The Gröbner fan of an An-module
Abstract
Let I be a non-zero left ideal of the Weyl algebra An of order n over a field k and let L:R2n→R be a linear form defined by L(α,β)=∑i=1neiαi+∑i=1nfiβi. If ei+fi≥0, then L defines a filtration F•L on An. Let grL(I) be the graded ideal associated with the filtration induced by F•L on I. Let finally U denote the set of all linear form L for which ei+fi≥0 for all 1≤i≤n. The aim of this paper is to study, by using the theory of Gröbner bases, the stability of grL(I) when L varies in U. In a previous paper, we obtained finiteness results for some particular linear forms (used in order to study the regularity of a D-module along a smooth hypersurface). Here we generalize these results by adapting the theory of Gröbner fan of Mora-Robbiano to the D-module case. Our main tool is the homogenization technique initiated in our previous paper, and recently clarified in a work by F. Castro-Jiménez and L. Narváez-Macarro.
Full text
The G obne an o an
A
n
-mo dule
A. Assi
, F.J. Cas o-Jimenez
y
and M. G ange
z
Abs ac
Le
I
b e a non-ze o le ideal o he Weyl algeb a
A
n
o o de
n
o e a eld
k
and le
L
:
R
2
n
?!
R
b e a linea o m dened by
L
(
;
) =
P
n
i
=1
e
i
i
+
P
n
i
=1
i
i
. I
e
i
+
i
0, hen
L
denes a l a ion
F
L
on
A
n
. Le g
L
(
I
) b e he g aded ideal asso cia ed o he l a ion
induced by
F
L
on
I
. Le nally
U
deno e he se o all linea o ms
L
o which
e
i
+
i
0
o all 1
i
n
. The aim o his pap e is o s udy, by using he heo y o G obne bases,
he s abili y o g
L
(
I
) when
L
a ies in
U
. In a p e ious pap e , we ob ained ni eness esul s
o some pa icula linea o ms (used in o de o s udy he egula i y o a
D
-mo dule along
a smo o h hyp e su ace). He e we gene alize hese esul s by adap ing he heo y o G obne
an o Mo a-Robbiano o he
D
-mo dule case. Ou main o ol is he homogeniza ion echnique
ini ia ed in ou p e ious pap e , and ecen ly cla ied in a wo k o F. Cas o-Jimenez and L.
Na aez-Maca o.
1991 Ma h. Sub j. Class: P ima y 35A27, Seconda y 13P10, 68Q40
1 In o duc ion
Le
A
n
(
k
) deno e he Weyl algeb a o o de
n
o e a eld
k
:
A
n
(
k
) (
A
n
o sho ) is he
cen al
k
-algeb a gene a ed by
x
i
; D
i
; i
= 1
;::: ;n
wi h ela ions [
x
i
; x
j
] = [
D
i
; D
j
] = 0 and
[
D
i
; x
j
] =
ij
. Le
P
=
P
;
p
;
x
D
b e a non-ze o elemen o
A
n
and deno e by
N
(
P
) he
New on diag am o
P
, namely
N
(
P
) =
(
;
)
2
N
2
n
;
p
;
6
= 0
g
:
I
L
:
R
2
n
?!
R
is he linea o m dened by
L
(
;
) =
P
n
i
=1
e
i
i
+
P
n
i
=1
i
i
, hen he
L
-
o de o d
L
(
P
) o
P
is dened o b e he maximal elemen in he se o
L
(
;
)
;
(
;
)
2 N
(
P
).
