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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
1296 M.-P Malliavin (Ed.)
Serninaire d'Algebre
Paul Dubreil et Marie-Paule Malliavin
Proceedings, Paris 1986
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
1296 M.-P Malliavin (Ed.)
Serninaire d'Algebre
Paul Dubreil et Marie-Paule Malliavin
Proceedings, Paris 1986
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Editeur
Marie-Paule Malliavin Universite Pierre et Marie Curie, Mathernatiques 10, rue Saint Louis en 1'lIe, 75004 Paris, France
Mathematics Subject Classification (1980): 12H05, 12L 10, 13C 15, 13H 10, 14005, 14H20, 14M 17, 15A33, 16A08, 16A 10, 16A 18, 16A34, 16A60, 16A64, 22E30, 43A85, 58G07 ISBN 3·540·18690·5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-18690-5 Springer-Verlag New York Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9,1965, in its version of June 24,1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1987 Printed in Germany
Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140·543210
III
Li ste des auteurs. K.Adjamagbo p . 1 - Y.Andre p. 28 - M. van den Bergh p, 228 M.Brion p, 177 - A.Debiard p, 42 - A.van den Essen p. 125 J.van Geel p. 193 - C.R.Hajarnavis p. 235 - J.Herzog p. 214 G.Krause p. 261 - B.Roux p. 276 • J.T.Stafford p. 247 S.P.Smith p. 158 - M.Zayed p. 312
TABLE DES TITRES K.ADJAMAGBO - Synthese de 1a Theorie du determinant sur un anneau finement f i l t r e ..........•.••..•.•.....••..••....•• 1 V.ANDRE - Quatre descriptions des groupes de Galois differentiels 28 A.DEBIARD - Systeme differentie1 hypergeometrique et parties radiales des operateurs invariants des espaces symetriques de type BC p .........••••..••.. " . . • ••• ••• .••• 42 A.Van den ESSEN - Modules with regular singularities over filtered rings and al.gebraic micro-localization ....•.•.•••.. 125 S.P.SMITH - Curves. differential operators and finite dimensional algebras •..............................•...•...•••• 158
* M.BRION - Sur 1 'image de 1 'application moment .........•••.••••• 177 J.van GEEL - Maximal orders over curves .......•....•......•••.. 193 J.HERZOG - Linear Cohen-Macaulay modules on integral quadrics 214
*
M.van den BERGH - Regular rings of dimension three .........•••• 228 C.R.HAJARNAVIS - Homological and Cohen-Macaulay properties in non-commutative noetherian rings ...•...•••.•.•••.•• 235 J.T.STAFFORD - Global dimension of semi-prime noetherian rings •• 247 261 G.KRAUSE - Chaines d'ideaux annulateurs d'un anneau noetherien B.ROUX - Anneaux de valuation discrete complets scindes. non commutatifs. en caracteristique zero ......•..•..••. 276 M.ZAYED - Ultra produits et modules sans facteurs directs de type fin i •.••.............................••...••.••..•. 312
IV
PREVIOUS VOLUMES OF THE "SEMINAIRE PAUL DUBREIL" WERE PUBLISHED IN THE LECTURE NOTES, VOLUMES 586 867
(1980), 924
(1976), 641
(1977), 740 (1978), 795
(1981), 1029 (1982), 1146
(1983-84)
(1979)
and 1220 (1985).
I
SYNTHESE DE LA THEORIE DU DETERHINANT SUR UN ANNEAU FINEY-ENT FILTRE
Kossivi ADJAHAGBO Universite de Paris VI, U.E.R. 47, U.A. 761
O. INTRODUCTION Nous presentons ici une synthese de nos resultats concernant "le determinant sur un anne au finement filtre", dans le prolongement de ceux de Jean Dieudonne concernant "les determinants sur un corps non commutatif" [12]. Dans cette synthese, nous nous sommes efforces de trouver le point de vue le plus "simple et concis" pour presenter les notions et resultats dont nous sommes tributaires, dans l'espoir que cet effort favorisera la comprehension mutuelle et meme la collaboration entre les specialistes des diverses disciplines mathematiques au carre four desquelles nous nous situons,
a
savoir l' a.lgebre non commutative, la
geometrie algebrique, la K-theorie algebrique et les equations aux derivees partielles. Pour les preuves des resultats originaux presentes ici, nous renvoyons les lecteurs Ln t e r e s s e s
a
[5] ou [ 6] .
Cette presentation a fait l'objet de deux exposes au Seminaire d'Algebre de l'Universite de Paris VI, en octobre 1984 puis en 1985, ainsi que d'un autre au Colloque de Mathematiques de l'Universite Catholique de Nijmegen (Pays-Bas) en avril 1986.
2
Qu'il nous soit done permis de remercier ici Madame Ie Professeur M.P. Malliavin pour l'interet qu'elle a porte
a
notre travail et pour
qu'elle nous a donnee d'exposer et de publier les resultats obtenus, Monsieur Ie Professeur A. Van Den Essen pour la fructueuse collaboration nee de mon invitation
a
Nijmegen sur son initiative,
ainsi que Monsieur Ie Professeur J. Vaillant pour ses constants encouragements qui nous ont permis de developper la presente theorie. Le plan que nous allons suivre est Ie suivant 1. Le concept du determinant. 2. L'ideal caracteristique d'un module de t.f. sur un anneau f.f. 3. Sur les matrices inversibles
a
coefficients dans un anne au
f.f.
4. Sur la K-theorie des anneaux f.f .
5. Sur les operateurs differentiels matriciels localement injectifs. 6. Sur les operateurs differentiels matriciels localement bijectifs.
3
1. LE CONCEPT DU DETERHINANT 1.1. Notation Par anneau nous entendrons unifere dont l'unite est notee Si
A
1.
est un anneau, nous noterons :
A*
A\ {O}
A*
{elements inversibles de A},
M (A) m,n
,
{matrices a
m
lignes et a
colonnes a coefficients
n
dans A}, M
n,n
(A),
(M (A»*, n
{matrices elementaires E M* (A)},
fh(A)
n
u
M(A)
n >
* (A)
0
u
M
n >
0
Si
a E M (A) nous noterons m,n de la colonne j de a.
Si
K
i
et
K, * c'est-a-dire
l'abelianise du groupe
IT
l'application canonique de
K
lJ
est un corps, nous noterons
K' K
a .. le coefficient de la ligne
le monoide
K'
K* dans
U {a} dont
0
K',
est un element absorbant.
1.2. Le determinant de Dieudonne Si K
est un corps, d'apres J. Dieudonne [12], il existe une
application unique de (i) Vn
E
IN* det
(if)
va
E
K
M(K)
,
M(K) dans
V(a,b)
a.b
,
(iii) Vn E N*, Va
E
det det
K
M (K)2 n
Mn(K) i=l K
n E IN*, detKIMn(K)
notee
det
telle que :
K,
,
K a.detKb
= 0 a
n IT IT
Si
K,
Iai j
r; H* (K)
o
pour
1
-
1.
(L (I-1)
.r M' oM)
est alors une
appele "la resolution libre canonique de
sont deux
R-modules, on appelle "groupe d'exten-
N", et on note
d 'homolog ie du complexe
(Hom (L R
(ExtR(M,N), ExtR(M,N)-f)
un , N),
Ie groupe
Hom (f M' N), Hom (oM' N) ) . R R
Pour toute resolution projective (P,f,o) (ExtR(M,N), ExtR(M,N)-f)
M".
de
M, Ie groupe gradue
est alros isomorphe au groupe d'homologie
du complexe (HomR(p,N), HomR(f,N), HomR(o,N».
= ExtR(M,N)-f(n)
On note habituellement entier
n. On a ainsi
o
ExtR(M,N)
=
pour tout
pour tout entier
{a}
n
< 0, et
= HomR(M,N).
