the holomorphic solutions of non
characteristic inverse images of D-modules.
First, we recall the classical Cauchy-Kowalewsky theorem for one operator.
Proposition 7.4.1 Let U , V be two open neighborhoods of 0 in C
I n and C
I respectively
and let P be an operator of degree d in Γ(U × V ; DU ×V ). Consider the embedding
i : U −→ U × V
u
7→
(u, 0)
and assume that
σ (P )((u, 0); (0, τ )) 6= 0
I ×. Then for any g ∈ Γ(U × V ; OU ×V ) and any g0 , . . . , gn ∈ Γ(U ; OU ), there is
for τ ∈ C
an open neighborhood W of U in U × V such that the Cauchy problem
(
Pf = g
◦ i = gk
(∂vk f )
has a unique holomorphic solution in W .
; k = 0,. . . , d − 1
36
Jean-Pierre Schneiders
Next, we consider the general case.
Definition 7.4.2 Let f : X −→ Y be a morphism of complex analytic manifolds. For
any pair of DY -modules N , N 0 , we have an obvious canonical morphism
f −1 RHom DY (N , N 0 ) −→ RHom DX (f ∗ N , f ∗ N 0 ).
Since we have
f ∗ OY ' OX ,
we get a canonical restriction morphism
f −1 RHom DY (N , OY ) −→ RHom DX (f ∗N , OX ).
Theorem 7.4.3 Let f : X −→ Y be a morphism of complex analytic manifolds and
let N be a coherent DY -module. Assume f is non-characteristic for N . Then the
canonical restriction morphism
f −1 RHom DY (N , OY ) −→ RHom DX (f ∗N , OX )
is an isomorphism in Db (X ).
Proof: As in the proof of Theorem 7.3.2, we may assume f is (case a) the embedding
i : U −→ U × V
u
7→
(u, 0)
or (case b) the projection
p:U ×V
(u, v )
−→ U
7→
u
where U , V are open neighborhoods of 0 in C
I n and C
I respectively.
For case (a), an iteration of the procedure used in the proof of Theorem 7.3.2 shows
that N has a projective resolution R· by finite sums of DU ×V -module of the type
DU ×V /DU ×V P where i is non-characteristic for the operator P . Hence, we may assume
from the beginning that N is of this special kind. On one hand, Proposition 7.3.1
shows that
∼ i∗ D
DUd −→
U ×V /DU ×V P,
and we get
∼ Od .
RHom DU (i∗ DU ×V /DU ×V P, OU ) −→
U
On the other hand,
i−1 RHom DU ×V (DU ×V /DU ×V P, OU ×V )
is represented by the complex
P|U ·
OU ×V |U −→ OU ×V |U
An Introduction to D-Modules
37
which, by Proposition 7.4.1, is quasi-isomorphic to ker(P|U ·). With the preceding
explicit representations, the restriction map becomes
ker(P|U ·) −→ OUd
h
7→
d−1
⊕ (∂vl h) ◦ i
l=0
and the conclusion follows from Proposition 7.4.1.
In case (b), we may assume N = DU . Then
p−1 RHom DU (N , OU ) ' p−1 OU .
Moreover, since
p∗DU ' DU ×V →V ,
Proposition 7.1.4 shows that
RHom DU ×V (p∗ DU , OU ×V )
is represented by the complex
∂
v
OU ×V −→
OU ×V
which is canonically quasi-isomorphic to p−1 OU . Using these explicit representations,
2
the restriction map becomes the identity on p−1 OU and the proof is complete.
8
Direct images of D -modules
In this section, it is convenient to work with right D-modules. The reader will easily
obtain the corresponding results for left D-modules by applying the side changing
functors.
8.1
Direct image functors
Definition 8.1.1 Let f : X −→ Y be a morphism of complex analytic manifolds.
Since the functors (·) ⊗DL DX →Y and Rf! (·) have bounded amplitude they define by
X
composition a functor
op
f ! : D(DX
) −→ D(DYop )
such that
f ! (M) = Rf! (M ⊗DLX DX →Y ).
op
for M in D(DX
). This is the direct image functor (with proper support) for right
DX -modules. It has bounded amplitude, and induces a functor
op
f ! : Db(DX
) −→ Db (DYop ).
The direct image functor is compatible with the composition of morphisms.
38
Jean-Pierre Schneiders
Proposition 8.1.2 Let f : X −→ Y , g : Y −→ Z be two morphisms of complex
analytic manifolds and set h = g ◦ f . Then we have the canonical isomorphism
∼ g ◦f .
h! −→
!
!
Proof: One has successively :
g ! ◦ f ! (M)
=
Rg! (Rf! (M ⊗DLX DX →Y ) ⊗DLY DY →Z )
∼
−→
Rg! Rf! (M ⊗DLX DX →Y ⊗fL−1 DY f −1 DY →Z )
∼
−→
Rh! (M ⊗DL DX →Z ),
X
where the last isomorphism comes from the first part of Proposition 7.1.3.
2
In general, the direct image of a coherent D-module is not coherent. This is however
the case for proper direct images provided that M is good as will be shown in the next
section.
8.2
Proper direct images
Definition 8.2.1 Let X be a complex analytic manifold.
A right coherent DX -module is generated by a coherent O-module if M has a coherent OX -submodule G for which the natural morphism
G ⊗OX DX −→ M
is an epimorphism.
A right coherent DX -module is good if, in a neighborhood of any compact subset
K of X , it has a finite filtration (Mk )k=1,...,N by right coherent DX -submodules such
that each quotient Mk /Mk−1 is generated by a coherent OX -module.
Good DX -modules form a thick subcategory of Coh(DX ) denoted by Good(DX ).
The associated full triangulated subcategory of Db (DX ) is denoted by Dbgood(DX ).
Theorem 8.2.2 Let f : X −→ Y be a morphism of complex analytic manifolds.
op
Assume M ∈ Dbgood(DX
) and f is proper on supp(M). Then
(i) f ! M is in Dbgood(DYop ),
(ii) char(f ! M) ⊂ $ρ−1 (charM).
Proof: Since the problem is local on Y and f is proper on supp M, we only need to
work in a neighborhood of a compact subset of X . Hence, we may assume that M
has a finite filtration (Mk )k=1,...,N by coherent DX -modules such that each quotient
Mk /Mk−1 is generated by a coherent OX -module. It is clear that the theorem will be
true for M if it is true for each quotient Mk /Mk−1 . Hence, we may assume from the
beginning that M is generated by a coherent OX -module G . We endow M with the
filtration induced by the order filtration of DX through the epimorphism
G ⊗OX DX −→ M
An Introduction to D-Modules
39
and denote by F M the resulting F DX -module.
For any coherent F DX -module F N and any compact subset K of X , there is an
integer k such that, in a neighborhood of K , the natural morphism
⊕ Fd N ⊗OL F DX (−d) −→ N
d≤k
X
becomes a strict epimorphism. Thus, we may assume that F M has a filtered projective
resolution F P · such that
F P k = ⊕ Gdk ⊗OX F DX (−d)
d∈Ik
where Ik is a finite subset of ZZ and Gdk is a coherent OX -module with f -proper support.
We denote by P · the underlying complex of DX -module.
Let R· be a bounded f -soft resolution of C
I X by sheaves of C
I -vector spaces. Since
j
k
R ⊗ Gd is f -soft, the simple complex associated to
S ·,· = f! (R· ⊗ P · ⊗DX DX →Y )
is quasi-isomorphic to f ! (R· ⊗ P · ) ' f ! M. Let us endow the DY -module
S j,k = ⊕ f! (Rj ⊗ Gdk ) ⊗OY DY
d∈Ik
with the structure of right F DY -module defined by setting
F S j,k = ⊕ f! (Rj ⊗ Gdk ) ⊗OY F DY (−d).
d∈Ik
A simple computation shows that the differentials of S ·,· are then filtered morphisms.
We denote by F S ·,· the corresponding double complex of F DY -modules and by F S ·
its associated simple complex.
By Grauert’s coherence theorem for proper direct images, Rf! (Gdk ) is quasi-isomorphic to a bounded complex of coherent OY -modules. Hence, F S · is quasi-isomorphic
to a bounded complex of coherent F DY -modules. Moreover, one checks easily that
G˜rF S · ' $∗ ρ−1 G˜rF R· .
From this formula, it follows that
char(S · ) ⊂ supp G˜rF R·
and Proposition 6.1.7 allows us to conclude.
8.3
2
The D-linear integration morphism
Let X and Y be complex analytic manifolds of complex dimension dX and dY respectively. In this section, we will associate to any morphism f : X −→ Y of complex
analytic manifolds a canonical integration morphism
f ! ΩX [dX ] −→ ΩY [dY ]
40
Jean-Pierre Schneiders
in Db(DYop ). This map will play for direct images a role which is similar to the role of
the restriction morphism
f ∗ OY −→ OX
in the study of inverse images.
p,q
Let X be a complex manifold. In the sequel, we denote by Dbp,q
X (resp. FX )
the sheaf of forms of type (p,q ) with distributions (resp. differentiable functions) as
·
coefficients. We also denote by DbX
(resp. FX· ) the associated Dolbeault complex with
differential d = ∂ + ∂ .