I u he mo e
e
i
+
i
0 o all 1
i
n
, hen o d
L
(
P :Q
) = o d
L
(
P
) + o d
L
(
Q
) o
all non-ze o elemen s
P ; Q
2
A
n
, in pa icula
L
denes a l a ion on
A
n
(whe e o all
k
2
R
; F
L
k
(
A
n
) =
P
2
A
n
; o d
L
(
P
)
k
g
). I
e
i
+
i
>
0 o all
i
= 1
;::: ;l
( esp.
e
i
+
i
= 0
o all
i
=
l
+ 1
;::: ;n
), hen he asso cia ed g aded algeb a is:
Uni e si e d'Ange s
y
Uni e sidad de Se illa. Pa ially supp o ed by he DGICYT PB94-1435
z
Uni e si e d'Ange s
1
g
L
(
A
n
)
'
k
[
x
1
;::: ;x
n
;
1
;::: ;
l
][
D
l
+1
;::: ;D
n
]
wi h ela ions:
x
i
x
j
=
x
j
x
i
; x
i
m
=
m
x
i
; x
i
D
p
=
D
p
x
i
?
ip
;
m
D
p
=
D
p
m
o all 1
i; j
n
, 1
m
l
and
l
+ 1
p
n
. The p incipal symb ol o
P
is he elemen o
g
L
(
A
n
),
L
(
P
) =
X
L
(
;
)=o d
L
(
P
)
p
x
1
1
l
l
D
l
+1
l
+1
D
n
n
Le us p oin ou ha in he commu a i e case, he e is no condi ion o he yp e
e
i
+
i
0.
He e, since
?
x
i
D
i
+
D
i
x
i
= 1, we mus equi e o d
L
(1)
o d
L
(
x
i
) + o d
L
(
D
i
).
Le
I
b e a non-ze o le ideal o
A
n
and le g
L
(
I
) b e he g aded ideal asso cia ed wi h
he l a ion induced by
F
L
on
I
. Le nally
U
deno e he se o all linea o m
L
o which
e
i
+
i
0 o all 1
i
n
. The aim o his pap e is o s udy, by using he heo y o s anda d
and G obne bases, he s abili y o g
L
(
I
) when
L
a ies in
U
.
Le
Y
b e he hyp e su ace o
k
n
dened by
x
1
= 0. Gi en wo non nega i e eals
p; q
, we dene
he linea o m
L
p;q
on
R
2
n
by
L
p;q
(
;
) =
p:
(
P
n
i
=1
i
) +
q :
(
1
?
1
): his is an in e p ola ion
b e ween he l a ion
F
by he o de o op e a o s (
q
= 0) and he
V
-l a ion o Malg ange-
Kashiwa a (
p
= 0). In [10], Y. Lau en p o ed, using 2-mic o die en ial op e a o s, ha he
adical ideal
p
g
L
p;q
(
I
) is no a (
F ; V
)- homogeneous ideal o only a ni e se o a ional
numb e s
=
p=q
(e en ually emp y), when
p; q
a y in
R
2
+
(in [11] and [13], an analy ic
in e p e a ion o hese numb e s is gi en). C. Sabbah and F. Cas o p o ed in [15 ] he same
esul by using a lo cal a ene . In [2] we ob ained, using he heo y o s anda d bases, a
cons uc i e p o o o his esul . This allowed us o gi e an algo i hm o he calcula ion o
hese numb e s. So, i was na u al o hink ab ou gene al ni eness esul s when
L
a ies in
U
.
Recall ha he heo y o G obne bases (c [5]) wo ks e y well in he Weyl algeb a
A
n
(c
[6],[7] and [8]). Howe e , when he co ecien s o he linea o m
L
2
U
a e nega i e, he
di ision p o cess in
A
n
can b e inni e. In [2], in o de o a oid his dicul y, we wo ked in
A
n
[
] by homogenizing wi h esp ec o he o al o de in a way inspi ed by [12]. Howe e , he
non commu a i i y o
A
n
[
] causes some dicul y, since di isions by homogeneous elemen s
do no p o duce necessa ily homogeneous emainde s. Al hough ou algo i hm (which consis s
in ehomogenizing he emainde s and i e a ing di ision) allows us he calcula ion o s anda d
bases wi h esp ec o any
L
2
U
, i do es no seem o b e adap ed o he ques ion we a e wo king
on. In [9], his dicul y is a oided in he ollowing na u al way: conside he g aded
k
-algeb a
B
, gene a ed by
x
i
; D
i
; i
= 1
;::: ;n
and
wi h homogeneous ela ions:
[
; x
i
] = [
; D
i
] = [
x
i
; x
j
] = [
D
i
; D
j
] = 0
;
[
D
i
; x
j
] =
ij
2
:
This
k
-algeb a coincides wi h he Rees algeb a asso cia ed wi h he Be ns ein l a ion on
A
n
.