6.2. Definition des operateurs differentiels matriciels localement bijectifs Dans toute la suite nous utiliserons les notations de Soient
m
et
n
que l'operateur differentiel matriciel
og
ment (resp. formellement) bijectif si pour tout tion
n
A Ix :
O£
+
m
O£
.
-n
(resp. Alx : O£
5.1.
a E M (D ) . Nous dirons m,n "n A : + est analytique-
des entiers positifs, et
+
-m
O£)
x
de
r/, l'applica-
est bijectif.
6.3. Theoreme de bijectivite locale Soient Dr/-module
m
et
n
des en tiers positifs
An/Am.a.
a E M (D ) m,n "n
,
et
M
Ie
Les propositions suivantes sont equivalentes (1)
L'Operateur differentiel matriciel
a
On r/
+
am r/
est analytique-
a
On r/
+
am r/
est formellement
ment localement bijectif. (2) L'operateur differentiel matriciel localement bijectif. (3) m = n
et
pour tout
x
E
r/,
Ext
D
(Mix, O£)
{O}.
(Mix, O£)
{O}.
£
(4) m = n
et
pour tout
x E r/, Ext
D
£
25
m
16.
n
et
vcar M ==
(6)
m == n
et
iCar p M == gr p D .
(7)
m
et
det p a
(5)
n
(8)
m
n
et
(9)
m
n
et
rz
Ie
}1
(gr
rz * F Drz)
{O} •
a E M*n (Die) •
6.4. Remarques Le theoreme de bijectivite locale resulte du theoreme d'inversibilite de matrices 3.1 et du theoreme d'injectivite locale 5.4, ce dernier resultant luimeme du theoreme classique de cauchyKowaleska. On peut deduire egalement Ie theoreme de bijectivite locale d'un theoreme de P. Schapira et M. Kashiwara ([ 13], 10.1.5)
I
comme l'a
indique ce premier lorsque la question de la bijectivite locale nous a ete posee par S. Mizohata en avril 1984, et comme nous l'avons explique dans [3] .
26
BIBLIOGRAPHIE [1]
K. ADJAMAGBO, Determinant sur des anneaux filtres, C.R.A.S., t.293, serie I, Paris, 1981, p.447-449.
[ 2]
, Theoremes d'indice pour les systemes generaux d'equations differentielles lineaires, dans Equations aux derivees partielles hyperbolio.ues et holomorphes, J. Vaillant ed., Hermann, 1984, p.134-165.
3]
, Sur les systemes lineaires d'equations aux derivees partielles localement inversibles, C.R.A.S., t.299, Serie I, n017, Paris, 1984, p.863-866.
[ 4]
, Sur la calculabilite des determinants de matrices d'operateurs differentiels, dans Actes du Colloque d'algebre 1985, Universite de Rennes.
[ 5]
, Theorie des determinants sur un domaine de Ore ou sur un anneau finement filtre, These, Universite de Paris VI
(a paraitre) . , The determinant over a finely filtered ring (a
6] paraltre) . [7]
J. -E. BJORK, Rings of differential operators, North-Holland Mat herna t Lo a L Library, Vol. 21, 1979.
[8]
N. BOURBAKI, Algebre commutative, chap. 7, Hermann, Paris, 1979.
[ 9J
, Groupes et algebres de Lie, Chap. 1,Hermann, Paris, 1960.
[10] [ 11]
, Algebre, chap. 10, Masson, Paris, 1980. M. DAVIS, E. WEYUKER, Computability, complexity and langages, Computer Science and applied Mathematics, Academic Press, 1983.
[12]
J. DIEUDONNE, Les determinants sur un corps non commutatif, Bull. Soc. Math. Fr., 71, 1943, p.27-45.
[ 13]
M. KASHIWARA, P. SCHAPIRA, Microlocal study of sheaves, Asterisque, 128, 1985.
[ 14]
J. MILNOR, Introduction to Algebraic K-theory, Annals of Mathematics Studies, n072, Princeton, 1971.
27
[15]
Higher Algebraic K-theory I, dans Algebraic K-theory I, Lecture-Notes in Mathematics, 341, Springer-Verlag 1973, p.77-139.
[ 16)
M. SATO, M. KASHIWARA, T. KAWAI, Hyperfunctions and pseudodifferential operators, Lecture Notes in Mathematics, 287, 1973, p.264-S24.
[17]
M. SATO, M. KASHIWARA, The determinant of matrices of pseudodofferential equations, Proc. Japon Acad., 51, 1975, p.17-19.
[ 18]
, Algebre locale. Multiplicites, Lectures Notes in Mathematics, 11, Springer-Verlag, 1965.
[ 19]
Microlocalisation algebrique, dans Seminaire d'Algebre Paul Dubreil et Marie-Paule Malliavin, Lecture Notes in Math., 1146, Springer, 1985, p.299-316.
[20]
A. SUSLIN, On the structure of the special linear group over polynomial rings, devant paraitre dans Izvestia Acad. Nauk. U.S.S.R.
[21]
A. VAND DEN ESSEN, Algebraic Microlocalization, dans Communications in Algebra, 14, 1986, p.971-1000.
QUATRE DESCRIPTIONS DES GROUPES DE GALOIS DIFFERENTIELS
Yves Andre U.A.763
du C.N.R.S.
Institut Poincare II, rue P. et M. Curie 75231 PARIS
Ce texte reproduit presque fidelement, mais sans demonstration, les exposes des 13 et 20 Octobre 1986, et presente quelques aspects -tour
a
tour algebriques, arithmetiques, topologiques et geometriques- de la
theorie de Galois des vectoriels
a
connexion integrable ; dans une
autre terminologie, il s'agit de proprietes galoisiennes generiques des
D-modules
O-coherents. Les preuves seront exposees ailleurs [1]
I. INTRODUCTION 1. Image classique
Soit
k
(Picard-Vessiot)
un sous-corps de
posons
K
k(x). On considere une
equation differentielle lineaire homogene (*)
au
y(n)
+ gn-1 y(n-l) + ... + goY = 0 ,
l'on suppose que les elements
Ainsi la place nulle de
K
Y1,·· "Y n
K=
de
(*)
dans
gi
de
K
n'ont pas de pole en
est ordinaire, et il existe
n
O.
solutions
k[[x]], lineairement independantes sur
I (n-1) F = K(Y 1""'Yn'Y 1""'Yn ) de k«x» est stable sous la derivation a = d/dx. Le groupe de Galois differentiel clas-
Le sous-corps ique de
F/K
est Ie groupe
AutaF/K
des automorphismes de
F
qui
k.
29
fixent
K
et commutent
de f Ln L sur
a
a
c'est un sous-groupe algebrique de GL n,
k,
Lorsque
k
K, ce qui
est algebriquement clos, on a
fournit une correspondance galoisienne. Exemples
quadratures
Y'
1.1)
gY
envoie
Y = exp f x g. Tout o E AutaF/K 1 * 0 ou aY 1 ' a E k ,done AutaF/K est (];m
adrnet la solution
sur un multiple
JIl p ' Faisons
k = Ql
et
Y 1
alors
g =
3 (l+x)
ce qui viole la correspondance precedente. On peut la retablir en adjoignant e 2 i IT/3 a k. 1.2)
Y" _ (g'/g)Y'
Y2 = f
x o g. Tout
done
Aut P/K = (];a
o
adrnet les solutions independantes
AutaP/K
0
ou
0
envoie
Y
2
sur
Y
+ exY
2
Yl
avec
1
1,
ex
k
dans ce cas.
1.3) On dit qu'une equation (*) est resoluble par quadratures, et que I' extension correspondante peuvent
F/K
est de
si ses solutions
construites en iterant les procedes
dents, i.e. formation de equivaut
a:
AutaF/K
et
exp
1.1) et 1.2)
prece-
On montre que cette condition
est resoluble (pour
k
algebriquement clos).
