Lectures on Algebraic Theory of D-Modules

149 Pages·2012·1.044 MB·English

by  Dragan Milicic

Lectures on Algebraic Theory of D-Modules
Dragan Miliˇci´c

Contents
Chapter I. Modules over rings of differential operators with polynomial
coefficients 1
1. Hilbert polynomials 1
2. Dimension of modules over local rings 6
3. Dimension of modules over filtered rings 9
4. Dimension of modules over polynomial rings 14
5. Rings of differential operators with polynomial coefficients 18
6. Modulesoverringsofdifferentialoperatorswithpolynomialcoefficients 24
7. Characteristic variety 25
8. Holonomic modules 32
9. Exterior tensor products 36
10. Inverse images 40
11. Direct images 46
12. Kashiwara’s theorem 49
13. Preservation of holonomicity 51
Chapter II. Sheaves of differential operators on smooth algebraic varieties 55
1. Differential operators on algebraic varieties 55
2. Smooth points of algebraic varieties 62
3. Sheaves of differential operators on smooth varieties 73
Chapter III. Modules over sheaves of differential operators
on smooth algebraic varieties 77
1. QuasicoherentD -modules 77
X
2. Coherent D -modules 79
X
3. Characteristic varieties 83
4. Coherentor 86
5. D-modules on projective spaces 90
Chapter IV. Direct and inverse images of D-modules 97
1. The bimodule D 97
X→Y
2. Inverse and direct images for affine varieties 103
3. Inverse image functor 105
4. Projection formula 108
5. Direct image functor 111
6. Direct images for immersions 114
7. Bernstein inequality 119
8. Closed immersions and Kashiwara’s theorem 120
9. Local cohomology of D-modules 124
10. Base change 128
iii
iv CONTENTS
Chapter V. Holonomic D-modules 133
1. Holonomic D-modules 133
2. Connections 134
3. Preservationof holonomicity under direct images 136
4. A classification of irreducible holonomic modules 138
5. Local cohomology of holonomic modules 140
6. Preservationof holonomicity under inverse images 140
Bibliography 143
CONTENTS v
These notes represent a brief introduction into algebraic theory of D-modules.
The originalversionwaswrittenin 1986whenI wasteachingayearlongcourseon
the subject. Minor revisions were done later when I was teaching similar courses.
A major reorganizationwas done in 1999.
I wouldliketo thankDanBarbaschandPavlePandˇzi´cforpointing outseveral
errors in previous versions of the manuscript.