The homogeniza ion p o cess b e ween
A
n
and
B
e i y he same p op e ies as in he commu a i e
case, in pa icula he no ion o he homogenized ideal
h
(
I
) o
I
is well dened. On he o he
2
hand, we can ge he die en g aded ideals g
L
(
I
) om calcula ions o he g aded ideals o
h
(
I
). Since he no ion o educed s anda d bases exis s o ideals in
B
, hen he na u al way
in o de o s udy ou ques ion is in adap ing o he
D
{mo dule case he heo y o G obne an
de elop ed by T. Mo a and L. Robbiano in [14].
Le us summa ize he s uc u e o he pap e : in Sec ion 2 we ecall some o he esul s o
[9] ela ed o he homogeniza ion p oblem. We also p o e ha a s anda d basis w. . . he
Be ns ein l a ion o an ideal
I
in
A
n
gi es us a gene a ing sys em o
h
(
I
) in
B
=
A
n
[
]. In
sec ion 3, pa ag aph 3.2., we ob ain ni eness esul s o he se o g aded ideals g
L
(
J
) whe e
J
is an homogeneous ideal o
A
n
[
]. The main o ol we use he e is he Hilb e unc ion o
J
. This
no ion has also b een used by one o us [1] in o de o p o e simila esul s in he commu a i e
case. The esul s o pa ag aph 3.2. a e hen applied in o de o p o e ha he se o g aded
ideals g
L
(
I
)
; L
2
U
is ni e (pa ag aph 3.3.). Finally, in sec ion 4, we s udy he epa i ion
o he g aded ideals g
L
(
h
(
I
)), whe e
L
a ies in
U
. We dene s he no ion o p i ileged
exp onen (o s ai s) Exp
L
(
h
(
I
)) o an ideal asso cia ed wi h a xed well o de ing on
N
1+2
n
(see
3.3.). Ou main esul is hen he ollowing, which gene alizes he esul s in he commu a i e
case as ound in [1], [14] and [17]:
Theo em 1.1
The e exis s a pa i ion
E
o
U
in o con ex a ional polyhed al cones, such ha
o al l elemen
2 E
,
g
L
(
h
(
I
))
and
Exp
L
(
h
(
I
))
do no depend on
L
2
(and he same is
ue o
g
L
(
I
)
).
Some esul s o his a icle has b een used in [16].
2 Homogeniza ion
We shall use he e he esul s o [9]. Le
A
n
[
] deno e he algeb a
A
n
[
] =
k
[
; x
][
D
] =
k
[
; x
1
;:::;x
n
][
D
1
;:::;D
n
]
wi h ela ions:
[
; x
i
] = [
; D
i
] = [
x
i
; x
j
] = [
D
i
; D
j
] = 0
;
[
D
i
; x
j
] =
ij
2
:
The algeb a
A
n
[
] is a g aded algeb a, he deg ee o he monomial
k
x
D
b eing
k
+
j
j
+
j
j
:
In ac , he
k
{algeb a
A
n
[
] is isomo phic o he Rees algeb a asso cia ed wi h he Be ns ein
l a ion on
A
n
. The algeb a
k
[
] is cen al in
A
n
[
], and he quo ien algeb a
A
n
[
]
=
h
?
1
i
is
isomo phic o
A
n
.