2. Critique La definition
AutaF/K
n'est pas satisfaisante, et
a
plusieurs
egards : 2.1) Le cas particulier
Y'
- 3(i+x) Y = 0
ci-dessus suggere que Ie
" v r ai" groupe de Galois differentiel devrait pe k
Aut
=
Ql.
aF/K=l
n'
est que Ie groupe des points de
2.2) La definition depend (s=O
fi13
a
et que Ie grou-
fi13
a
valeurs dans
priori du choix de la place ordinaire
ci-dessus) et de la construction de solutions. On souhaiterait
une definition
a
partir des seules donnees
K/k
et
(*).
2.3) La definition, posee dans Ie cadre (delicat) de l'algebre differentielle, ne tire pas assez parti de la linearite de (*). La definition que nous utiliserons, inspiree de la dualite tannakienne [16], est une variante de celIe de [10]
; elle est intrinseque
30
et ramene l'algebre differentielle issue d' equations aux derivees partielles lineaires, neaire.
un systeme integrable d' de l'algebre multili-
a
3. Le groupe de Galois differentiel "intrinseque". Soit
k
un corps de caracteristique nulle et
de type fini dans laquelle Der(K/k)
k
l'algebre de Lie des derivations de
Ie
K/k
une extension
est algebriquement ferme. On note K
annulant
k, et
Kespace dual. Soit
MIC la categorie des Kvectoriels K/ k a connexion integrable : un objet !: = (M, v) est forme d' un Kespace vectoriel de dimension finie M et d' une application 1
klineaire v: M M@ verifiant la regIe de Leibniz et telle que l'application Klineaire deduite de v (et notee encore v) Der(K/k)
EndkM
soit un morphisme
d'
algebre de Lie; un morphisme
de
MIC est une application Klineaire qui commute aux connexions, K/k si bien que MICK/ est une categorie klineaire. k Pour Ie formalisme des connexions integrables, on renvoie le lecteur
aux premiers paragraphes de [8J. Soit M E ob MIC On note M la plus petite souscategorie K/ k. pleine de MIC contenant l'objet et stable par @, 0, A et K/k sousobj ets. On note la plus petite souscategorie pleine de MICK/k contenant M et stable par dualite ( )Y, c'est une petite categorie abelienne dont on note l'ensemble des elements , de ses objets pour lesquels
v,
=
0
(tenseurs horizontaux) .
Definition : Le groupe de Galois differentiel sateur dahs des objets de p61ynomiales de GL (cf (101) M Autrement dit
(M)
G (M) est Ie stabiligal dans les diverses representations
est Ie plus grand sousschema en groupes
ferme de GL qui stabilise les sousKvectoriels M les constructions standard 0, A sur M.
vstables dans
reprenons le cas particulier suivant 1.1 : k = ill, 1 est le a 3 O+x) = . On verifie que Gg a l (M) K- groupe algebrique llr Dans le but de simplifier l'expositioI'!, nous
Exemple M
=
K
:
ill (x) , v (0)
31
supposerons desormais que le degre de transcendance de
K/k
est au
plus un. 4. Descriptions a) Algebriques. Dans la suite, Fix(.)
designe Ie fixateur dans
GL
M
de (.). On montre que 4.1)
G 1 (M) = ga V' et = Fix(M)
xK(G g a l (M)) So it Soit
si Ie groupe des caracteres
K-rationnels
est fini.
K le complete de K en une place rationnelle ordinaire s. s F la K-algebre engendree par HomV' (KS,d}) (le k-espace
des solutions locales de
(M,
¥'
M), sur laquelle l'action de
Der(K/k) se
prolonge naturellement ; Ie lien entre les paragraphes 1 et 3
est
donne par l'isomorphisme 4.2}
G
ga
1 (M)
(i< s )
AutDer(K/k} (F/K)
b} Arithmetique. On peut reduire nombre premier
Ks
M modulo
p, pour presque tout
p, faisant apparaitre l'operateur de
II existe un groupe
Gcurv(M)
p-courbure
(voir §III) dont l'algebre de Lie est
minimale vis-a-vis de la propriete de contenir (apres reduction mod. p)
les
pour presque tout
ture de N. Katz [10]
4.3)
o lIM} Gga
2
p. Dans ce contexte, la conjec-
s'ecrit :
Gcurv(M),
ou l'exposant nul designe la composante neutre. L'inclusion
est facile
l'egalite est verifiee lorsque
CO
est resoluble (voir §IV) • c} Topologique. On suppose
k =
normale de corps de fractions
;
K, et
soient s
A
une
k-algebre integre
une place de
Gmono(M,s} l'adherence de Zariski de l'image de
gal
A. On note
IT l (A,s)
dans
V' ). La correspondance de Riemann-Hilbert donne pour GL((M @ K s) regulier (fuchsien)
M
32
d)
Geometrique. On suppose ici que
M est la cohomologie de De Rham
* HDR(X) d'une variete complete lisse sur
K i munie de la connexion de
Gauss-Manin V. On propose la conjecture suivante : G;al (M)
4.5}
G;al (N)
n'
§VIII}, et
DO
M,
l'ensemble des cycles de Hodge absolus sur
X
(voir
la composante neutre du groupe derive.
c:
L'inclusion Fix(M h. a )
de
est trivial.
Mh. a
On a note
N
si pour aucun facteur
est toujours verifiee. On montre l'egalite lorsque abelien, pour la fibre reduite modulo l'ideal maximal
d'une place de bonne reduction
s.
ga l(M)
II. G
1. Fonctorialites.
Boit
1.1}
Gg a l (M)
M. Alors Ie morphisme naturel
N E ob
sur
G (N) • gal
1.2) A toute extension des scalaires
objet 1.3)
kIM de Boit
GL M
MIC /k I' On a Kk,
K /k
une extension de
k'/k G (k I M) ga l
GL N
est associee G
ga
1 (M)
e
K
envoie
a M
un
Kk I .
"corps de fonctions". D'apres Katz
[ 101, on a un isomorphisme sur les composantes neutres :
GOgal associe
(M}@v .. ' " O G O ( M ) '
gal
K
a M
K'
, ou
MK"
designe l'objet de
MICK'/k
par extension de la base.
De plus, il existe
K'/K
eg a l
, ) soit connexe. (La K preuve utilise la description topologique et la reduction mod.p). Inversement, s L on part un objet
ResK'/KN
d'
tel que
un objet
(N,vI K) de
(M
N de
MICK'/k' on lui associe
MICK/ k ,et on a : (N})
e
K'.
K
2. Le probleme des sous-espaces stables.
Tout sous-espace GLM
v-stable d'une representation polynomiale de
est, par definition, stable sous
G liM) ga
; la reciproque est
33
=
fausse. A titre de contrexemple, soit module a connexion triviale , et soit dans
(l,9,)
avec
@
9,
E K \
L
k , Alors
1
(K , d : K
le
la droite engendree par Lest fixee par
mais n'est pas un objet de
1
Le probleme vient de ce que la categorie des representations de Gg a l (M)
est
K-lineaire, contrairement
a
M
: Les assertions suivantes sont equivalentes i) Gg a l (M)
agit homothetiquement sur
M,
ii) End M est trivial (i.e. somme de
M
iii)
dans
est somme directe de sous-objets de rang un tous isomorphes,
et sont impliquees par : iv)
M est le plus petit objet de
sous
M con tenant une droite
L
stable
(M) •
s
si Gg a l (M) et seulement s'il est combinaison lineaire a coefficients dans K V• d'elements de M On en tire la description 1.4.1. On deduit de la qu'un tenseur
III.