Let f : X −→ Y be a morphism of complex analytic manifolds. Recall that we
have a pull-back morphism
f ∗ : f −1 FYp,q −→ FXp,q
and a push-forward morphism
f! : f! DbpX+dX ,q +dX −→ DbpY+dY ,q +dY .
Both are compatible with the differentials. Moreover, if g : Y −→ Z is another
morphism of complex analytic manifolds, we have (g ◦ f )∗ = f ∗ ◦ g ∗ and (g ◦ f )! = g! ◦ f! .
The two morphisms are linked by the formula
(f! u) ∧ ω = f! (u ∧ f ∗ ω ).
Definition 8.3.1 Let M be a left DX -module. We set
p,q
Dbp,q
X (M) = DbX ⊗OX M.
The differentials
p+1,q
(M )
∂ p,q : Dbp,q
X (M) −→ DbX
u⊗P
→
7
∂ p,q u ⊗ P +
x
X
dzi ∧ u ⊗ Dzi P
i=1
∂
p,q
p,q +1
: Dbp,q
(M )
X (M) −→ DbX
p,q
u ⊗ P −→ ∂ u ⊗ P
·
do not depend on the local coordinate system (z : U −→ C
I n ). We denote DbX
(M )
·, ·
the simple complex associated to the bi-graded sheaf DbX
(M) and the differential
d = ∂ + ∂ . We call it the distributional de Rham complex of M.
·
Lemma 8.3.2 The differential of DbX
(DX ) is compatible with the right DX -module
op
b
structure of its components, and, in D (DX
), one has a canonical isomorphism :
·
∼ Ω .
D bX
(DX )[dX ] −→
X
·
Proof: The compatibility of the differential of DbX
(DX ) with the right DX -module
structure of its components is a direct consequence of the local forms of ∂ and ∂ .
An Introduction to D-Modules
41
Since DX is flat over OX , the Dolbeault quasi-isomorphism
∼ Dbp,·
ΩpX −→
X
induces the quasi-isomorphisms
p,·
∼ Dbp,· ⊗
∼
ΩpX ⊗OX DX −→
X
OX DX −→ DbX (DX ).
Hence, Weil’s lemma shows that the natural morphism
∼ Db· (D )
DR·X (DX ) −→
X
X
from the holomorphic to the distributional de Rham complex of DX is a quasi-isomorphism of complexes of right DX -modules and the conclusion follows from the quasi-isomorphism
DR·X (DX ) ' ΩX [−dX ].
2
of Proposition 5.2.4.
Lemma 8.3.3 To any morphism f : X −→ Y of analytic manifolds is associated a
canonical integration morphism
f! : f! Db·X (DX →Y )[2dX ] −→ Db·Y (DY )[2dY ].
in the category of bounded complexes of right DY -modules.
Proof: At the level of components the integration morphism is obtained as the following chain of morphisms :
∼
f! DbpX+dX ,q+dX (DX →Y ) −→
f!(DbpX+dX ,q+dX ⊗f −1OY f −1 DY )
∼
−→
f!DbpX+dX ,q+dX ⊗OY DY
−→ DbpY+dY ,q +dY ⊗OY DY
∼
−→
DbpY+dY ,q +dY (DY ).
To get the second morphism one has used the projection formula, the fact that DbX is
a soft sheaf and the fact that DY is locally free over OY . The third arrow is deduced
from the push-forward of distributions along f .
To conclude, we need to show that the integration morphism is compatible with the
differentials of the complexes involved. Thanks to the local forms of the differentials,
2
this is an easy computational verification and we leave it to the reader.
Lemma 8.3.4 Let f : X −→ Y and g : Y −→ Z be morphisms of complex analytic
manifolds. One has the following commutative diagram:
1
g! (f! Db·X (DX →Y )[2
d ] ⊗DY DY →Z ) −→ g! (Db·Y (DY )[2dY ] ⊗DY DY →Z )
 X

y2
g! f!Db·X (DX →Z )[2dX ]

y3
4
−→
Db·Z (DZ )[2dZ ].
In this diagram, arrow (1) is deduced from f! by tensor product and proper direct
image, arrow (2) is an isomorphism deduced from the projection formula, arrow (3) is
g! and arrow (4) is equal to (g ◦ f )! .
42
Jean-Pierre Schneiders
Proof: Going back to the definition of the various morphisms, one sees easily that the
commutativity of the preceding diagram is a consequence of the Fubini theorem for
distributions, that is, the formula
(g ◦ f )! (u) = g! (f! (u))
where g∗ and f∗ denotes the push-forward of distributions along g and f respectively,
u being a distribution with g ◦ f proper support.
2
Proposition 8.3.5 To any morphism f : X −→ Y of complex analytic manifolds is
associated a canonical integration arrow
R
f
: f ! (ΩX [dX ]) −→ ΩY [dY ]
in Db(DYop ). Moreover, if g : Y −→ Z is a second morphism of complex analytic
manifolds then
R
R
R
g ◦f = g ◦ g ! ( f )
Proof: One gets the arrow
R
f
by composing the morphisms :
∼
f ! (ΩX [dX ]) −→
Rf! (ΩX [dX ] ⊗DLX DX →Y )
·
∼
(DX →Y )[2dX ])
Rf! (DbX
−→
·
∼
(DX →Y )[2dX ])
f!(DbX
−→
−→ DbY· (DY )[2dY ]
∼
ΩY [dY ]
−→
Let us point out that the second and last isomorphisms come from Lemma 8.3.2, that
the third one is deduced from the fact that Dbp,q
X (DX →Y ) is c-soft and that the fourth
arrow is given by Lemma 8.3.3.
The compatibility of integration with composition is then a direct consequence of
Lemma 8.3.4.
2
Corollary 8.3.6 To any morphism f : X −→ Y of complex analytic manifolds and
op
any M ∈ Db (DX
) is associated a morphism
Rf∗ RHom DX (M, ΩX [dX ]) −→ RHom DY (f ! M, ΩY [dY ])
which is functorial in M and compatible with composition in f .
op
Proof: For any M, N ∈ Db(DX
), we have a canonical
Rf∗ RHom DX (M, N ) −→ RHom DY (f ! M, f ! N ).
which is functorial both in M and in N and which is compatible with composition in
f . To end the construction, we compose the preceding morphism for N = ΩX [dX ] with
the morphism
RHom DY (f ! M, f ! ΩX [dX ]) −→ RHom DY (f ! M, ΩY [dY ])
deduced from the integration morphism.
2
An Introduction to D-Modules
43
Proposition 8.3.7 Let f : X −→ Y be a morphism of complex analytic manifolds.
The canonical section 1X →Y of DX →Y induces a morphism
Rf! ΩX [dX ] −→ f ! ΩX [dX ]
which, by composition with the D-linear integration morphism
f ! ΩX [dX ] −→ ΩY [dY ],
gives the O-linear integration morphism
Rf! ΩX [dX ] −→ ΩY [dY ].
Proof: Since the O-linear integration morphism is deduced from the push-forward
f! : f ! DbdX ,· [dX ] −→ DbdY ,· [dY ]
through the Dolbeault quasi-isomorphisms
∼ DbdX ,·
ΩX −→
;
∼ DbdY ,· ,
ΩY −→
the conclusion follows easily from the explicit construction of the D-linear integration
morphism.
2
Corollary 8.3.8 Let f : X −→ Y be a morphism of complex analytic manifolds.
Assume G is an object of Db (OX ). Then, we have the commutative diagram
Rf∗ RHom DX (G ⊗OX DX , ΩX [dX ])
w
RHom DY (f ! (G ⊗OX DX ), ΩY [dY ])
u
u
Rf∗ RHom OX (G , ΩX [dX ])
w
RHom OY (Rf! (G ), ΩY [dY ])
where the horizontal arrows are deduced from the D-linear and O-linear integration
morphisms, the vertical ones being canonical isomorphisms.
8.4
Holomorphic solutions of proper direct images
Proposition 8.4.1 Let f : X −→ Y be a morphism of complex analytic manifolds.
op
Assume M is an object of Dbgood(DX
) with f -proper support. Then the canonical
integration morphism
Rf∗ RHom DX (M, ΩX [dX ]) −→ RHom DY (f ! M, ΩY [dY ])
is an isomorphism in Db (Y ).
Proof: Working as in Theorem 8.2.2, we may assume that
M = G ⊗OX DX
where G is a coherent OX -module with f -proper support. Hence, we are reduced to
prove that
Rf∗ RHom DX (G ⊗OX DX , ΩX [dX ]) −→ RHom DY (f ! (G ⊗OX DX ), ΩY [dY ])
is an isomorphism. Thanks to Corollary 8.3.8, this is nothing but the well-known
duality theorem of analytic geometry.
2
44
9
9.1
Jean-Pierre Schneiders
Non-singular D -modules
Definition
Set
T˙ ∗X = T ∗X \ TX∗ X
and denote by
p : T˙ ∗X −→ P ∗ X
the canonical projection to the cotangent projective bundle. Denote by
q : P ∗ X −→ X
the projection to the base manifold. For any coherent DX -module M, char(M) is a
conic analytic subvariety of T ∗ X . Therefore, it is possible to find an analytic subvariety
V ⊂ P ∗ X such that p−1 (V ) = char(M) ∩ T˙ ∗X . Since q is proper, q (V ) is an analytic
subvariety of X . This is the singular support of M; we denote it by sing(M). This is
obviously a subset of supp(M).