Le
P
=
P
;
p
;
x
D
b e a non-ze o op e a o o
A
n
. We deno e by
N
(
P
) he New on
diag am o
P
,
N
(
P
) =
(
;
)
2
N
2
n
;
p
;
6
= 0
g
;
hen we deno e by o d
T
(
P
) he o al o de o
P
3
o d
T
(
P
) = max
j
j
+
j
j
;
p
;
2 N
(
P
)
g
:
The die en ial op e a o
h
(
P
) =
X
;
p
;
o d
T
(
P
)
?j
j?j
j
x
D
2
A
n
[
]
is called he homogeniza ion o
P
. I
H
=
P
k ;;
h
k ;;
k
x
D
is an elemen o
A
n
[
], we deno e
by
H
j
=1
he op e a o o
A
n
H
j
=1
=
X
k ;;
h
k ;;
x
D
:
Wi h he no a ions ab o e, o all
P ; Q
2
A
n
and o all homogeneous elemen
H
2
A
n
[
],
1.
h
(
P Q
) =
h
(
P
)
h
(
Q
).
2. The e exis s
k ; l ; m
2
N
such ha
k
h
(
P
+
Q
) =
l
h
(
P
) +
m
h
(
Q
).
3. The e exis s
k
2
N
such ha
k
h
(
H
j
=1
) =
H
.
Le
<
b e a o al o de ing on
N
2
n
(no necessa ily a well o de ing), compa ible wi h sums. We
ecall ha he ex ension o
<
, deno ed by
<
h
, is he o al well o de ing on
N
1+2
n
(compa ible
wi h sums) dened by:
(
k ; ;
)
<
h
(
k
0
;
0
;
0
)
()
8
<
:
k
+
j
j
+
j
j
< k
0
+
j
0
j
+
j
0
j
o
k
+
j
j
+
j
j
=
k
0
+
j
0
j
+
j
0
j
and
(
;
)
<
(
0
;
0
)
Since
<
h
is a o al well o de ing compa ible wi h sums, we ha e o all non-ze o elemen
G
=
P
a;;
g
(
a;;
)
a
x
D
he no ion o p i ileged exp onen o
G
w. . .
<
h
, which we deno e by
exp
<
h
(
G
): I
N
(
G
) =
(
a; ;
);
g
(
a;;
)
6
= 0
g
deno e he New on diag am o
G
, hen exp
<
h
(
G
) =
max
<
h
N
(
G
). Also we ha e o all non-ze o ideal
J
o
A
n
[
], he no ion o G obne (o s anda d)
basis o
J
, namely, i we deno e by
Exp
<
h
(
J
) =
exp
<
h
(
P
)
j
P
2
J
g
;
hen
P
1
;::: ;P
g
J
is a s anda d basis o
J
i
Exp
<
h
(
J
) =
[
i
=1
(exp
<
h
(
P
i
) +
N
1+2
n
)
:
We ha e nally a di ision heo em in
A
n
[
], analogous o ha in he ing o p olynomials o in
he Weyl algeb a
A
n
. Fo mo e de ails, see [9]. Le
:
N
1+2
n
=
N
N
2
n
!
N
2
n
deno e he
na u al p o jec ion, hen we ha e:
1. I
P
2
A
n
, hen
(exp
<
h
(
h
(
P
))) = exp
<
(
P
).
4
2. Mo e gene ally, i
H
is an homogeneous elemen o
A
n
[
], hen
(exp
<
h
(
H
)) =
(exp
<
h
(
h
(
H
j
=1
))) = exp
<
(
H
j
=1
)
:
Le
I
b e a le ideal o
A
n
. We deno e by
h
(
I
) he homogeneous ideal o
A
n
[
], gene a ed
by
h
(
P
)
j
P
2
I
g
. We call
h
(
I
) he homogenized ideal o
I
. Wi h hese no a ions we ha e he
ollowing (see [9]):
1.
(Exp
<
h
(
h
(
I
))) = Exp
<
(
I
)
:
2. Le
P
1
;::: ;P
m
g
b e a gene a ing sys em o
I
and le
e
I
b e he ideal gene a ed by
h
(
P
1
)
;::: ;h
(
P
m
)
g
in
A
n
[
]. Then
(Exp
<
h
(
e
I
)) = Exp
<
(
I
).