E @ 0 M
est fixe par
(M)
On considere de nouveau l'equation differentielle (*),
1.
ou l' on suppose que forme
d'
Lorsque
p
l/m] (x). On peut aussi l' ecrire sous la
E
8Y + GY
O.
est un nombre premier ne divisant pas
reduire (*) modulo coefficients dans La
gi
un systeme differentie1
m, on peut
p, d'ou une equation differentie1le (*)p IT
P
a
(x'
I
= {d/dx + G)p mod. p , est un operateur lineai-
p-courbure
re (et non differentiel). E1le se calcule par recurrence [ 10] G(l)
D'apres P. Cartier, IF'p (x) ,
(*)
=
G, G{k+l)
p
8G(k) + G.G{k)
o
(*)
p
,
=
G{p)
mod. p.
est entierement soluble dans
(Grothendieck) : 5i = 0 pour presque tout p, alors p est entierement soluble par des fonctions a1gebriques sur m{x).
34
Pour plus de precisions, on renvoie le lecteur aux articles de Katz [8] [ 9] .
2. L'algebre de Lie de Katz Par epaississement de
5 cu r v M
=
de The k $
-+
[lO]IX.
' on entend la donnee (M,V) E Ob MIC K/ k
un sous-anneau de type fini sur
Spec
Ik
une
Ik-variete lisse, de corps de fonctions
satisfaisant a
IK
((\
k =
Ik un module localement libre
(IM,V)
IK
K, et
a
connexion integrable sur
$
avec (IM,W) @ Spec K = (M,V).
s
Un epaississement permet de reduire modulo Apparaissent les 1jJ
pour presque tout
p.
p-courbures:
:IM@IF-+IM@
p
p
p
*
1 @ F abs (r/$ @ IF
IF P
p
)
,
donnees par
Soit M l'ensemble des sous K-vectoriels V d'objets de -curv tels qu'il existe un epaississement avec 1jJ (D) V e W mod.p pour
M,
p
presque tout p. lrcurv(M)
=
l:"
Stab
e End M.
Lie Gga 1 (M) • La conjecture de Grothendieck mentionnee plus haut equivaut au cas particulier):c
=
(14)
(Mc u r v ) =
O.
Reciproquement (voir [lO)X), si la conjecture de Grothendieck est satisfaite pour tout objet de
M,
il en est de meme de celle de Katz.
3. Le probleme de sous-espaces stables. Soit
we
M
-curv
V , d'objets de
des applications tout
la categorie dont les objets sont des quotients V/W,
p
Mc u r v
et dont les morphismes
K-lineaires commutant aux
p
V/W -+ V'/W'
sont
p-courbures pour presque
apres epaississement et reduction modulo
p.
35 Theoreme 1.
a
est equivalente
la categorie
representations d'un sous schema en groupe i) Gcurv(M)
RePK Gcurv(M) des
ferrne de
GL verifiant M
est connexe,
ii) Lie Gcurv(M) iii) G (M) curv
5curv(M)
est un sous-groupe normal de
Ggal(M).
11 suit de ce theoreme une bijection entre les sous-espaces des
representations polynomiales de
Mc u r v
part, et les objets de
GL stables par M d'autre part.
Gcurv(M) d'une
,(I -curv
Cette difference notable avec categorie
11 vient de ce que 2 K-lineaire (contrairement a
est une
Modulo ce theoreme, la conjecture de Katz se reecrit comme en
IV. COMPARAISON DE
GO gal
ET
1.4.3.
G curv
1. Le critere d' algebricite de Chudno iTski [21 •
Soit
en
Hypotheses : i) Uniformisation :
f (x)
avec
d'ordre fini
U i U
i
-
meromorphe sur =
(2)
= U
0
o.
, xi =
(u ) -
i=l, •.. ,j
J(n) ::>I(n} for all n E
I
Choose Po E G(J(N».
So Q(n,N}p
o
I O. Put
p := gr(R)PO' By lemma 2.18 below we get (*)
Q(n,N(P}} # 0
all n E
implying Ep(M} I 0, whence p ::> J(M}
(by prop. 2.11). Since M is
holonomic this gives p::> p' for some P' E G(J(M»
with htpl
lJ • So A
htp > lJ . Now observe that p is involutive and homogeneous by th.2.16 and A lemma 2.12, so htp .::..
lJ
A
. Consequently htp
Then (2.16) gives 2(n,N(p}} So N E
as desired!
=0
= lJ
A
implying p
=
P' E G(J (M)) .
for all n > some nO' contradicting (*).
148
Lemma 2.18. Let
Po
P
E Spec(grO(R)),
:= gr(R)P
O.
Then
PE
Spec(gr(R)) and
the canonical morphism of grO(R)-modules.
Q(n,N) + Q(n,N(p))
can be extended uniquely to a gr (R)
o
X: of grO(R)
Po
Q(n,N)
Po
Po
-morphism
+ Q(n,N(p))
-modules and is an isomorphism.
For the proof of this lemma, which is based on the properties 1°, 2° mentioned in §1, we refer to [4], lemma 4.10.
The algebraic analogon of theorem 2.9; the general case. Let A be a filtered ring with filtration FA satisfying the conditions a) and b)
above and let M E
As before we have EX(A[X]) and EX(M[X]). The
crucial result linking the general case with the E-ring case is:
Proposition 2.19. M is an A-module with R.S. iff EX(M[X]) is an EX(A[X])module with R.S.
Proof. i) Suppose FM is very good on M. Then one easily verifies that FM[X] is very good on M[X]. Hence
LX (FM)
is very good on E (M[X]) by prop. 2.8.
X
ii) Conversely, suppose F is a very good filtration on EX(M[X]). By Cor. 1.9 G(F)
(=(j
-1
(Fn))nEZ) is good on M. We show now:
J(M) C Ann gr(M) i.e. G(F) is very good. Therefore let ala) E J(M), say veal = k E Z and m E G(F) (n) = j-l(F ) i.e. n
¢X(m) E F
n·
We must show: am E G(F)
However, ala) E J(M)
(n+k-l)
i.e. ¢(a)¢x(m) E F
n+ k- 1.
implies T .- ¢(X)-(k-l)¢(a) E J(EX(M[X])) and VeT)
i.e. T E Since F is very good and ¢x(m) E F
n
we get
149
so
X
(m) E F
k l' as desired. n+ -
We also need to know how holonomic modules behave under M + Ex (M[X]) •
Proposition 2.20.
([4], Cor. 6.10). There is equivalence between
i) M is a holonomic A-module iff EX(M[X]) is a holonomic E (A[X])-module X ii)
=
X
(A[X])'
This result follows from the following fact:
(2.21) If J(M) J(EX(M[X]»
e
where p.
=
= Pi n...n Pr n... n
P;
- --1 gr(A)[X,X
]p.
=
Proposition 2.22. VA
= sup pEl
htp, I
(the minimal prime decomposition of J(M»
then
is the minimal prime decomposition of J(EX(M[X]», and furthermore htp;
([4], prop. 6.11)
e
htp., all i.
(A[X]) iff
=
= VA'
where
X
the set of all involutive primes in gr(A) •
Corollary 2.23. If
=
VA' then M is holonomic iff Ex(M[X]) is holonomic.
Now we can state the announced analogon of theorem 2.9:
Theorem 2.24. Suppose
=
VA' Let M be a holonomic A-module.
Then there is equivalence between: i) M is an A-module with R.S. ii)
E
P
(M)
is an Ep(A)-module with R,S'
I
for all p
iii) Ep(M) is an Ep(A)-modUle with R.S., for all p
Spec(gr(A».
E
G(J(M».
Proof. i) + ii) follows from prop. 2.8.ii) + iii) is obVious, so it remains to prove iii) + i). If so by prop. 2.19 we get
P
E G(J(M»
then Ep(M) is an Ep(A)-module with R.S.
150
(2.25)
EX(Ep(M) [X])
is an E (E (A) [X])-module with R.S., all p E G(J). X P -
We have to prove M is an A-module with R.S., so by prop. 2.19 we must show
(2.26)
EX(M[X])
is an E (A[X])-module with R.S. X --
By Cor. 2.23 EX(M[X])
is a holonomic Ex(A[x])-mOdule, so we can use tho 2.14
of the E-ring case to prove (2.26) E'1(E
X(M[X]))
i.e. it suffices to prove that
is an EO] (Ex(A[X]»-module withR.S. for allO]EG(J(Ex(M[X]»).