A non singular DX -module is a coherent DX -module M such that sing(M) = ∅. In
other words, a coherent DX -module M is non-singular if and only if char(M) ⊂ TX∗ X .
9.2
Linear holomorphic connections
To refresh the reader’s memory, we start with an informal survey of the basic definitions
in the theory of linear connections.
Let p : E −→ X be a holomorphic vector bundle on the complex analytic manifold
X . Here, we will identify a (linear) holomorphic connection on E with its covariant
differential
∇ : E −→ T ∗X ⊗ E
which is a first order differential operator such that
∇(he) = dh ⊗ e + h∇e
for any holomorphic function h and any holomorphic section e of E . A holomorphic
section e of E is horizontal if ∇e = 0. For any holomorphic vector field θ, we define
the covariant derivative of a holomorphic section e of E along θ by the formula
∇θ e = hθ, ∇ei .
Note that
∇θ (he) = Lθ h ⊗ e + h∇θ e
for any holomorphic function h and any holomorphic section e of E . Hence, for any
θ, ψ ∈ ΘX , the operator
[∇θ , ∇ψ ] − ∇[θ,ψ] : E −→ E
An Introduction to D-Modules
45
has order 0. Since it is also OX -linear and antisymmetric in θ and ψ , it defines a vector
bundle homomorphism
R∇ : E −→
2
^
T ∗X ⊗ E.
This is the curvature of ∇. A flat holomorphic connection is a connection with 0
curvature (i.e. R∇ = 0). The covariant differential has a unique extension
∇p :
p
^
T ∗X ⊗ E −→
p^
+1
T ∗X ⊗ E
such that
∇p(ω ⊗ e) = dω ⊗ e + (−1)pω ∧ ∇e
for any holomorphic p-form ω and any holomorphic section e of E . A simple computation shows that
∇p+1 ◦ ∇p(ω ⊗ e) = (−1)p ω ∧ R∇ (e).
The dual of a holomorphic connection ∇ on E is the only holomorphic connection ∇∗
on E ∗ such that
d he∗, ei = h∇∗e∗, ei + he∗, ∇ei
for any holomorphic section e∗ , e of E ∗ and E respectively. A simple computation
shows that
R∇∗ = −R∗∇ .
9.3
Non-singular D-modules and flat holomorphic connections
Definition 9.3.1 Let E be a holomorphic vector bundle on X endowed with a flat
holomorphic connection ∇. Since R∇ = 0,
∇[θ,ψ] = [∇θ , ∇ψ ]
for any holomorphic vector fields θ , ψ . Therefore, by Proposition 3.1.3 there is a unique
structure of left DX -module on the sheaf E of holomorphic sections of E which extends
its structure of OX -module in such a way that
θ · e = ∇θ e
for any θ ∈ ΘX and e ∈ E . We denote E∇ the corresponding DX -module.
Proposition 9.3.2 Let E be a holomorphic vector bundle endowed with a flat holomorphic connection ∇. Then
(i) E∇ is a coherent DX -module,
(ii) charE∇ ⊂ TX∗ X ,
(iii) Ω· (E∇ ) is the complex
∇
∇1
∇2
1
2
⊗OX E −→ ΩX
⊗OX E −→ · · · −→ ΩX ⊗OX E −→ 0.
0 −→ E −→ ΩX
46
Jean-Pierre Schneiders
Proof: We know by Proposition 4.1.3 that
(DX ⊗OX E∇ ) ⊗DX OX ' DX ⊗DX (E∇ ⊗OX OX ).
Therefore,
(DX ⊗OX E∇ ) ⊗DX SP X
·
is a resolution of E∇ by left DX -modules. Since E is a locally free OX -module, the
components of this resolution are locally free DX -modules. Hence, E∇ is coherent and
the filtration
(
E∇ k ≥ 0
Fk E∇ =
0 k<0
on E∇ is good. Since G˜rF E∇ is supported by TX∗ X , it follows that char(E∇ ) ⊂ TX∗ X .
2
Part (iii) being a direct consequence of the definitions, the proof is complete.
Proposition 9.3.3 Let E be a holomorphic vector bundle on X endowed with a flat
holomorphic connection ∇. Then, as DX -modules, we have
∗
Hom OX (E∇ , OX ) ' E∇
∗.
Hence, in Db (X ),
∗
RHom DX (E∇ , OX ) ' Ω· (E∇
∗ ).
Proof: The first isomorphism is clear. To get the second one, we note that
RHom DX (OX ⊗OX E∇ , OX ) ' RHom DX (OX , Hom OX (E∇ , OX )).
2
Corollary 9.3.4 Let E be a holomorphic vector bundle on X endowed with a flat
∗
holomorphic connection ∇. Then the DX -module E∇
∗ represents the homogeneous
system
∇u = 0
defining the horizontal sections of E .
Proposition 9.3.5 For a coherent DX -module M, the following conditions are equivalent:
(a) M is non-singular,
(b) charM ⊂ TX∗ X ,
(c) M is OX -coherent,
(d) M is the DX -module associated to a flat holomorphic connection.
An Introduction to D-Modules
47
Proof: By definition, (a) and (b) are equivalent.
(b) ⇒ (c). The problem being local on X , we may also assume that M is endowed
with a good filtration FM and that X is an open subset of C
I n . Since
char(M) ⊂ TX∗ X,
we may find integers α1 , . . . , αn such that
ξkαk GM = 0
for k = 1, . . . , n. Let x be a point in X . Since GM is a coherent GDX -module, we may
find a neighborhood V of x and generators m1 , . . . , mp of order d1 , . . . , dp of GM|V .
Choose
n
X
k ≥ sup(d1 , . . . , dp ) +
αk
k=1
and let m be a germ of order ` of GM at y ∈ V . By construction, we may find germs
s1, . . . , sp of order ` − d1 , . . . , ` − dp of GD X such that
m = s1 m 1 + · · · + s p m p .
The s`’s may be written as
s` =
X
aβ` ξ β
|β |=`−di
with aβ` ∈ OX . In each term of this sum we have βj ≥ αj for at least a j ∈ {1, . . . , n}.
From this fact we deduce that the s` ’s annihilate GM; hence m = 0. By the preceding
discussion we know that locally Gk M = 0 for k 0. This means that the filtration
FM is locally stationary and thus M is OX coherent.
(c) ⇒ (d). Note that it is sufficient to prove that M is a locally free OX -module.
As a matter of fact, M is then the sheaf of holomorphic section of a holomorphic vector
bundle E on X and the action
θ· : M −→ M
of a holomorphic vector field θ ∈ ΘX may clearly be interpreted as the covariant
derivative along θ associated to a flat holomorphic connection ∇ on E .
Since the problem is local on X , we may assume X is an open neighborhood of 0
in C
I n and prove only that M is a locally free OX -module near 0.
Let us choose m1, . . . , mp ∈ M0 such that the associated classes [m1], . . . , [mp]
form a basis of the finite dimensional vector space
M0 /(z 1 , . . . , z n )M0.
By Nakayama’s lemma, m1, . . . , mp generate M0 on (OX )0 .
Let us define the order of a relation
p
X
k=1
bk mk = 0
(b 1 , . . . , b p ∈ (O X ) 0 )
48
Jean-Pierre Schneiders
as the minimum of the vanishing orders of the bi ’s at 0.
Suppose we have a non trivial relation
p
X
bk m k = 0
k=1
of order δ . By construction, we have also
∂z ` mk =
p
X
ar`k mr
r =1
for some
ar`k
∈ (OX )0. Applying ∂z` to the relation we started with gives us the relation
p
X
(∂ z ` br +
r =1
p
X
ar`k bk )mr = 0
k=1
which is clearly of order δ − 1 for a suitable `.
Hence, if we can find a non trivial relation of order δ we can also find one which
has order 0. Such a non trivial relation is impossible since by construction [m1 ], . . . ,
[mp] form a basis.
2
9.4
Local systems
Here, by a local system on X we mean a locally constant sheaf of C
I -vector spaces with
finite dimensional fibers. Recall that a locally constant sheaf on [0, 1]n is constant. In
particular, if γ : [0, 1] −→ X is a path in X from x0 to x1 , the sheaf γ −1 L is constant
on [0, 1]. Therefore, we have a canonical monodromy isomorphism
∼ L .
mγ : Lx0 −→
x1
Moreover, for any path γ 0 joining x1 to x2 , we get
mγ 0 γ = mγ 0 ◦ mγ .
Let
h : [0, 1]2 −→ X
be a homotopy between two paths γ0 , γ1 joining x0 to x1 . Since the sheaf h−1 L is
constant on [0, 1]2 , we get
mγ0 = mγ1 .
Recall that the Poincaré groupoid of X is the category πX which has the points of X
as objects; a morphism from x0 to x1 being a homotopy class of paths from x0 to x1 .
The preceding discussion shows that a local system L on X induces a functor from the
Poincaré groupoid to the category of finite dimensional C
I -vector spaces. It is easily
checked that this process induces an equivalence between the corresponding categories.
Let us fix a point x0 in X . Since
Hom πX (x0, x0) = π1(X, x0 ),
we also get an equivalence between the category of local systems on X and the category
of finite dimensional C
I -linear representations of the Poincaré group π1 (X, x0 ) of X .
An Introduction to D-Modules
9.5
49
Non-singular D-modules and local systems
We will now establish a link between local systems and non-singular D-modules.