Le
B
(
A
n
) deno e he Be ns ein l a ion on
A
n
( ha is he case wi h
e
i
=
i
= 1 o all
i
= 1
;::: ;n
). I
P
is a die en ial op e a o in
A
n
, hen we deno e by
B
(
P
) he p incipal
symb ol o
P
w. . . he Be ns ein l a ion. I
I
is an ideal o
A
n
, hen we deno e by g
B
(
I
) he
g aded ideal asso cia ed wi h he induced Be ns ein l a ion on
I
.
A s anda d basis w. . . he Be ns ein l a ion has he ollowing in e es ing p op e y:
Lemma 2.1
Le
I
be a non-ze o le ideal o
A
n
and le
P
1
;::: ;P
m
g
be a amily o die en ial
ope a o s o
I
. The ol lowing asse ions a e equi alen :
i)
h
(
I
) = (
h
(
P
1
)
;::: ;h
(
P
m
))
:
ii)
g
B
(
I
) = (
B
(
P
1
)
; : : : ;
B
(
P
m
))
:
P o o .
The p o o is classical and uses he s uc u e o g aded algeb a o
A
n
[
] (see o de ails
[3]). Rema k ha a s anda d basis wi h esp ec o he Be ns ein l a ion sa ises ii), bu he
con e se is in gene al alse.
3 Fini eness esul s
Le
L
2
U
(see 1) and conside he ex ension o
L
o
R
R
2
n
(by abuse o no a ion we con inue
o w i e
L
and
U
in
R
R
2
n
),
L
:
R
R
2
n
!
R
, such ha
L
(
a; ;
) =
P
n
i
=1
e
i
i
+
P
n
i
=1
i
i
.
Recall in pa icula ha
e
i
+
i
0 o all 1
i
n
.
Le
P
b e a non-ze o die en ial op e a o o
A
n
[
]. We dene he
L
{o de o
P
in he usual
way (we deno e his elemen by o d
L
(
P
)). I
P ; Q
2
A
n
[
], hen o d
L
(
P Q
) = o d
L
(
P
) + o d
L
(
Q
),
consequen ly he
L
{o de denes a l a ion on
A
n
[
], which we shall call he
L
-l a ion and we
shall deno e by
F
L
(
A
n
[
]). We deno e by
L
(
P
) he p incipal symb ol o
P
w. . . he
L
-o de ,
p ecisely, i
P
=
P
p
(
)
x
D
, hen
L
(
P
) =
P
L
(
;
)=o d
L
(
P
)
p
(
)
x
1
1
l
l
D
l
+1
l
+1
D
n
n
wi h
l
is as dened in he in o duc ion. I
J
is a non-ze o homogeneous ideal o
A
n
[
], we deno e
by g
L
(
J
) he g aded ideal asso cia ed wi h he induced
L
{l a ion on
J
(i.e. g
L
(
J
) is he
ideal o g
L
(
A
n
[
]) gene a ed by
L
(
P
)
j
P
2
J
g
). In his sec ion we shall p o e ha , i he
co ecien s
e
i
;
i
a y in
R
, hen he se o g
L
(
J
) is ni e. We shall use in he p o o he Hilb e
unc ion, he e o e we shall s a by ecalling some o i s p op e ies.
5
3.1 Hilb e unc ion
Le
E
N
1+2
n
such ha
E
+
N
1+2
n
=
E
. We dene he Hilb e unc ion o
E
(and we deno e
i by
H
E
) o b e he map
H
E
:
N
7?!
N
:
H
E
(
k
) =
]
(
a; ;
)
2
N
1+2
n
n
E
;
a
+
j
j
+
j
j
=
k
g
;
8
k
2
N
:
Le
J
b e an homogeneous ideal o
A
n
[
] =
k
2
N
A
n
[
]
k
, whe e
A
n
[
]
k
is he
k
{ ec o space
gene a ed by he monomials
a
x
D
o o al deg ee
a
+
j
j
+
j
j
=
k
. We se
J
k
=
A
n
[
]
k
J
.
Le
b e a o al well o de ing on
N
1+2
n
compa ible wi h sums, and le
E
= Exp
(
J
).