Observe that by (2.21) the elements of G(J(EX(H[X]») are all of the form pe (= p gr(A)[x,X-
1])
for some p E G(J(M). SO it suffices to prove:
(2.27)
To make the link with (2.25) consider --1 gr(Ep(A))[X,X ]
o(S)
p
-1
- --1 gr(A)[X,X ]
and put
P
:=
O(Sp)
-1
-
--1
gr(A)[X,X
]p.
Then p is a prime ideal in gr(EX(Ep(A) [X]») and from (2.25) and the implication i) + ii) we get
(2.28)
E (E (E (M)[X]»)
""
P
X
P
is an E (E (E (A) [X]»-module with R.S. "" X P
P
for all p E G(J(M».
Finally using the following lemma,(2.28) implies (2.27):
Lemma 2.29. i) There is an isomorphism of filtered rings
y : Epe(Ex(A[X]»
': E (E (E (A) [X]»
""
P
X
p
ii) There is an isomorphism of filtered E e(E (A[X]»-modules
p
y
X
151
We refer to [4], lemma 6.15 for the proof.
Proof of theorem 2.16. i) Let
Po E
G(J(N». It suffices to prove
Therefore put and grO(Ep(R»
p:
gr(R)P
O'
Po
is involutive in grO(R).
By lemma 2.18 (with N=R) the rings grO(R)
Po
are isomorphic. We identify them. The Poisson product on
grO(R) can be extended to grO(R)
Po
and equals the Poisson product on
gro(Ep{R» . I f pogr (R)
o
Po
i.e.
Po
is involutive in grO(R)p
o
is involutive. However J := J(N)
J(n)
Po
then the contracted ideal in grO(R) J(n) for all n
nO hence
J(Q(n,N(p»)(by lemma 2.18)
J(N(P»
.
So it suffices to prove
J(N(P»
is involutive in grO(Ep(R».
Furthermore, since
Po
E G{J{N»
= G(J{n»
ifn>
-
"o'
Q{n,N)
Po
isagrO{R)
Po
-module
of finite length, so by lemma 2.18 we get
Q(n,N(p»
is a gro{Ep(R»-mOdUle of finite length.
Therefore we may replace the triple (N,M,(Mn)n) by (N(P) ,Ep{N),
(M»n)
and can additionaly assume: ii) Q{n,N) is a grO{R)-module of finite length, for all n
nO'
Since left multiplication by s-l gives an injection of Q(n+l,N) into Q(n,N) (see the proof of tho 2.15) we conclude there exists n multiplication by s n
n
1•
-1
Putting N{k)
1
nO such that left
gives an isomorphism of Q(n+l,N) onto Q(n,N) for all := N -1
n
this gives N(n) = s-l N(n+l)+N(n-l).
Put A := R u := s +R_ E A, M' := N{n+l)/N(n-l). Then u is a central 2 O/R_2, 2 R /R Q(n+l,N) M'/uM' and uM' Ker M'. element in A with u = 0, A/uA + 0 -1' u
152
Finally uA
§3.
J (n+l)
KeruA, whence J
J(M'/uM') is involutive by [6], Th.II.
A CONSTRUCTION OF REDUCED FILTRATIONS
The main result of this section (theorem 3.1) gives a new characterization of A-modules M with regular singularities along J(M) which enables us to construct reduced filtrations from generators of the A-module and generators of its characteristic ideal.
Theorem 3.1. There is equivalence between i) M has regular singularities along J(M). ii) For every m
E
M and every D
E
A with
E
o(D)
J(M) there exists p
E
such that p-l
(3.2)
DPm
E E
A«p-i) (r-l))D
im,
where r
.- v(D).
i=O More precisely, let m ... ,m generate the A-module M and O(D l),···,O(Dq) l l, the ideal J(M). Suppose ii) is satisfied, so there exists p E (3.2) holds for all D , all m Then the filtration j t.
r (n)
.=
such that
r on M defined by
1 E E t=l j d
is a reduced filtration on M.
Lemma 3.2. Let M = Am and 0 iDE A with v(D) filtration on M such that DF*M C n
F M n
E A(n-(r-l)i)D i=O
iM
=
rEz.
z.
1M all n E
F*M is a good
Suppo Then
defines a good filtration on M.
Proof. By prop. 1.8 it suffices to prove that (F M) and (F*M) are equivalent
n
First, m
E
F* M for some nO nO
E
E
M. O+r-l . ( l)M implying A(n-(r-l)i)D im C F* M whence rq n+n O
More general Dim E F*
Z, so Dm
n
DF* M C F* nO n
FMC F* M, all n E Z. Secondly, F*M = E A(n-v.)m. for some m. E M, n n+n O i=l n v. E Z. There exists
Co
E Z such that m. E F
Co
Mall i, whence
153
A(n-v.)m. C F
,...,
n+cO-v
M. Let nO := max CO-Vi' then
F
C
i
Corollary 3.3. Situation as in lemma 3.2. There exists N N
im, i) F M = E A(n-(r-1)i)D all n n i=O N N+1m ii) D E E A( (N+1-i) (r-1) )Dim. i=O
E
Proof. i) There exist m ... ,m q 1,
N
J
E
E A(v.-(r-1)i)D J i=O
M, v
im,
z.
O
E
such that
E z.
1,
So m , E F M. Hence there exist N E J v. J m.
E
M, all n n+n
... ,v
q
EZ
such that FnM
EA(n-v.)m .. J J
with
all j.
So N
F M n
E A(n-v.)m. J J
C
E A(n-(r-1)i)D i=O
im
C
F M n
as desired. ii) Take n := (r-1) (N+1). Then 1 N+ 1m D
E
co
E
E A(n-(r-1)i)D i=O
A(O)
im
A(n-(r-1) (N+1)) gives
F M n
N
(by i))
N
E A(n- (r-1) i)Dim i=O
E A( (N+1-i) (r-1) )Dim. i=O
To prove theorem 3.1 we use the strongly filtered ring AX := A[X,X filtration (A (n)) defined by A .(n)
X
X
= EA(n-i)X i,
-1
] with
and the filtered Ax-module
-1
i
M := M[X,X ] with filtration given by M (n) = EM .X Observe that X X i. M @MX Furthermore one easily verifies: if (M ) is good on M, then n x - --1 - --1 (Mx(n)) is good on M = gr(A) [x,x ], gr(M = gr(M) [X,X ] which x; x) - --1 = J(M) [x,x ] (cf.§l). implies J(M X) Proof of theorem 3.1. i) + ii) Let m
E
M and D
E
A with o(D)
E
J(M), v(D)
=
r
E
Z. Since M has
R.S. there exists a good filtration FM on M such that DFnM C F n
E
Z. Put M*
=
Am. Since FA is noetherian, the filtration F*M = M*
is good on M* and satisfies (3.2)
n+r_ 1M
(with p=N+1).
C
all
n FM
Then apply Cor. 3.3 which gives
154
ii)
+
i) Assume ii) is satisfied. Consider the filtration f' as defined in
theorem 3.1. It is obviously good, so it remains to show Df'(n) c f'(n+v(D)-l) for all DE A with O(D) E J(M). Since the O(D ) , ... ,O(D 1
- ( r • -1)
suffices to check this for each D .. Put T. := X By i)
+
ii) there exists p E
J
J
J
q)
generate J(M) it
D.; r. := v(D.). J J J
such that (3.2) holds for all D
j,
all m t
hence all j,t.
(3.4)
Observe that J(M)
=
(O(D ) , ... ,O(D » ) implies J(M 1
q
X)
(0 (T 1) , ••• ,0 (T q) )
hence A (0)T X
The involutivity of J (M (3.5)
l r ,T ] E p
X
implies that
A (O)T. + AX(O),
l;
q
X)
+ ... + A (O)T
1
X
J
e;I
q
+ AX(O).
is a Lie-algebra so
all p,q.