CHAPTER I
Modules over rings of differential operators with
polynomial coefficients
1. Hilbert polynomials
Let A = ∞ An be a graded noetherian commutative ring with identity 1
n∈Z
contained in A0. Then A0 is a commutative ring with identity 1. Assume that
L
An =0 for n<0.
1.1. Lemma. (i) A0 is a noetherian ring.
(ii) A is a finitely generated A0-algebra.
Proof. (i) Put A = ∞ An. Then A is an ideal in A and A0 =A/A .
+ n=1 + +
(ii) A is finitely generated. Let x ,x ,...,x be a set of homogeneous gen-
+ 1 2 s
L
erators of A and denote d = degx , 1 ≤ i ≤ s. Let B be the A -subalgebra
+ i i 0
generated by x ,x ,...,x . We claim that An ⊆ B, n ∈ Z . Clearly, A0 ⊆ B.
1 2 s +
Assume that n > 0 and y ∈ An. Then y ∈ A and therefore y = s y x where
+ i=1 i i
yi ∈An−di. It follows that the induction assumption applies to yi, 1≤i≤s. This
implies that y ∈B. P (cid:3)
The converse of 1.1 follows from Hilbert’s theorem which states that the poly-
nomial ring A0[X ,X ,...,X ] is noetherian if the ring A0 is noetherian.
1 2 n
Let M = Mn be a finitely generated graded A-module. Then each Mn,
n∈Z
n∈Z, is an A0-module. Also, Mn =0 for sufficiently negative n∈Z.
L
1.2. Lemma. The A0-modules Mn, n∈Z, are finitely generated.
Proof. Letm ,1≤i≤k,behomogeneousgeneratorsofM anddeg m =r ,
i i i
1≤i≤k. Forj ∈Z denote byz (j), 1≤i≤ℓ(j), allhomogeneousmonomialsin
+ i
x1,x2,...,xs of degree j. Let m ∈ Mn. Then m = ki=1yimi where yi ∈ An−ri,
i ≤ i ≤ k. By 1.1, y = a z (n − r ), with a ∈ A0. This implies that
i j ij j i Pij
m= a z (n−r )m ;henceMnisgeneratedby(z (n−r )m ;1≤j ≤ℓ(n−r ),
i,j ij j i i P j i i i
1≤i≤k). (cid:3)
P
Let M (A0) be the category of finitely generated A0-modules. Let λ be a
fg
function on M (A0) with values in Z. The function λ is called additive if for any
fg
short exact sequence:
0−→M′ −→M −→M′′ −→0
we have
λ(M)=λ(M′)+λ(M′′).
Clearly, additivity implies that λ(0)=0.
1
2 I.DIFFERENTIAL OPERATORS WITH POLYNOMIAL COEFFICIENTS
1.3. Lemma. Let
0→M →M →M →···→M →0
0 1 2 n
be an exact sequence in M (A0). Then
fg
n
(−1)iλ(M )=0.
i
i=0
X
Proof. Evident. (cid:3)
Let Z[[t]] be the ring of formal power series in t with coefficients in Z. Denote
by Z((t)) the localization of Z[[t]] with respect to the multiplicative system {tn |
n∈Z }.
+
Let M be a finitely generated graded A-module. Then the Poincar´e series
P(M,t) of M (with respect to λ) is
P(M,t)= λ(Mn)tn ∈Z((t)).
n∈Z
X
For example, let A = k[X ,X ,...,X ] be the algebra of polynomials in s
1 2 s
variableswithcoefficientsinafieldk gradedbythetotaldegree. Then, A0 =k and
for every finitely generated graded A-module M, we have dim M < ∞. Hence,
k n
we can define the Poincar´e series for λ=dim . In particular, for the A-module A
k
itself, we have
∞
s+n−1 1
P(A,t)= dim Antn = tn = .
k s−1 (1−t)s
n∈Z n=0(cid:18) (cid:19)
X X
The next result shows that Poincar´eseries in general have an analogous form.
1.4.Theorem (Hilbert, Serre). For any finitely generated graded A-module M
we have
f(t)
P(M,t)=
s (1−tdi)
i=1
where f(t)∈Z[t,t−1]. Q
Proof. We prove the theorem by induction in s. If s = 0, A = A and M is
0
a finitely generated A0-module. This implies that Mn = 0 for sufficiently large n.
Therefore, λ(Mn)=0 except for finitely many n∈Z and P(M,t) is in Z[t,t−1].
Assume now that s > 0. The multiplication by x defines an A-module endo-
s
morphism f of M. Let K = kerf, I = imf and L = M/I. Then K, I and L are
graded A-modules and we have an exact sequence
f
0−→K −→M −→M −→L−→0.
This implies that
0−→Kn −→Mn −x→s Mn+ds −→Ln+ds −→0
is an exact sequence of A0-modules for all n∈Z. In particular, by 1.3,
λ(Kn)−λ(Mn)+λ(Mn+ds)−λ(Ln+ds)=0,
1. HILBERT POLYNOMIALS 3
for all n∈Z. This implies that
(1−tds)P(M,t)= λ(Mn)tn− λ(Mn)tn+ds
n∈Z n∈Z
X X
= (λ(Mn+ds)−λ(Mn))tn+ds
n∈Z