Lemma 3.1
Fo al l
k
2
N
, we ha e:
dim
k
(
A
n
[
]
k
=J
k
) =
]
(
a; ;
)
2
N
1+2
n
n
E
;
a
+
j
j
+
j
j
=
k
g
=
H
E
(
k
)
P o o .
Le
P
1
;::: ;P
m
g
b e a amily o homogeneous op e a o s o
J
such ha :
E
=
m
[
i
=1
(exp
(
P
i
) +
N
1+2
n
)
:
I we deno e by
k
i
= o d
T
(
P
i
), hen o all
P
2
A
n
[
]
k
, he e exis s a amily o homogeneous
elemen s
Q
1
;::: ;Q
m
; R
o
A
n
[
] such ha :
1.
P
=
P
m
i
=1
Q
i
P
i
+
R
.
2. o d
T
(
Q
i
) =
k
?
k
i
;
o d
T
(
R
) =
k
.
3. I
R
6
= 0, hen he New on diag am
N
(
R
)
N
2
n
+1
n
E
. Thus
P
2
J
k
()
R
= 0.
In pa icula ,
P
+
J
k
=
R
+
J
k
. This p o es ha he classes, mo dulo
J
k
, o he monomials
a
x
D
, wi h
a
+
j
j
+
j
j
=
k
, (
a; ;
)
62
E
o m a basis o
A
n
[
]
k
=J
k
o e
k
. This p o es
ou asse ion.
Le , o all
k
2
N
,
H
J
(
k
) = dim
k
(
A
n
[
]
k
=J
k
). This denes a map
H
J
:
N
!
N
which we
call he Hilb e unc ion o
J
. By Lemma 3.1,
H
J
=
H
E
do es no dep end on
.
3.2 Fini eness Theo ems o homogeneous ideals
Le
O
(
N
1+2
n
) deno e he se o o al well o de ing on
N
1+2
n
compa ibles wi h sums ( o such
an o de , 0 is he smalles elemen , his implies in pa icula ha exp
(
P Q
) = exp
(
P
) +
exp
(
Q
))).
Theo em 3.2
Le
J
be a non-ze o homogeneous ideal o
A
n
[
]
. Then
Exp
(
J
)
j 2 O
(
N
1+2
n
)
g
is a ni e se .
6
P o o .
By Lemma 3.1, i suces o p o e ha he se o subse s
E
N
1+2
n
such ha :
1.
E
+
N
1+2
n
=
E
.
2.
H
E
=
H
J
.
is ni e. Deno e his se by
E
and assume ha
E
is inni e. Gi en an elemen
E
o
E
and an
in ege
k
2
N
, we se
E
(
k
)
=
2
E
;
j
j
k
g
:
Le
k
0
2
N
b e he smalles in ege o which
H
J
(
k
)
<
dim
k
(
A
n
[
]
k
) (such an in ege exis s
b ecause
J
6
= (0)). Since
N
1+2
n
(
k
0
)
is a ni e se , one o he p ossible choices o
E
(
k
0
)
o ccu s o all
E
in an inni e subse
E
1
=
E
i
g
i
1
o
E
. Thus, he e a e elemen s
i
2
N
1+2
n
(
k
0
)
;
1
i
such
ha
E
i;
(
k
0
)
= (
[
i
=1
(
i
+
N
1+2
n
))
(
k
0
)
o all
i
1
:
Assume, wi hou loss o gene ali y, ha
E
1
=
E
and se
S
0
=
[
i
=1
(
i
+
N
1+2
n
)
:
Clea ly
S
0
E
i
o all
i
1, on he o he hand
E
i
6
=
E
j
o all
i
6
=
j
. In pa icula
H
J
6
=
H
S
0
.