Proposition 9.5.1 Let X be a complex analytic manifold and let M be a coherent
DX -module. Assume M is non-singular. Then, locally,
p
M ' OX
(p ∈ IN)
as a left DX -module.
Proof: Let x be a point of X and denote by i : {x} −→ X the canonical embedding.
Since M is non-singular, char(M) ⊂ TX∗ X . Therefore, i is non-characteristic for M
and, by Theorem 7.3.2, i∗ M is a coherent D{x} -module. Since D{x} is nothing but the
field of complex numbers, there is an integer p ∈ IN and an isomorphism
∼ C
α : i∗M −→
I p.
By the Cauchy-Kowalewsky-Kashiwara theorem,
p
RHom DX (M, OX
)x ' Hom(i∗ M, C
I p ).
Therefore, in a neighborhood of x, we can find a morphism of DX -modules
p
β : M −→ OX
such that α = i∗ β . Denote N0 is kernel and N1 its cokernel. Both are non-singular
DX -modules and coherent OX -modules. Since i∗ N1 is the cokernel of α, we have
i∗ N1 = 0
and Nakayama’s lemma shows that N1 = 0 in a neighborhood of x. Since the sequence
0 −→ i∗ N0 −→ i∗M −→ C
I p −→ 0
α
is exact, i∗ N0 = 0 and N0 = 0 in a neighborhood of x. Combining the preceding
2
results, we see that β is an isomorphism in some neighborhood of x.
Remark 9.5.2 In the language of connections, the preceding result means that any
holomorphic vector bundle E endowed with a flat holomorphic connection has horizontal local frames.
Proposition 9.5.3 Let X be a complex analytic manifold. Assume M is a nonsingular coherent DX -module. Then,
RHom DX (M, OX ) ' L
where L is a local system on X . Moreover, the functor
M 7→ Hom DX (M, OX )
induces an equivalence between the category of non-singular coherent DX -modules and
the category of local systems on X with a quasi-inverse given by the functor
L 7→ Hom(L, OX ).
50
Jean-Pierre Schneiders
Proof: We know that, locally,
p
M ' OX
.
Therefore, locally,
p
RHom DX (M, OX ) ' RHom DX (OX
, OX ) ' C
Ip
and the first part of the proposition is clear. The canonical map
M −→ Hom(Hom DX (M, OX ), OX )
is obviously an isomorphism for M = OX ; therefore it is also an isomorphism for any
non-singular DX -module M. In the same way, we see that the canonical map
L −→ Hom DX (Hom(L, OX ), OX ),
is an isomorphism for local systems on X since it is so for L = C
I X.
10
2
Appendix
To split the difficulties, we first recall some basic results of linear algebra. Then, we
develop filtered algebra, and finally we switch to its sheaf theoretical counterpart.
10.1
Linear algebra
Let A be a ring with unit. Without explicit mention of the contrary, an A-module is
a left A-module. With this convention, we identify Aop -modules and right A-modules.
10.1.1
Homological dimension
The homological dimension of an A-module M is the smallest integer n ∈ IN such that
one of the following equivalent conditions is satisfied.
(a) There is a resolution
0 −→ Pn −→ Pn−1 −→ · · · −→ P0 −→ M −→ 0
of M by projective A-modules.
(b) For any A-module N
ExtjA (M, N ) = 0
for j > n.
We denote it by hdA (M ).
The global homological dimension of the ring A to the smallest integer n ∈ IN such
that one of the following equivalent conditions is satisfied
An Introduction to D-Modules
51
(a) Any A-module M has a projective resolution
0 −→ Pn −→ Pn−1 −→ · · · −→ P0 −→ M −→ 0
of length n.
(b) Any A-module N has an injective resolution
0 −→ N −→ I0 −→ I1 −→ · · · −→ In −→ 0
of length n.
(c) For any A-modules M , N we have
ExtjA (M, N ) = 0
for j > n.
We will denote it by glhd(A). Note that, since an A-module N is injective if and only
if
ExtjA (A/I, N ) = 0
for any j > 0 and any left ideal I of A, we have
glhd(A) = sup{hdA (A/I ) : I left ideal of A}.
So the global homological dimension depends only on cyclic A-modules.
Proposition 10.1.1 Let A be a ring and denote by A[x] the ring of polynomials in
one unknown with coefficients in A. Then
glhd(A[x]) ≤ glhd(A) + 1.
Proof: Thanks to the exact sequence
·x
0 −→ A[x] −→ A[x] −→ A −→ 0,
the result is a direct consequence of the formula
RHom A (A ⊗AL[x] M, N ) ' RHom A[x] (M, N )
holding for every A[x]-module M and N .
10.1.2
2
Noether property
We say that an A-module M is of finite type if it is generated by a finite number of
elements. It is finite free if it is isomorphic to Ap for some p ∈ IN.
For a ring A the following conditions are equivalent
(a) Any submodule of an A-module of finite type is of finite type.
52
Jean-Pierre Schneiders
(b) Any ideal of A is finitely generated.
(c) Any increasing sequence of submodules of an A-module of finite type is stationary.
(d) The subcategory of A-modules of finite type is stable by kernels, cokernels and
extensions.
A ring satisfying one of these conditions is said to be Noetherian. In some sense,
Noetherian rings are the only rings for which finite type modules are well-behaved.
If A is Noetherian, any A-module of finite type M has an infinite resolution
dn−1
d
n
· · · −→ Ln −→
Ln−1 −→ · · · −→ L0 −→ M −→ 0
by finite free A-modules. If hdA (M ) ≤ n, then Pn = ker dn−1 is a projective A-module
of finite type and we have the finite projective resolution
0 −→ Pn −→ Ln−1 −→ · · · −→ L0 −→ M −→ 0.
To compute the homological dimension of an A-module of finite type, we have the
following proposition.
Proposition 10.1.2 Assume A is Noetherian and glhd(A) < +∞. Then, for any
A-module of finite type M , the homological dimension hdA (M ) is the smallest integer
n ∈ IN such that
ExtiA (M, A) = 0
for i > n.
Proof: Let N be any A-module. We have an exact sequence
0 −→ R −→ A(I ) −→ N −→ 0
where I is some set of generators of N . Since M has a resolution by finite free Amodules
ExtiA (M, A(I )) = ExtiA (M, A)(I ) = 0
for i > n. Then, the exact sequence
ExtiA (M, A(I ) ) −→ ExtiA (M, N ) −→ ExtiA+1 (M, R) −→ ExtiA+1 (M, A(I ) )
shows that
ExtiA (M, N ) ' ExtiA+1 (M, R)
for i > n. By induction, we find for any p ∈ IN an A-module Rp such that
ExtiA (M, N ) = ExtiA+p (M, Rp )
for i > n. As soon as i > n and i + p > glhd(A), we get
ExtiA (M, N ) = 0,
and the conclusion follows.
2
An Introduction to D-Modules
10.1.3
53
Syzygies
Let us now investigate when any A-module of finite type has a finite resolution by finite
free A-modules.
Proposition 10.1.3 Assume the ring A is Noetherian. Then, the following conditions
are equivalent:
(a) any A-module of finite type M has a finite resolution
0 −→ Ln −→ Ln−1 −→ · · · −→ L0 −→ M −→ 0
by finite free A-modules,
(b) any A-module of finite type M has finite homological dimension and any finite
type projective A-module P is stably finite free (i.e. there is a finite free A-module
L such that P ⊕ L is finite free).
Proof: (a) ⇒ (b). Obviously, M has a finite projective resolution and hdA (M ) < +∞.
We will prove by induction on n that a finite type projective A-module P which has
a finite free resolution of length n is stably finite free. For n = 0, it is obvious. For
n = 1, the exact sequence
0 −→ L1 −→ L0 −→ P −→ 0
splits since P is projective; hence L0 ' L1 ⊕ P and the conclusion follows. For n > 1,
we have the resolution
d
d
d
n
1
0
0 −→ Ln −→
Ln−1 −→ · · · −→ L1 −→
L0 −→
P −→ 0
denote P1 the kernel of d0 . Since the sequence
0 −→ P1 −→ L0 −→ P −→ 0
is exact, the module P1 is projective. Moreover, it has a finite free resolution of length
n−1
0 −→ Ln −→ Ln−1 −→ · · · −→ L1 −→ P1 −→ 0.
The induction hypothesis gives us a finite free A-module L10 such that L10 ⊕ P1 is finite
free. Since the sequence
0 −→ L10 ⊕ P1 −→ L10 ⊕ L0 −→ P −→ 0
is exact, the conclusion follows from the case n = 1.
(b) ⇒ (a). Let M be an A-module of finite type. Let n = sup(1, hdA (M )). We
have a resolution
i
n
Ln−1 −→ · · · −→ L0 −→ M −→ 0
0 −→ Pn −→
54
Jean-Pierre Schneiders
where Lk is finite free and Pn is a projective module of finite type. Let Ln0 be a finite
free A-module such that Ln0 ⊕ Pn is finite free. Since the sequence
idL0 ⊕in
n
Ln0 ⊕ Ln−1 −→ · · · −→ L0 −→ M −→ 0
0 −→ Ln0 ⊕ Pn −→
2
is still exact, the proof is complete.
A ring A is syzygic when it satisfies the equivalent conditions of the preceding
proposition and has finite global homological dimension.