It is not difficult to verify that (3.4) and (3.5) imply
j
is an AX (0) -module of finite type with AXM By prop. 2.4. f'x(n)
:
Ax(n)M
O
O
=
and
0;} MO C
MO'
defines a reduced filtration on M so in X'
particular
(3.6)
T.f' (n) J X
f'X(n),
C
all n,j
Observe that
l;
t
Using M x
l;
j
0(X)
AS, perhaps, suggested by (e) A
j9(X)
.
THEOREM
(c)
be a curve.
k-algebra and is right and left noetherian. Although J9(X)
(a)
X
= 8(X)
or
,
A
= t9:x,x
, the key to understandingJ9(A)
is to compare
,vIA)
and fl)(A)
for
where A denotes
the integral closure of A in Fract A , its field of fractions. Def i.ne £)(A,A)
{D
E
E(A) I D (f)
E
A
for all
f
E A:}
•
This is a non-zero right ideal of .fj (A) and a left ideal of .,t)(A) Since 08(A)
is a simple, hereditary ring
progenerator in where and
T
T
Mod- .,D(A)
.v(A,A)
is necessarily a
. Thus we have ,£!(A) :: End rf)(A)J)(A,A)
is Morita equivalent to
•
The relation between
=
T
J)(A)
depends on the fact that they have a common left ideal, name-
ly J)(A,A). A key lemma is that 9(A)
T
i f and only i f J}(A,A)'f A=A ,
161
where
denotes the linear span of all
D E .f)(A,A) and
D(f)
such that
f EA. It is these observations which are exploited
to obtain the above results. Through part understand
JB
(e)
(X)
of Theorem 3
we can, in a sense, say that we
completely when
is simple. So from now on we
concentrate on what happens when JD(X)
is not simple. However, there
is one question still of interest when
is simple; give a pro-
cedure for obtaining generators for that J7(X)
, or find the least
n
such
is generated by differential operators of order < n
To understand
J)(X)
tv
when
n : X
+
X
is not injective one is led
to prove. THEOREM 4 [6]. The factor
J)(X)
H(X)
constains a unique minimal nonzero ideal J (X)
:= J9(X)/J(X)
is a finite dimensional
kalgebra,
and H(X) = $x ESingxHx is a direct sum of algebras Hx one for each singular point x . The structure of H depends only on the local x Eig£ &x,x . In fact Jlx,x has a unique minimal nonzero ideal JX,x and H =e'b /J . x x I X ,xX The relationship between the ideal structure of J9(X) submodule structure of
is illustrated by
THEOREM 5 [6]. Consider &' (X) (a)
Y(X)
(b)
and the
as a
J)(X)
module. Then
has Hni te length has a unique simple submodule, namely
J (X) • !9'(X)
C (X)
=
cD (X, X)* &(X)
19'(X) /J (X) .
then
C (X)
is a faithful
H (X) module;
(c)
If
(d)
C(X) $x SingxCx is a direct sum of local algebras, one for each singular point of X
(e)
Cx
c
=
X, x /J X, x . [1;X, x
(Z
tr(X)
and is a faithful
Clearly one would like to dimensional algebras H and x note that, since H and C x x from Theorem 3 that H and x It is not difficult to observe then kl
C(X)
Hx:---"-'-== module
understand the structure of the finite C ' and so H(X) and C(X) . First x depend only on &X ' it will follow ,x -1 Cx are zero precisely when # IT (x)=l.
that if is a homomorphic image of
k[t1,···,tJ /(f1, ... ,f r)
, ... ,t]/(f1, ... ,fr ,8f./8t.) because J(X).(9'(X) con t adn s vthe n J conductor of &(X) in f)-t:X) and the image of each 8 f t belongs i/8 j to the conductor
162
§ 3. THE ALGEBRAS
H x
.AND
x
In this section A = (JX,x
and we set
C
x
is a curve with a unique singular point B = A
and
0
x,
This section is a collection of
and C We H x x the ring Hx may be either 0 , or M (k ) n k , or the ring of lower triangular
examples illustratins some of the possibilities for will give examples where of
n
n
x
matrices over k
0
0
2 x 2 matrices , or (k' k) In these examples Cx is respectively n), 0, /(t /(t 2) , and k[s,tl/(s,t)2 We have no general re0
0
sult, but these examples do give some clues as to what should be expected in general. We denote the maximal ideal of
A
by
ring with Jacobson radical denoted -1
# IT
o
Bv [ 6, § 7.4] a (t)
x
if and only if
there exists
and
Bt . If bEB we shall write
cor-
is a local ring.
&x,x
t E
B
if and only if
2 E DerkB
such that
It is an easy exercise to see that this forces
1
and
=0
H
(x) = 1 , we may rephrase this as
PROPOSITION 1.
is a semi-local
The maximal ideals of
rr- 1 (x ) . Since
respond to the points
m. B
DerkB
Ba,
a (b)
b'
We shall assume in all the examples we construct that
IT
N
X
X
is unramified at all points. The reason for this restriction is because we can make use of the following result to simplify the calculations. THEOREM 2 (WoC.Brown [3J )
IV
X
HIT
Thus we have , locally J)(A) examples where
Hx
Mn(k)
in
eX,x If I
Proof. Since # valent , so J) (B ,A) I = JJ(B,A) k .
H x
is unramified at all points
*
B . But
= B[a]
# rr-
1
(x )
>
1 , J7(A) A/I
n
=
4"
vX,x '
and
B '" A . However, I ::. k
First we construct
0
1
.
1 . Let I denote the conductor
is a maximal ideal of
n- 1 (x ) t
*
J)(B)
The easiest case is
PROPOSITION 3. Suppose that of
X
J9(X)
then a8(X)
then
H x '" k
.
are not Morita equi,A)
is now a faithful
whence Hx-moduleo Hence
c
163
This explains [6 , Theorem 4.4] since under the hypotheses of that
GX ,x ' since II ,x which is a product of fields.
theorem one must have I a maximal ideal of a looal ring contained in
&.X
,x II
is
It is possible for H to equal k without the hypothesis of x Proposition 3 being satisfied. Indeed, if J7(B,A) *B is a maximal ideal of
A
then
H
k . This is illustrated by the following
x
EXAMPLE 4 [6, § 5.7] Take X = /AI , and /J'('X) = kI t] . Define X by 2(t-l) 4(t-l)k[t] 4(t-llk[tl 19'(X) k + kt + t . The conductor is t which is not a maximal ideal of
&(X)
. It is shown in [6]
J) (B ,A) * B = !!! , the unique maximal ideal of fJ: unique singular point of Hx-module so
with
X
Hz
X)
.
Again
x ,x
AI J3 (B ,A)
(here * B
x
that is the
is a faithful
k
This example may be understood as follows . Let X' be the curve 2k[tl 9(X') = k + t . We have a factorisation of IT as X'
X
i) (X' ,X)
with
:::J
Hence J3
the conductor of PIDPJSITION 5. §;x.,x
t
n =
injective. Hence 2(t-ll k+t B'(X'} and 2(t-l)J7(X,X') 2 t (t-l) J7 (X') . Thus cD (X,X) :::J 0\9 (X' ,X)J7(}"X ') :J t . 2 * crIX) .=: t (t-l) &'(X') =!!! . The is that is- injective, and
fJ(X)
J3(X,X'}*
= &(X')
8' (X)
and
• However,
in et(X'}
SUpp?se that # n
e-X,x
&'(X) =
is a maximal ideal of 8(X) . -1
(x) > 1. SUpp?se that the Jacobson radical of n n+l' and that fk.x ,x = k+kt+ •.•+kt +t cY. .x • Then - Hx '" Mn+1 (k) x
Proof. Let !!! be the maximal ideal of A = . Then Theor-sm 2,!Jj (A) c ri3 (B) . The sarne argument as [ 6 , Lemna 5.3] J)(B,A)
tn+l $(B) , whence C = A/tn+lB
is a faithful
x
result will follow i f we can show that
= tB, and by shows that
H -rrodul.e , Thus , the
x
is a simple Hx-rocdule, or equiva-
is generated by 1 , and that n+ A/t I B is a simple 37 (A) -m:xlule n) it will suffice to show that there exists D E J) (A) such that D(t = 1 . we proceed to show that D := (t8 - 1) ... (t8 - n) 8n belongs to /}(A) ; since n) (-I)n(nl) -2D(t = 1 this will canplete the proof of the Proposition.