Le consequen ly
k
1
> k
0
b e he smalles in ege o which
H
J
(
k
1
)
< H
S
0
(
k
1
). Fo all
j
2,
he e exis s
j
2
E
j
n
S
0
such ha
j
j
j
=
k
1
. The se
N
1+2
n
(
k
1
)
b eing ni e, he e is an inni e
subse
E
2
E
and elemen s
+
i
;
1
i
+
1
, in (
N
1+2
n
n
S
0
)
(
k
1
)
such ha :
E
j;
(
k
1
)
= (
[
+
1
i
=1
(
j
+
N
1+2
n
)
(
k
1
)
o all
E
j
2 E
2
:
Le
S
1
=
[
+
1
i
=1
(
j
+
N
1+2
n
)
;
hen
S
0
S
1
. Now ep ea he same a gumen wi h
E
2
and
S
1
,... We cons uc his way an
inni e sequence
S
0
S
1
:::
o subse s o
N
1+2
n
wi h
S
i
+
N
1+2
n
=
S
i
o all
i
0. This is
imp ossible.
As a consequence o Theo em 3.2. we ge he ollowing esul :
Theo em 3.3
Le
J
be a non-ze o homogeneous ideal o
A
n
[
]
. Then
g
L
(
J
);
L
2
U
g
is a
ni e se .
7
P o o .
Fix
2 O
(
N
1+2
n
), hen o any
L
2
U
, deno e by
L
he o al o de ing on
N
1+2
n
such
ha :
(
k ; ;
)
L
(
k
0
;
0
;
0
)
()
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
k
+
j
j
+
j
j
< k
0
+
j
0
j
+
j
0
j
o
k
+
j
j
+
j
j
=
=
k
0
+
j
0
j
+
j
0
j
and
8
>
>
<
>
>
:
L
(
k ; ;
)
< L
(
k
0
;
0
;
0
)
o
L
(
k ; ;
) =
L
(
k
0
;
0
;
0
) and
(
k ; ;
)
(
k
0
;
0
;
0
)
(Whe e we ecall ha
L
(
k ; ;
) =
P
n
i
=1
e
i
i
+
P
n
i
=1
i
i
). Clea ly
L
2 O
(
N
1+2
n
). On he
o he hand, by 3.2,
Exp
L
(
J
)
j
L
2
U
g
is a ni e se . Consequen ly we ha e only o p o e
ha , i
E
N
1+2
n
wi h
E
+
N
1+2
n
=
E
, hen
g
L
(
J
)
j
Exp
L
(
J
) =
E ; L
2
U
g
is a ni e
se . Fix o his end
E
and le
L
2
U
b e such ha
E
= Exp
L
(
J
). Then conside a educed
s anda d basis
B
=
Q
1
; : : : ; Q
m
g
o
J
w. . .
L
(i.e.
[
m
i
=1
(exp
L
(
Q
i
) +
N
1+2
n
) =
E
and
N
(
Q
i
)
n
exp
L
(
Q
i
)
g
N
1+2
n
n
E
, o all 1
i
m
, whe e
N
(
Q
i
) is he New on diag am
o
Q
i
). Clea ly
B
is also a educed s anda d basis o
J
w. . .
L
0
, o all
L
0
2
U
such ha
Exp
L
0
(
J
) =
E
(indeed, i exp
L
0
(
Q
i
)
6
= exp
L
(
Q
i
), we would ha e exp
L
0
(
Q
i
)
=
2
E
). In
pa icula , as p o ed in [2], Lemma 1.3.3.,
L
0
(
Q
1
)
;::: ;
L
0
(
Q
m
)
g
gene a es g
L
0
(
J
) o all
L
0
2
U
such ha Exp
L
0
(
J
) =
E
. E e y
N
(
Q
i
) b eing ni e, we ha e only a ni e numb e o
p ossibili ies. This p o es ou asse ion.