Corollary 10.1.4 Assume the ring A is syzygic. Then, any finite type A-module M
has a finite free resolution of length sup(1, hdA (M )).
10.1.4
Local rings
Recall that a ring A is local if it has a unique maximal ideal.
Lemma 10.1.5 (Nakayama’s lemma) Assume A is a local ring with maximal ideal
m and let M be an A-module of finite type. Then, M = 0 if and only if M/mM = 0.
Proof: The proof will work by induction on the number of generators of M . Denote
this number by p and assume g1 , . . . , gp are these generators. If p = 1 then M = mM
implies that there is m1 ∈ m such that g1 = m1g1 . Hence (1 − m1 )g1 = 0 in M and
since 1 − m1 is invertible in A, g1 = 0 and M = 0. If p > 1 then there are mij ∈ m for
i = 1, . . . , p; j = 1, . . . , p such that
g1 = m11 g1 + · · · + m1pgp
..
..
.
.
gp = mp1g1 + · · · + mpp gp
From the first line we see that g1 is a linear combination of g2 , . . . , gp ; hence the
2
conclusion by the induction hypothesis.
Proposition 10.1.6 Assume A is a Noetherian local ring with maximal ideal m and
M is an A-module of finite type. Then, a map
u
Ap −→ M
is surjective if and only if so is the associated map
v
(A/m)p −→ M/mM.
If, moreover, TorA1 (A/m, M ) = 0, then u is bijective if and only if so is v .
An Introduction to D-Modules
55
Proof: Denote by N0 and N1 the kernel and the cokernel of u. Clearly, v is surjective
if and only if N1 /mN1 = 0 since the sequence
A/m ⊗A Ap −→ A/m ⊗A M −→ A/m ⊗A N1 −→ 0
is exact. If v is bijective and TorA1 (A/m, M ) = 0, the sequence
0 −→ A/m ⊗A N0 −→ A/m ⊗A M −→ A/m ⊗A M −→ 0
is exact and N0 /mN0 = 0; hence N0 = 0.
2
Corollary 10.1.7 Assume A is a Noetherian local ring and M is a finite type Amodule. Then
(a) TorA1 (A/m, M ) = 0 implies M finite free,
(b) TorAj (A/m, M ) = 0 for j > n implies hdA (M ) ≤ n.
In particular, if glhd(A) < +∞, A is syzygic.
Proof: (a) is obvious from the preceding proposition since A/m is a field.
(b) Let
d
n
Ln−1 −→ · · · −→ L0 −→ M −→ 0
· · · −→ Ln −→
be a finite free resolution of M . Let Fn = ker dn . By construction, TorA1 (A/m, Fn) = 0
2
and Fn is of finite type hence Fn is finite free by (a) and hdA (M ) ≤ n.
10.2
Graded linear algebra
A graded ring GA is the data of a ring with unit A and a family of subgroups (Gk A)k∈ZZ
of A satisfying:
a. A =
L
k∈ZZ Gk A,
b. 1 ∈ G0 A,
c. Gk A.G` A ⊂ Gk+` A.
Here we always also assume that Gk A = 0 for k < 0 (i.e. GA is positively graded).
A graded module GM over the graded ring GA is the data of an A-module M and
a family of subgroups (Gk M )k∈ZZ of M such that
a. M =
L
k∈ZZ Gk M
b. Gk A.G` M ⊂ Gk+` M
56
Jean-Pierre Schneiders
Here we will always also assume that Gk M = 0 for k 0.
A morphism of a graded GA-module GM to a graded GA-module GN is a morphism
of A-modules
f : M −→ N
such that f (Gk M ) ⊂ Gk N .
These morphisms form a group denoted by Hom GA (GM, GN ).
With this notion of morphisms the category of graded GA-modules is Abelian.
The reader will easily find the form of the kernel, cokernel, image, and coimage of a
morphism in this category.
For r ∈ ZZ and any GA-module GM we define the shifted GA-module GM (r) to be
the A-module M endowed with the graduation defined by
Gk M (r) = Gk+r M.
Let us denote by GHom GA (GM, GN ) the graded group defined by setting
Gk Hom GA (GM, GN ) = Hom GA (GM, GN (k )).
10.2.1
Noether property
A GA-module GL is finite free if there are integers d1 , . . . , dn such that
n
GL ' ⊕ GA(−di ).
i=1
A finite free presentation of length s of a GA-module GM is an exact sequence
GLs −→ GLs−1 −→ · · · −→ GL0 −→ GM −→ 0
where GLi is finite free.
A GA-module GM is of finite type if it has a finite free presentation of length 0.
This means that there are homogeneous elements m1 ∈ Gd1 M, . . . , mn ∈ Gdn M such
that any m ∈ Gd M may be written as
m=
n
X
ad−di mdi
i=1
where ad−di ∈ Gd−di A.
A graded ring GA is Noetherian if any GA-submodule of a GA-module of finite
type is of finite type. An equivalent condition is that the subcategory of GA-modules
of finite type is stable by kernels, cokernels and extensions.
If GA is Noetherian then any GA-module of finite type has finite free presentations
of arbitrary length.
An Introduction to D-Modules
10.2.2
57
Homological dimension
The homological dimension of a GA-module GM is the smallest integer n such that
ExtjGA (GM, GN ) = 0 for j > n and any GA-module GN . It is also the smallest n
such that GM has a projective resolution of length n. We denote it by hdGA (GM ).
The global homological dimension of GA is the supremum of the homological dimensions of all GA-modules. It is denoted by glhd(GA). It is also the supremum of
the homological dimensions of finite type GA-modules.
We denote by Σ the functor which associates to any GA-module its underlying
A-module. We have
ΣGM =
M
Gk M
k∈ZZ
and Σ is an exact functor.
Proposition 10.2.1 If GA is Noetherian and GM is a GA-module of finite type then
ΣGExtiGA (GM, GN ) = ExtiΣGA (ΣGM, ΣGN )
for i ∈ IN. In particular hdGA (GM ) = hdΣGA (ΣGM ) and glhd(GA) = glhd(ΣGA).
Proof: Since GA is Noetherian, we have a resolution
· · · −→ GLn −→ GLn−1 −→ · · · −→ GL0 −→ GM −→ 0
of GM by finite free GA-modules. If d1 , . . . , dp are integers, we have
GHom GA
p
p
p
i=1
i=1
⊕ GA(−di ), GN = ⊕ GHom GA (GA, GN )(di ) = ⊕ GN (di ).
i=1
Thus,
ΣGHom GA (GL, GN ) ' Hom ΣGA (ΣGL, ΣGN )
when GL is a finite free GA-module. Since ΣGL is a finite free ΣGA-module, the
isomorphism
ΣGHom·GA (GL., GN ) ' Hom·ΣGA (ΣGL· , ΣGN )
gives the requested result in cohomology.
2
Corollary 10.2.2 Assume GA is Noetherian and glhd(GA) is finite. Then, for any
finite type GA-module GM , hdGA (GM ) is the smallest integer n such that
GExtiGA (GM, GA) = 0
for i > n.
58
10.2.3
Jean-Pierre Schneiders
Syzygies
The study of finite free resolutions of finite length in the preceding section may be
reproduced in the graded case and we get
Proposition 10.2.3 Assume GA is Noetherian. Then, the following conditions are
equivalent :
(a) Any finite type GA-module GM has a finite free resolution of finite length.
(b) Any finite type GA-module GM has finite homological dimension and finite type
projective GA-modules are stably finite free.
A ring GA with finite global homological dimension satisfying the equivalent conditions of the preceding proposition is called syzygic.
Proposition 10.2.4 Let GA be a graded ring. Assume that projective G0 A-modules
of finite type are finite free (resp. stably finite free). Then, so are the projective
GA-modules of finite type.
Proof: Consider the ring morphism
p0 : GA −→ G0 A.
If p0 is an isomorphism, we have nothing to prove. Otherwise, denote by GI its kernel.
We have
G0 A ⊗GA GM = GM/GI.GM
for any GA-module GM . If GM is of finite type then
GM = GI.GM
if and only if GM = 0. As a matter of fact, assuming the contrary there is a lowest
integer k0 ∈ ZZ such that Gk0 M 6= 0. A non zero element of Gk0 M is thus in GI.GM
and should be of degree greater than k0 + 1; hence a contradiction.
Now, let GP be a finitely generated projective GA-module. It is clear that G0 A ⊗GA
GP is a finitely generated projective G0 A-module. Hence it is finite free (resp. stably
finite free) and (up to adding to GP a finite free GA-module) there is an isomorphism
∼ G A⊗
G0 Ap −→
0
GA GP.
This isomorphism comes by, scalar extension, from a morphism
GL −→ GP
where GL is a finite free GA-module. Denote by GN0 the kernel and by GN1 the
cokernel of this morphism. We have the exact sequence
G0 A ⊗GA GL −→ G0 A ⊗GA GP −→ G0 A ⊗GA GN1 −→ 0
An Introduction to D-Modules
59
hence G0 A ⊗GA GN1 = 0 and GN1 = 0. The exact sequence
0 −→ GN0 −→ GL −→ GP −→ 0
splits since GP is projective. Hence we have the exact sequence
0 −→ G0 A ⊗GA GN0 −→ G0 A ⊗GA GL −→ G0 A ⊗GA GP −→ 0
and GN0 = 0. The conclusion follows.