lently is a simple E(A)-m:xlule. Notice that n kt is an essential A-subrrodule . Thus, to show
Since !f) (B) =B [ d] we have DE J:) (B) . The action of D on A annihilates nk + kt + ... + kt 1 , so it rrmafns to shaw that D* (Btn+l) C Btn +! . First, notin ce that 8 * (Btn+ 1) Bt . Secondly, notice that, for all j-E:N, j) (ta -j) * (Bt C Btj+l . Hence (t8 - n) ..• (t8 -1) * (Bt) C Btn+ 1 and thus
n+l n+l D* (Bt ) C Bt
and
1 . Suppose that the Jacobson radical of &-x where
yE
IX
is
ex, x \ t &- x, x
tB'x
IX
,and
,Q.nd that f):x y
IX
= k + kt + kty + t
is not a unit. Then
Hx
2
k 0
()
(kl.k)
X,x
0
165
Proof. The argument is very similar to that in Proposition 6. One 2,e(B) computes li(t 2B) = B + Bta + B(ta l)a + t , checks that 1,t,ty,ta ED(A) and that their images in H are linearly indepenx dent. And finally one shows that if D v(ya l)a + uta belongs to 2 £J(A) with u,v E B , then D E kta + t £1(B) ; where Hx is spanned by 1,t,ty,ta these elements on
and the result follows by considering the action of A/t 2B . 0
This completes the list of examples stated at the beginning of , and this section. Notice in the examples where H is M2 ( k ) x 2 is (k 0) that C is isomorphic to k[t] l(t ) in both cases x k k Cx Z ex,x/I where I is the conductor of &x,x . In particular, kno wing
Cx and &X II ,x
does not determine
Hx
In the above examples H is always an indecomposable algebra, in x the sense that H cannot be written a direct product of two nonzero x subalgebras. More generally we have PROPOSITION algebra.
X., and any
x EX, H x
is an indecomposable
Proof. Suppose
H is a direct product of nonzero subalgebras. x Then there exist nonzero central orthogonal idempotents e,f EH x
with
1
tion of
=
e + f
Cx
as a
C = HxeC x e HxfC . However, this decomposix x J7x,xmodule is also a decomposition of Cx as an
Then
and hence as a
Cxmodule
But
o
hence indecomposable. Hence either possibility contradicts the fact that
Cx
is a local algebra,
or fC x = 0 • But, either is a faithful Hxmodule. 0
§ 4.CONSTANT COEFFICIENT DIFFERENTIAL OPERATORS AND THE SPACE OF
POLYNOMIAL SOLUTIONS. Let
J3
= J)(R)
R
be the polynomial ring in two variables, and C[x, y, a , d] the ring of differential operators on R x y
=
Let D E.f} and set S (f EO RID (f) o}, the space of polynomial solutions. Observe that if P,Q EJ) with DP = QD , and f E S then P(f) E S
also. Define, the idealiser ofc1)D,
This is a subring of J9, containing
li(,l)D) = {p E,vIDP e.J)D}.
as a two sided ideal. The
above observation says that S is a left it is annihilated by.l)D , so S is a left
li($D)module. Furthermore li(J)D)/J7Dmodule.
166
Let
0: J)-+J) be the anti-automorphism given by
o (x )
8etting 0 (D) Thus
8
0 ( d x)
d ' X
=
x
=
0 (y)
DO , we have
=
0 ($'D)
=
ay
=
DO J)
0 (d y )
f.Q' = Q(f)
for
Q' E
= II. (DO£> ) . OJ7-module lI(D°j1)/D by
II (D°J))
where
Now consider for example, the case where D
=
d y
23 -
d X
+ dX)
023 D = Y - x
• Then
y .
, and 0 (II (cf) D))
can be given the structure of right
defining
=
D
=
Q = o(Q'). d 2 -
y
ax 3
(resp.
023 (resp. D = Y - x + x)
and the space of polynomial solutions is a right where g = y2 _ x 3 (resp. g = y2 _ x 3 + x ) . But by [6, § 1.6) II(gJ))/gJ) "$(R/gR) = g
E
a: [x, y)
Thus
8
J)(C)
where
C
is the curve defined by
is a right E(c) -module . We will show below
(for both the given examples, and more generally whenever is injective) that cribe all of action of whence
8
8
is a simple right
on
1. Je(C)
8
-+
C
and the
8 . In the given examples it is clear that
1 E 8,
. 80 the problem of describing all polynomial so-
lutions of the differential equation ask for a description of 2.2 , that )9(C)
C
J)(C)-module. Thus to des-
we need know only one non-zero element of
iJ(C)
=
8
n :
J) (C)
D(f) =
a
leads us naturally to
(for example, once we know by Theorem
is finitely generated, we want to know the genera-
tors) and a description of the action of i)(C)
on
8 .
The procedure we shall adopt in order to describe all of be to first describe
f)(C)
(through its relationship with
8 , will
93 (C)
as
outlined in Theorem 2.3)
and thence to obtain a description of 3 2 and so act on 8 . For example in the case D = ay - a. x , 2,t 3) we have = a:[t and as in [6, Remark 3.12) , 2 3 2 :fJ(C) = a: [ t ,t ,td ,t a ,(td - 1) d,t -1 (td - 2) a) and (after the detai2 led considerations below) t a gives rise to the element Q = 2Xd + 3Yd 2 E II (,pD) and t -1 (td - 2) a gives rise to the eleII
(J7D) /
£>D
y
ment
P
that
8
X
4x 2 a + 12xyd + 9y2d 2 - 2x E II (£)D) It will be shown x y x a:[P) .1 + Qa:[P). 1 and thus we obtain all elements of 8
by starting with
1 E 8 and acting by
Q,P
as follows. Thp diagram
indicates how solutions are obtained from previous ones by applying P and Q (we ignore scalar multiples, so although Q(x 3 + 3y2) = 24xy we just write
j y
x 3 + 3y2
g.
3 3 2 P Q
x
j: Y
.. xy
xy) 4
. 2 P
5
--·x -x y
-x y+y
2 2
Y
--L. .... -;- •...
167
One can continue to apply
P
Q
and
to obtain more solutions, and
Proposition 6 below shows that one will in this way obtain a basis for
a: Ix,y]
{f E
I (8/
- 8}) (f)
= O}
.
Two points should be observed. First, to find elements such as P, Q E
:n: (iJ D)
simply by computing inside
:P
seems an impossibly dif-
ficult task • Even if one can find elements of
TI(J)D)
one needs to
know whether one has found enough elements to generate all of TI(gJD)/J9D
(hence the importance of Theorem 2.2
saying that J)(C) is
finitely generated , and hence the importance of trying to obtain a
.
procedure to find generators of
J'(C»
polynomial solutions belong to
1.j1(C)
Secondly, to know that all (in the case when II:
is injective) one needs to show (as we do below) that right
S
E+
C
is a simple
(C)-module.
It is no problem to extend the above analysis to the more general situation described in the following Proposition. First note that there is a natural anti-automorphism a on the ring n) £)(A a:[t l, •.. ,t .•. ,d where d j = d/dt n] j n,8 l, a (t.) 0(8. ) 8. t. for all j J J J J PROPOSITION 1 . Let contained in Set J') = J)(R) sider
R • :
and thus
a: [ t l' ... , tnl
, and let
J
be an ideal of
• Then
S
S = {f E HID (f) = 0
TI( a (J 33) ) -submodule annihilated by
is a left
TI( a (J9))/a (JJ) S
- -.....