We shall nally gi e a b ound o he ca dinali y o
O
(
J
) =
Exp
(
J
);
2 O
(
N
1+2
n
)
g
. Le
o all
E
2 O
(
J
),
J
E
= (
y
1
;::: ;y
s
)
k
[
y
1
;::: ;y
2
n
+1
], whe e
y
1
;::: ;y
2
n
+1
a e inde e mina es
and
1
;::: ;
s
g
is he minimal b ounda y o
E
, ha is
E
=
[
s
i
=1
(
i
+
N
1+2
n
) and o all
k
= 1
;::: ;s
,
k
=
2 [
i
6
=
k
(
i
+
N
1+2
n
). Clea ly
H
J
E
=
H
E
, hen we ha e:
]
O
(
J
) =
]
J
E
;
E
2 O
(
J
)
g
]
M
k
[
y
1
;::: ;y
2
n
+1
] monomial ideal ;
H
M
=
H
J
g
Le
d
(
J
) deno e he maximal deg ee o he elemen s a ising in he minimal b ounda ies o
Exp
(
J
)
;
2 O
(
N
2
n
+1
)
g
. I (
d
1
; d
2
;:::
) deno e he alues o he Hilb e unc ion o
J
, hen
we ha e:
P op osi ion 3.4
]
O
(
J
)
d
(
J
)
Y
k
=1
C
a
k
a
k
?
d
k
;
whe e
a
k
= dim
k
A
n
[
]
k
=
C
2
n
+
k
k
and
C
a
b
is he binomial coecien .
8
P o o .
The numb e o p oin s in
E
which a e exp onen s o monomials o deg ee
k
is exac ly
a
k
?
d
k
. This p o es ou asse ion.
3.3 Fini eness Theo ems o ideals in
A
n
Le
I
b e non-ze o le ideal o
A
n
. The aim o his pa ag aph i o gi e o
I
analogous esul s
o hose o 3.2. Le o his end
<
b e a o al well o de ing on
N
2
n
, compa ible wi h sums, and
deno e, o all
L
2
U
, by
<
L
he o al o de ing on
N
2
n
such ha :
(
;
)
<
L
(
0
;
0
)
,
8
<
:
L
(
;
)
< L
(
0
;
0
)
o
L
(
;
) =
L
(
0
;
0
) and (
;
)
<
(
0
;
0
)
Le
P
2
A
n
b e a non-ze o die en ial op e a o . We deno e by exp
<
L
(
P
) he p i ileged
exp onen o
P
w. . .
<
L
, i.e. exp
<
L
(
P
) = max
<
L
N
(
P
) (See [2] o he main p op e ies o he
p i ileged exp onen o an op e a o ). We also se
Exp
<
L
(
I
) =
exp
<
L
(
P
)
j
P
2
I
n
0
gg
:
Clea ly Exp
<
L
(
I
) +
N
2
n
= Exp
<
L
(
I
).
Theo em 3.5
Fo a gi en o al wel l o de ing
<
on
N
2
n
, compa ible wi h sums,
Exp
<
L
(
I
)
j
L
2
U
g
is a ni e se .
P o o .
This esul s ollows om Theo em 3.2 as ollows: s ly we ema k ha , wi h he
no a ions o sec ion 2,
L
=
<
h
L
, o he ollowing choice o
,
(
k ; ;
)
(
k
0
;
0
;
0
)
,
8
<
:
(
;
)
<
(
0
;
0
)
o
(
;
) = (
0
;
0
) e
k < k
0
Now apply
(Exp
<
h
(
h
(
I
))) = Exp
<
(
I
), o he o de
<
=
<
L
.
Theo em 3.6
g
L
(
I
)
j
L
2
U
g
is a ni e se .
P o o .
Le
h
(
I
) b e he homogenized ideal o
I
in
A
n
[
]. The asso cia ed g aded ideal g
L
(
h
(
I
))
is an ideal o he ing g
L
(
A
n
[
])
'
(g
L
(
A
n
))[
] (whe e [
x
i
;
i
] = 0 i
e
i
+
i
>
0 and [
D
i
; x
i
] =
2
i
e
i
+
i
= 0). Le
:
A
n
[
]
7?!
A
n
;
(
H
) =
H
j
=1
deno e he deshomogeniza ion mo phism. I
L
2
U
,
gi es ise o a mo phism
L
: g
L
(
A
n
[
])
7?!
g
L
(
A
n
)
'
g
L
(
A
n
[
])
=
(
?
1)
:
9