2
Corollary 10.2.5 Assume A is syzygic then the graded ring A[x] is syzygic.
10.3
Filtered linear algebra
A filtered ring F A is the data of a ring with unit A and a family of subgroups (Fk A)k∈ZZ
of A satisfying
a.
S
k∈ZZ
Fk A = A,
b. Fk A ⊂ Fk+1 A (k ∈ Z ),
c. Fk A.F` A ⊂ Fk+` A (k, ` ∈ ZZ).
d. 1 ∈ F0A.
Here, we always also assume that Fk A = 0 for k < 0 (i.e. F A is positively filtered).
The order of a non zero element a ∈ A is the smallest integer k ∈ ZZ such that
a ∈ Fk A. We denote it by ord(a).
A (filtered) module over the filtered ring F A is the data of an A-module M and a
family (Fk M )k∈ZZ of subgroups of M such that
a.
S
k∈ZZ
Fk M = M
b. Fk M ⊂ Fk+1 M
;
c. Fk A.F` M ⊂ Fk+` M
k ∈ ZZ
;
k, ` ∈ ZZ
here we will always also assume that Fk M = 0 for k 0.
The order of a non zero element of M is the smallest integer k ∈ ZZ such that a ∈
Fk M . We denote it by ord(m). By convention ord(0) = −∞. Of course, ord(m.n) ≤
ord(m) + ord(n) and ord(m + n) ≤ sup(ord(n), ord(n)).
A morphism
F u : F M −→ F N
of F A-modules is a morphism u : M −→ N of the underlying A-modules such that
u(Fk M ) ⊂ Fk N.
We denote by Fk u the morphism u|Fk M : Fk M −→ Fk N .
60
Jean-Pierre Schneiders
Morphisms from an F A-module F M to an F A-module F N form a group which is
denoted by Hom F A (F M, F N ). With this notion of morphisms the category of F Amodules is an additive category. Consider an arbitrary morphism
F u : F M −→ F N
having u : M −→ N as underlying morphism. The kernel of F u is the kernel of
u filtered by the family (ker u ∩ Fk M )k∈ZZ . The cokernel of F u is the cokernel of u
filtered by the family (p(Fk N ))k∈ZZ where p : N −→ coker u is the canonical projection.
As usual, the image of F u is the kernel of its cokernel. Hence, it is the A-module
im u filtered with the filtration (im u ∩ Fk N )k∈ZZ . As for the coimage of F u, it is the
cokernel of the kernel of F u. Hence, it is the A-module im u endowed with the filtration
(u(Fk M ))k∈ZZ .
Note that the canonical filtered morphism
coim F u −→ im F u
is a bijection but is not in general an isomorphism.
The morphisms for which this map is an isomorphism are called strict.
The category of filtered F A-modules is not Abelian. However, one can show that
it is an exact category in the sense of Quillen [14]. Hence, it is suitable for homological
algebra. For the sake of brevity we will avoid this point of view here and refer the
interested reader to Laumon [13].
An exact sequence of F A-modules is a sequence
Fu
Fv
F M −→ F N −→ F P
such that ker Fk v = im Fk u. It follows from this definition that F u is strict. If moreover
F v is strict we say that it is a strict exact sequence.
For any r ∈ ZZ and any F A-module F M , we define the shifted module F M (r) as
the module M endowed with the filtration (Fk+r M )k∈ZZ .
Finally, we introduce for any couple F M, F N of F A-modules the filtered group
FHom F A (F M, F N )
by setting
Fk Hom F A (F M, F N ) = Hom F A (F M, F N (k )).
To any filtered ring F A we associate a graded ring GA defined by setting
Gk A = Fk A/Fk−1 A
the multiplication being induced by that of F A.
To a filtered F A-module F N we associate a graded GA-module GM defined by
setting
Gk M = Fk M/Fk−1 M
An Introduction to D-Modules
61
the action of GA on GM being induced by that of F A on F M .
The image of m ∈ Fk M (resp. a ∈ Fk A) in Gk M (resp. Gk A) is its symbol of order
k . We denote it by σk (m) (resp. σk (a)). The principal symbol of m (resp. a) is its
symbol of order ord(m) (resp. ord(a)).
A filtered morphism of F A-modules
F u : F M −→ F N
induces a graded morphism of GA-modules
Gu : GM −→ GN.
The preceding construction clearly defines a functor from the category of F Amodules to the category of GA-modules. We denote this functor by Gr.
The last part of this section will be devoted to the study of this functor. In particular, we will investigate how finiteness and dimensionality properties of GA induces
similar properties on F A and on A.
Proposition 10.3.1 Let F A be a filtered ring and consider two morphisms
F u : F M −→ F N,
F v : F N −→ F P
such that F v ◦ F u = 0. Then, the sequence
Fu
Fv
F M −→ F N −→ F P
is a strict exact sequence if and only if the sequence
Gu
Gv
GM −→ GN −→ GP
is exact.
Proof: The condition is necessary.
Let nk ∈ Gk N be such that Gk v (nk ) = 0. There is n0k ∈ Fk N such that nk =
σk (n0k ). Hence we have v (n0k ) ∈ Fk−1 P . Since F v is strict we find n00k−1 ∈ Fk−1 N such
that v (n00k−1 ) = v (n0k ). Then v (n0k − n00k−1 ) = 0 and there is mk ∈ Fk M such that
u(mk ) = n0k − n00k−1 . This shows that
Gk u(σk (mk )) = σk (n0k ) = nk
and the conclusion follows.
The condition is sufficient.
Let us prove that F v is strict. Assume pk ∈ Fk P ∩ im v . Let ` be the smallest
integer such that pk = v (n`) with n` ∈ F` N . We need to show that ` ≤ k . Assume the
contrary. Then
G` v (σ` (n` )) = σ` (pk ) = 0.
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Jean-Pierre Schneiders
Hence, there is an m` ∈ F` M such that
G` u(σ`(m` )) = σ` (n` ).
As a consequence, we get n` − u(m` ) ∈ F`−1 M . But v (n` − u(m` )) = pk and we get a
contradiction.
Let us prove now that ker Fk v = im Fk u by increasing induction on k . For k 0,
Fk N = 0 so the result is obvious. Let nk ∈ ker Fk v . Since Gk v (σk (nk )) = 0, there is
mk ∈ Fk M such that
Gk u(σk (mk )) = σk (nk ).
Hence nk − u(mk ) ∈ Fk−1 N . By the induction hypothesis there is mk−1 ∈ Fk−1 M such
that
u(mk−1 ) = nk − u(mk )
2
and the proof is complete.
Corollary 10.3.2 Let F A be a filtered ring and F u : F M −→ F N be a morphism of
F A-modules. Then Gr ker F u ⊂ ker GrF u and im GrF u ⊂ Gr im F u. Moreover, the
following conditions are equivalent:
(a) F u is strict,
(b) Gr ker F u = ker GrF u,
(c) im GrF u = Gr im F u.
Proof: The equivalence of (a), (b) and (c) is a consequence of the preceding proposition. To conclude we just have to note that
Grk ker F u = ker u ∩ Fk M/ ker u ∩ Fk−1M
ker Grk F u = Fk M ∩ u−1 (Fk−1 N )/Fk−1 M
and that
im Grk F u = u(Fk M )/Fk−1 N
Grk im F u = im u ∩ Fk N/ im u ∩ Fk−1 N.
2
An F A-module F M is finite free if it is isomorphic to
p
⊕ F A(−di)
i=1
where d1 , . . . , dp are integers. An F A-module F M is of finite type if there is a strict
epimorphism
F L −→ F M
where F L is finite free. This means that we can find m1 ∈ Fd1 M, . . . , mp ∈ Fdp M such
that any m ∈ Fd M may be written as
m=
p
X
i=1
where ad−di ∈ Fd−di A.
ad−di mi
An Introduction to D-Modules
10.3.1
63
Noether property
For a filtered ring F A, the following conditions are equivalent.
• Any filtered submodule of a finite type F A-module is of finite type.
• Any filtered ideal of F A is of finite type.
• Any increasing sequence of filtered submodules of a finite free F A-module is
stationary.
Let us emphasize that in the preceding conditions we deal with non necessarily strict
submodules and ideals.
A filtered ring F A satisfying the above equivalent conditions is said to be (filtered)
Noetherian. From this definition, it is clear that if F A is filtered Noetherian then the
underlying ring A is itself Noetherian.
Proposition 10.3.3 Let F A be a filtered ring and denote by GA its associated graded
ring. Then F A is filtered Noetherian if and only if GA is graded Noetherian.
Proof: The condition is sufficient.
We need to prove that a filtered submodule F M 0 of a finitely generated F A-module
F M is finitely generated.
If F M 0 is strict, then the associated GA-module GM 0 is a submodule of the GAmodule GM associated to F M . Since GA is Noetherian and GM is finitely generated
so is GM 0 and the conclusion follows.
To prove the general case, we may thus assume that the image of the inclusion
F M 0 −→ F M
is equal to F M . In this case, using a finite system of generators of F M , it is easy to
find an integer ` such that:
Fk M 0 ⊂ Fk M ⊂ Fk+` M 0 .
We will prove the result by increasing induction on `.