X
= (λ(Ln+ds)−λ(Kn))tn+ds
n∈Z
X
=P(L,t)−P(K,t)tds,
i.e.,
(1−tds)P(M,t)=P(L,t)−tdsP(K,t).
Fromtheconstructionitfollowsthatx actasmultiplicationby0onLandK,i.e.,
s
we can view them as A/(x )-modules. Hence, the induction assumption applies to
s
them. This immediately implies the assertion. (cid:3)
Since the Poincar´e series P(M,t) a rational function, we can talk about the
order of its pole at a point. Let d (M) be the order of the pole of P(M,t) at 1.
λ
Bythe theorem, f(t)= a tk witha ∈Zanda =0forallk ∈Zexcept
k∈Z k k k
finitely many. Let p be the order of zero of f at 1. Assume that p > 0. Then
f(t) = (1−t)g(t) where g(tP) = b tk, with b ∈ Q and b = 0 for all k ∈ Z
k∈Z k k k
exceptfinitely many. Moreover,wehavea =b −b forallk ∈Z. Byinduction
k k k−1
P
in k this implies that b ∈Z. By repeating this procedure if necessary, we see that
k
f(t) = (1−t)pg(t) where g(t) = b tk, with b ∈ Z and b = 0 for all k ∈ Z
k∈Z k k k
except finitely many. Moreover, g(1)6=0.
P
1.5. Corollary. If d = 1 for 1 ≤ i ≤ s, the function n 7−→ λ(Mn) is equal
i
to a polynomial with rational coefficients of degree d (M)−1 for sufficiently large
λ
n∈Z.
Proof. Let p be the order of zero of f at 1. Then we can write f(t) =
(1−t)pg(t) with g(1)6=0. In addition, we put d=d (M)=s−p, hence
λ
g(t)
P(M,t)= .
(1−t)d
Now,
∞ ∞
d(d+1)...(d+k−1) d+k−1
(1−t)−d = tk = tk,
k! d−1
k=0 k=0(cid:18) (cid:19)
X X
and if we put g(t)= N a tk we get
k=−N k
P N
d+n−k−1
λ(Mn)= a
k
d−1
k=−N (cid:18) (cid:19)
X
for all n≥N. This is equal to
N N
(d+n−k−1)! (n−k+1)(n−k+2)...(n−k+d−1)
a = a ,
k k
(d−1)!(n−k)! (d−1)!
k=−N k=−N
X X
4 I.DIFFERENTIAL OPERATORS WITH POLYNOMIAL COEFFICIENTS
hence λ(Mn) is a polynomial in n with the leading term
N nd−1 nd−1
a =g(1) 6=0.
k
(d−1)! (d−1)!
!
k=−N
X
(cid:3)
We call the polynomial which gives λ(Mn) for large n ∈Z the Hilbert polyno-
mial of M (with respect to λ). From the proof we see that the leading coefficient
of the Hilbert polynomial of M is equal to g(1) .
(d−1)!
Returning to our example of A=k[X ,X ,...,X ], we see that
1 2 s
s+n−1 ns−1
dim An = = +....
k
s−1 (s−1)!
(cid:18) (cid:19)
Hence, the degree of the Hilbert polynomial for A = k[X ,X ,...,X ] is equal to
1 2 s
s−1.
Now we are going to prove a characterizationof polynomials (with coefficients
inafieldofcharacteristic0)havingintegralvaluesforlargepositiveintegers. First,
we remark that, for any s∈Z and q ≥s, we have
+
q
qs =s! +Q(q)
s
(cid:18) (cid:19)
where Q is a polynomial of degree s−1. Therefore any polynomial P of degree d,
for large q, can be uniquely written as
q q q
P(q)=c +c +...+c +c ,
0 1 d−1 d
d d−1 1
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
with suitable coefficients c , 0 ≤ i ≤ d. Since binomial coefficients are integers, if
i
c , 0≤i≤d, are integers, the polynomial P has integral values for integers n≥d.
i
The next result is a converse of this observation.
1.6. Lemma. If the polynomial
q q q
q 7−→P(q)=c +c +...+c +c
0 1 d−1 d
d d−1 1
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
takes integral values P(n) for large n ∈ Z, all its coefficients c , 0 ≤ i ≤ d, are
i
integers.
Proof. We prove the statement by induction in d. If d = 0 the assertion is
obvious. Also
d d
q+1 q
P(q+1)−P(q)= c − c
i i
d−i d−i
i=0 (cid:18) (cid:19) i=0 (cid:18) (cid:19)
X X
d d−1
q+1 q q
= c − = c ,
i i
d−i d−i d−i−1
i=0 (cid:18)(cid:18) (cid:19) (cid:18) (cid:19)(cid:19) i=0 (cid:18) (cid:19)
X X
using the identity
q+1 q q
= +
s s s−1
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
for q ≥ s ≥ 1. Therefore, q 7−→ P(q+1)−P(q) is a polynomial with coefficients
c ,c ,...,c , and P(n)∈Z for large n∈Z. By the induction assumption all c ,
0 1 d−1 i
0≤i≤d−1, are integers. This immediately implies that c is an integer too. (cid:3)
d