For ` = 1, let us introduce the auxiliary GA-modules
GK0 =
GK1 =
⊕ Fk M 0 /Fk−1 M
k∈ZZ
⊕ Fk M/Fk M 0 .
k∈ZZ
These modules satisfy the exact sequences
0 −→ GK0 −→ GM −→ GK1 −→ 0
0 −→ GK1 (−1) −→ GM 0 −→ GK0 −→ 0.
Since GM is a finite type GA-module, so are GK0 and GK1 . Hence GM 0 is also finitely
generated and the conclusion follows.
64
Jean-Pierre Schneiders
For ` > 1, we define the auxiliary F A-module F M 00 by setting
Fk M 00 = Fk−1 M + Fk M 0 .
Since we have
Fk M 00 ⊂ Fk M ⊂ Fk+1 M 00 ,
the preceding discussion shows that F M 00 is finitely generated. Moreover,
Fk M 0 ⊂ Fk M 00 ⊂ Fk+(`−1) M 0 ,
and the conclusion follows from the induction hypothesis.
The condition is necessary.
To F A we associate the graded ring
ΣF A = ⊕ Fk A.
k∈ZZ
Since F0A ⊂ F1 A the identity 1 ∈ F1A and defines a central element T of degree 1 in
ΣF A. To any F A-module F M we associate the ΣF A-module
ΣF M = ⊕ Fk M
k∈ZZ
Obviously, any morphism F u : F M −→ F N induces a morphism
ΣF u : ΣF M −→ ΣF N
of graded ΣF A-modules. Hence, we get a functor Σ from the category of filtered F Amodules to the category of graded ΣF A-modules. One checks easily that this functor
is fully faithful and that it transforms exact sequences into exact sequences. Moreover,
for any inclusion
u0
M 0 −→ L0
of ΣF A-modules where L0 is finite free there is a strict monomorphism
Fu
F M −→ F L
of F A-modules where F L is finite free and a commutative diagram
u0
M 0 −→ L0
↑
↑
ΣF u
ΣF M −→ ΣF L
It is thus clear that ΣF A is graded Noetherian if and only if F A is filtered Noetherian.
The conclusion follows from the formula
ΣF A/ΣF A.T = GA.
2
An Introduction to D-Modules
65
Proposition 10.3.4 Assume F A is Noetherian and F M is an F A-module. Then
(a) F M is an F A-module of finite type if and only if GM is a GA-module of finite
type,
(b) F M is a finite free F A-module if and only if GM is a finite free GA-module,
(c) any F A-module of finite type has an infinite resolution by finite free F A-modules
(i.e. there is an exact sequence
· · · −→ F Ls −→ F Ls−1 −→ · · · −→ F L0 −→ F M −→ 0
where each F Ls is a finite free F A-module.
Proposition 10.3.5 Assume F A is Noetherian. Then, for any F A-module of finite
type F M and any F A-module F N ,
GExtjGA (GM, GN ) = 0 ⇒ ExtjA (M, N ) = 0.
In particular, hdA (M ) ≤ hdGA (GM ) and glhd(A) ≤ glhd(GA).
Proof: Let
· · · −→ F Ln −→ F Ln−1 −→ · · · −→ F L0 −→ F M −→ 0
be a filtered resolution of F M by finite free F A-modules. Applying the graduation
functor we get a resolution
· · · −→ GLn −→ GLn−1 −→ · · · −→ GL0 −→ GM −→ 0
of GM by finite free GA-modules. Assuming GExtjGA (GM, GN ) = 0 means that the
sequence
GHom GA (GLj −1 , GN ) −→ GHom GA (GLj , GN ) −→ GHom GA (GLj +1 , GN )
is an exact sequence of GA-modules. The natural map
GrFHom F A (F L, F N ) −→ GHom GA (GL, GN )
is an isomorphism when F L is finite free. Hence the sequence
FHom F A (F Lj −1 , F N ) −→ FHom F A (F Lj , F N ) −→ FHom F A (F Lj +1 , F N )
is a strict exact sequence of F A-modules. When F L is finite free the underlying module
of FHom F A (F L, F N ) is Hom A (L, N ). Hence the conclusion.
2
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Jean-Pierre Schneiders
10.3.2
Syzygies
Proposition 10.3.6 Assume F A is a filtered ring and GA is syzygic then any F Amodule of finite type F M has a finite free resolution
0 −→ F Ln −→ F Ln−1 −→ · · · −→ F L0 −→ F M −→ 0
of length n = sup(1, hdGA GM ). In particular, A is syzygic.
Proof: Let
dn−1
d
n
· · · −→ F Ln −→
F Ln−1 −→ · · · −→ F L0 −→ F M −→ 0
be a finite free resolution of F M . Set F Pn = ker dn−1 . Since the sequence
0 −→ GPn −→ GLn−1 −→ · · · −→ GL0 −→ GM −→ 0
is exact, hdGA (GPn ) ≤ 0 and GPn is projective. It is thus stably finite free. Let GL0n
be a finite free GA-module such that GL0n ⊕ GPn is finite free. Let F L0n be a finite
free F A-module which has GL0n as associated graded GA-module. We have the exact
sequence
0 −→ F Pn ⊕ F L0n −→ F Ln−1 ⊕ F Ln −→ · · · −→ F L0 −→ F M −→ 0
and F Pn ⊕ F L0n is finite free since its associated GA-module GPn ⊕ GL0n is finite free.
To conclude, note that any finite type A-module M is the underlying A-module of
2
a finite type F A-module F M .
10.3.3
Gabber’s theorem
Let F A be a filtered ring and denote by GA its associated graded ring.
For any ak ∈ Fk A, b` ∈ F` A we define the commutator [ak , a`] of ak and a` as usual
by setting
[ak , a` ] = ak · a` − a` · ak .
Obviously, GA is commutative if and only if
[Fk A, F`A] ⊂ Fk+`−1 A.
In this case, one checks easily that
σk+`−1 [ak , a` ] = σk+`−1 [a0k , a0` ]
if ak , a0k ∈ Fk A (resp. a` , a0` ∈ F` A) have the same symbol of order k (resp. `). This
property allows us to define the Poisson bracket of two symbols sk ∈ Gk A and sl ∈ G` A
by setting
{sk , s` } = σk+`−1 ([ak , a` ])
An Introduction to D-Modules
67
where ak ∈ Fk A and a` ∈ F` A are such that σk (ak ) = sk and σ` (a` ) = s` . The Poisson
bracket satisfies the following easily checked identities:
{s, s0} = −{s0, s}
{s, s0 s00 } = {s, s0}s00 + s0 {s, s00 }
{s, {s0 , s00 }} = {{s, s0}, s00 } + {s0 , {s, s00}}
Now, let F M be an F A-module and denote by F I its annihilating ideal. Obviously,
if ak ∈ Fk I and a` ∈ F` I we have
[ ak , a` ] F M = 0
and [ak , a` ] ∈ Fk+`−1 I . Hence the graded
√ ideal GI associated to F I is stable by {·, ·}.
However, it is not obvious at all that GI is also stable by {·, ·}. This problem was
solved by Gabber in [5]. His proof is rather technical, so we skip it here and refer the
interested reader to the original paper.
Theorem 10.3.7 Let F A be a filtered ring and assume GA is a commutative Noetherian Q
I -algebra. Then, for any finite type F A-module F M we have
√
√
√
{ Ann GM , Ann GM } ⊂ Ann GM
where GM is the GA-module associated with F M .
10.4
Transition to sheaves
The reader will easily extend the definition of graded and filtered rings and modules
given above to the case of filtered and graded sheaves of rings and modules on a topological space X . The notion of graded or filtered morphisms are also easily extended
to sheaves as is the notion of strict filtered morphism and of graded or filtered exact
sequences.
Let GA (resp. FA) be a graded (resp. filtered) sheaf of rings and GM, GN (resp.
FM, FN ) two GA (resp. FA) modules. The functors
U −→ Hom GA|U (GM|U , GN |U );
U 7→ GHom GA|U (GM|U , GN |U )
U −→ Hom FA|U (FM|U , FN |U );
U 7→ FHom FA|U (FM|U , FN |U )
(resp.
) clearly define sheaves of groups and graded (resp. filtered) groups on X . We denote
then by
Hom GA (GM, GN ); GHom GA (GM, GN )
(resp.
Hom FA (FM, FN );
)
F Hom FA (FM, FN )
68
Jean-Pierre Schneiders
The definition of the graduation functor will also be easily extended to sheaves of
filtered rings and modules. Finally, the reader will generalize easily to sheaves the
notion of finite free and finite type modules and that of a finite free s-presentation.
The finiteness conditions require more attention since the category of modules of
finite type over a sheaf of rings is almost never well-behaved. To get satisfactory
finiteness conditions on A (resp. GA, FA) we need to go a step further and consider
modules having locally finite free 1-presentations.
Definition 10.4.1 The ring A (resp. GA, FA) is coherent if the category of modules
which admits locally finite free 1-presentations is stable by kernels, cokernels and extensions as a subcategory of the category of modules. A module on a coherent ring
(resp. graded ring, filtered ring) is coherent if it has locally a finite free 1-presentation.
A sheaf of rings A is coherent if and only if for any open subset U of X the kernel
of a morphism of free AU -modules is locally of finite type.
An analogous criterion holds for sheaves of graded rings.
For a sheaf of filtered rings FA, the criterion is that for any open subset U of X
the filtered kernel and the filtered image of a morphism of finite free FAU -modules are
both locally of finite type.
Remark 10.4.2 Let FA be a filtered ring on X and denote by GA the associated
graded ring. Then, an FA-module FM is locally of finite type (resp. finite free), if
and only if the GA-module GM is locally of finite type (resp. finite free).
Proposition 10.4.3 Let FA be a coherent filtered ring on X and denote by GA the
associated graded ring and by A the underlying ring. Then
(a) the ring A and the graded ring GA are coherent,
(b) an FA-module FM is coherent if and only if GM is GA coherent,
(c) a strict submodule FN of a coherent FA-module FM is coherent if and only if
the underlying A-module N is locally of finite type,
(d) an FA-module locally of finite type FM is coherent if and only if the underlying
A-module M is coherent.
Proof: (a) Let us prove that A is coherent. Let U be an open subset of X and consider
a morphism
u
Ap|U −→ Aq|U
This is the underlying morphism of a filtered morphism
p
⊕ FA(−di ) −→ FAq
i=1
We know that the kernel of this morphism is an FA-module locally of finite type. Since
its underlying A-module N is the kernel of u, the conclusion follows.
To prove that GA is coherent, we proceed as in Proposition 10.3.3.
An Introduction to D-Modules
69
(b) Proceed as in Proposition 10.3.4.
(c) Let FN be a strict submodule of FM and assume N is locally of finite type.
Since the local generators of N have locally a finite order in FM, we may view locally
FN as the filtered image of a morphism
FL|U −→ FN |U
where FL is a finite free FA-module. Hence the conclusion.
(d) Assume that FM is locally of finite type and that M is coherent. Locally, we
have a strict epimorphism
FL|U −→ FM|U −→ 0
where FL is finite free. Denote by FN its kernel. By hypothesis we know that N is
2
coherent; hence the conclusion by (c).
Theorem 10.4.4 Assume FA is a filtered ring on X and denote by GA the associated
graded ring. Then, the following conditions are equivalent:
(a) FA is coherent and FAx is Noetherian for every x ∈ X ,
(b) GA is coherent and GAx is Noetherian for every x ∈ X .
Proof: (a) ⇒ (b) is a consequence of the preceding proposition and of Proposition 10.3.3.
(b) ⇒ (a) We know already that FAx is Noetherian. It remains to prove that the
kernel and the image of a morphism
Fu
FL1 −→ FL0
of finite free FA-modules are locally of finite type.
Assume F u is strict at x ∈ X . Then Gr ker F ux = ker G rF ux . Since ker G rF u
is a coherent GA-module and G r ker F u ⊂ ker G rF u, Gr ker F uy = ker GrF uy for y
in a neighborhood of x. Hence F u is strict in a neighborhood V of x and ker F u is
locally of finite type since G r ker F u|V = ker G rF u|V is GA|V coherent. Moreover, in a
neighborhood of x, G r im F u|V = im G rF u|V is a GA|V coherent module and im F u is
locally of finite type.
Assume F u is not strict at x ∈ X . Since im F ux is a sub FAx -module of (FL0)x
and FAx is Noetherian, it is finitely generated. It is thus possible to find locally a
morphism
Fv
FL2 −→ FL0
of finite free FA-modules such that im F v ⊂ im F u and for which
Fv
x
(FL2 )x −→
im F ux −→ 0
is a strict epimorphism. The morphism
Fv
FL1 ⊕ FL2 −→ FL0
(α, β )
7→
(u(α) + v (β ))
70
Jean-Pierre Schneiders
is strict at x and such that locally im F w = im F u. By the preceding discussion, this
shows that im F u and ker F w are locally of finite type at x. Since im F v ⊂ im F u,
there is locally a non filtered morphism of A-modules
f : L2 −→ L1
such that u ◦ f = v . This gives us the morphism of A-modules
ker w −→ ker u
(α, β )
7→
α + f (β )
Hence ker u is locally of finite type. This gives us a finite free FA-module FL3 and a
morphism
FL3 −→ FL2
having ker F u as filtered image. Since FA is coherent, this shows that ker F u is an
2
FA-module of finite type and the proof is complete.
Assume the sheaf of ring A is coherent and let M be a coherent A-module then the
homological dimension of M is the smallest integer n such that the following equivalent
properties hold
• for any A-module N and any j > n
ExtjA (M, N ) = 0
• for any A-module N , any x ∈ X and any j > n
ExtjAx (Mx , Nx ) = 0
From this definition, we find that
hdA (M) = sup hdAx (Mx ).
x∈ X
We define the global homological dimension of A as the supremum of
{hdA M : M coherent A-module}.
We introduce similar definitions for a coherent graded ring X and its coherent graded
modules.
Proposition 10.4.5 Assume the filtered ring FA on X is coherent. Then, for any
coherent FA-module FM and any FA-module FN
GE xtjGA (GM, GN ) = 0 ⇒ ExtjA (M, N ) = 0
where M, N are the underlying A-modules of FM, FN and GM, GN are the
corresponding graded modules. In particular, we have hdA (M) ≤ hdGA (GM) and
glhd(A) ≤ glhd(GA).
An Introduction to D-Modules
Proof: Proceed as for the proof of 10.3.5.
71
2
A sheaf of rings (resp. graded rings) is syzygic if it is coherent, has finite global
homological dimension and syzygic fibers. Since a coherent module M on a coherent
ring A is finite free in a neighborhood of x ∈ X if and only if Mx is finite free we get
the following proposition.
Proposition 10.4.6 Assume A is syzygic. Then, any coherent A-module M has a
finite free resolution
0 −→ Ln −→ Ln−1 −→ · · · −→ L0 −→ M −→ 0
of length n = sup(1, hdA (M)).
In the same way, we get :
Proposition 10.4.7 Let FA be a filtered ring on X and denote by GA its associated
graded ring. Assume GA is syzygic. Then, any coherent FA-module FM has locally
a finite free resolution
0 −→ FLn −→ FLn−1 −→ · · · −→ FL0 −→ FM −→ 0
of length sup(1, hdGA GM). In particular, A is syzygic.
A sheaf of rings (resp. graded rings, filtered rings) is Noetherian if it is coherent
with Noetherian fibers and if locally any increasing sequence of coherent submodules
of a coherent module is locally stationary.
Proposition 10.4.8 Assume FA is a filtered ring on X and denote by GA and A its
associated graded ring and its underlying ring. Then, FA is Noetherian if and only if
GA is Noetherian. In this case A is also Noetherian.
Proof: Assume GA is Noetherian. By Theorem 10.4.4 we already know that FA is
coherent and that FAx is Noetherian for any x ∈ X . Assume that (FMk )k∈IN is an
increasing sequence of coherent strict submodules of the coherent FA|U module FM|U
where U is some open subset of X . Then, (GMk )k∈IN is an increasing sequence of
coherent GA-submodules of the coherent GA-module GM. Since GA is Noetherian,
every point x ∈ U has a neighborhood V ⊂ U for which there is an integer k0 such
that (GMk0 )|U = (GMk )|U for k ≥ k0 . This implies (FMk0 )|V = (FMk )|V for k ≥ k0 .
We obtain the same result for an increasing sequence of general filtered submodules of
FM|U by proceeding as in 10.3.3. Hence FA is Noetherian.
Assume now FA Noetherian, working as in the proof of Proposition 10.3.3, we see
easily that GA is Noetherian. Since any coherent A-module may be viewed as the
underlying module of a coherent FA-module and since a submodule FN of a coherent
FA-module FM is coherent when its underlying A-module is coherent, we get easily
2
the Noetherianity of A.
72
Jean-Pierre Schneiders
Corollary 10.4.9 A graded ring GA on X is Noetherian if and only if the underlying
ring ΣGA is Noetherian.
Proposition 10.4.10 If the sheaf of rings A is Noetherian then so is the sheaf of rings
A[x] .
Proof: By the preceding corollary, it is sufficient to prove that A[x] is Noetherian as
a graded ring on X .
First we show that A[x] is graded coherent. Let
Gu
GL1 −→ GL0
be a graded morphism of finite free graded A[x]-modules.
Denote GN its kernel. Since
p
GL1 = ⊕ A[x](di).
i=1
Gk N is a coherent A-submodule of Gk L1 ' Ap and since GN .x ⊂ GN we get Gk N ⊂
Gk+1 N as submodules of Ap . Since A is Noetherian, this sequence is locally stationary
and for any x ∈ X there is an open neighborhood U of x and an integer k0 such that
(Gk0 N )|U = (Gk N )|U
for k ≥ k0 . Hence N|U is generated by Gk0 N as an A|U [x]-module. The GA-module
GN is thus locally finitely generated on A[x].
To prove that A[x] is Noetherian, we have to prove that if (GN ` )`∈IN is an increasing
sequence of coherent A[x]-submodules of a finite free A[x]-module GL then (GN ` )`∈IN
is locally stationary. Working as above we see that
Gk N` ⊂ Gk0 N`0
if ` ≤ `0 , k ≤ k 0 . Hence, locally, there are integers `0 , k0 such that Gk0 N`0 = Gk N` for
2
k ≥ k0 , ` ≥ `0 . This implies that GN ` = GN `0 for ` ≥ `0 ; hence the conclusion.
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J.-P. Schneiders
Mathématiques
Université Paris XIII
Avenue J. B. Clément
93430 Villetaneuse
FRANCE
e-mail: [email protected]