TI(Ji)/J:J)"'JJ(A)
may be given the structure of a right
As a
is isomorphic to :J)(A,A/r.E.)
Proof. It is straightforward computation to see that TI( a (Jr!3» -submodule of ,f}(A)
-x
for all
There is an anti-automorphism
right .$ (A) -modUle, S
phism
R
. Set A (t l, ... ,t R/J, an
R
given by
R, annihilated by a (J j:)
is of course induced by TI(Ji)/J$7 is just [6,
S
is an
• The anti-isomor-
a , and the fact that
§ 1.6] . Thus it remains to prove the
final asertion. Apply the left exact functor $R( exact sequence of
R-modules
0--+ J
-+
, R/(t1, ... ,t R-+ A-+ 0
to the short n) to obtain the exact
sequence 0 - .t'R (A,R/ (t l, •.. , t n) ) - - J?R (R,R/ (t l, .•. ,t n) ) --+ J';)R (J ,R/ (t l, ... , t n)).
168
Thus ,f)A(A,A!!:i2)
='
j)R(A,R!(t1, .. ·,t n » {Q E,iJIQJ
Now consider
=
S
t 1J9+···+trr}ft 1J:'+···+t n J) . + ... +Jf)0n . By definition
S C R
{p E.f) III (J
.v) p
J)A (A,A!!:i2)
-+
It is clear that
+... +
j) 0 1
Through the anti-isomorphism Let 'It : S
, S
,1-
£J (A) -module map, let (p +»3 + ... +173 J 1 n
e E II (
(J
(J
S "'!lJ (A,A!!:i2)
5
S , and
E
and
,
d
(P)
+ t
='
d
E
E(A)
(e(e) + Jj71
'It
Suppose
for
1
JJ +
••• + t j)j .] n
(J
(e)
+ J"t1]== 'It
(s) .d.
as required .
A
Remark. It is easier to consider S
as a left
and
this is what we shall do in practice. That is ,QD(A)OP tified with
j)(Ar:ra1ule.
. Then
j}»
[ 11
Thus
is made into a right
is a vector space isomorphism. To see that
'It
that
='
an}! j) ;) 1+ ..• + fJ ;) n .
:J]
be defined by
is a right s
J= O}
='
will be iden-
and the action of this ring on
U(a(Ji»)!a (JJ)
S
will be obtained through the restriction of the usual action of differential operators on PROPOSITION 2. Let
n : '" C
-+
R C
='
[[x,yl
be an irreducible
C is injective. Let
£!(A,A!!:i2) Proof. Let By Theorem 2.3 ,
m
be a waximal ideal of
is a simple right
A
affine curve, such that
J)(A)
-module.
denote the integral closure of
.D(A)
and
;@(A)
A ==
A
in Fract
A .
are Morita equivalent. The proge-
nerators giving the Morita equivalence are
j9(A,A) and
p(A,A). Con-
sider the following natural maps obtained by taking composition of differential operators :
169
Since
J)(A,A)
given by composition is an isomor-
-+
phism, the above map is also an isomorphism. In particular, jJ (A,A/!!!) corresponds to
under the Morita equivalence. Hence to prove
the result, it is enough to show that
is a simple right
3)(A) -module. However, by Proposition 4 below, is the unique maximal ideal of .11(A,A/!!!')
A
$A(A,A/!!!') where
containing
m'
!!!. By [6, § 1.3e J,
, and this is a simple right .$(A)-module [6,
§l.4g J • Hence the result.
0
The next two results are required to complete the proof of Proposition 4.2. LEMMA 3.
A
ble variety ideal of
\!!l
A e m
C
\!!l
I
1
N
be
A-modules, and
ker (u : A \!!l k A
J
A
k
m. Hence
a - a
and
o , then for all
Write cation map. As {1
M
A . If
ments of
be the co-ordinate ring of an affine irreduci-
&(X)
X. Let
m0 A + J
a
!!!'
A
-+
a maximal
n
A)
where
Jl
is generated as a
is generated as an ideal by
J
m} • In particular, J C A \!!l m + m . Thus In C A mn + m '" A and A I!> E
o .
As
mN "" 0 , if G E Hom (M,N) then k In.G(M) (A Qll !!!n) .G(M) AG(!!!nM), and G(!!!n+1M)
=0
dition that
if and only if
e
PROPOSITION 4. reducible curve
A , and {!!!Ie I Ie E A} is an isomorphism
In+1. G
=o ,
A
A C
r
and also l8>
A + In.
Thus
D
=
be the co-ordinate tV
C , and set
B = &(C)
Let
of an affine ir-
m m . Then there
s , 0 leE.j3B(B,B/!!!Ie}
the direct sum. Define
Qll
which is precisely the con-
n(M,N) .
Proof. For each Ie , fix an If, for each
is the multipli-
k-algebra by ele-
A-module isomorphism then write L
Gte) = LIe'PleGte •
te
'{!
Ie : B/!!!Ie -+ A/!!!
Gte for the element in
170
First
is a map to
8 A c: .BB (B that
::.
because each
(B
whence
.p A 8 A E $lA (B
. I t is clear
is a rightJB(B)-module map. However, a word of warning is re-
quired : cl!(B) thatJ7
means
97 B (B , B)
not JJA(B,B)
so
B(B,B)
To see that
, and one must observe
really is a right J}(B)-module.
is injective, first observe that $A
is
a direct sum of non-isomorphic simple right J7(B)-modules [6,Corollary 4.3
and § 1.4g]
contained in since
E\ = 0
Hence i f
ke r
But if 8,\ EkeI'
0 } .
That this is now the injective envelope of =
follows from [Bourbaki, Algebre Homo-
logique, § 1, Ex. 29-32 I . That an earlier proof of this result could
171 o
be replaced by this reference was pointed out in [8].
We now return to the examples at the beginning of this section. In fact we will first discuss the example where
=
a 2 - a 3 (the other y x example is somewhat simpler since the corresponding curve is non-singu0
lar, and we shall comment on that in the remarks at the end of this section) 2xa PROPOSITION 3. Set Then
tn .
S = C
Proof. Set
=
0
1
+
g = y
A = C[x,y)/g[[x,y)
a 2 - a 3 y x
Q[ [p)
2
- x
and
3
, so
where
m
E : A
x
-
-
2a 2) a , Q = t We view P,Q as elements of 2,t 3) c CI:[ t) . Since 'J3 (A) " II (g:I1) with A = [[t we can find elements PI,Q' E lI(g which map to P and Q respectively. Such Set
t-1(ta
P
-
-
elements are p'
Q'
4x a 2 + 12ya a + x x y 2ya + 3x 2a x y
Notice that
P
O(PI), Q
=
o(Q')
•
Hence to prove the Proposition it is sufficient to show
=
+ E.[[P)Q • Recall that
a} . n n+1 We identify £) A with Hom[ Set j 'j3 = {.t [o j 2n+1, j -f 1} . This is a basis for
s:
=
{Epjl0.2.j.2.n} U kPjO!2.2.j.2.n+1} • Check that
scalar multiple) n+1
=
basis to
Ak
. EP (t J ) = 07k,j
is
and
AkA
j
_
Set (up to a non-zero
EP Q(t ) - 02k-1,j
Hence
(up to non-zero scalar multiples) the dual
13. In particular, it follows that o
172
Remarks (1). Proposition 6 allows one to routinely produce a basis for
S; in fact the proof essentially shows that
z.
{pj(1) Ij.-:O} U {QPj(1)!j
2} gives a basis for
S
this verifies
the claims made at the start of this section. (2). The elements as follows. We have in t8 E £l(A)
satisfies
P'
and
= [[t 2,t 3]
A
(t8) (y)
3y ,
=
Q'
(t8) (x)
.2)-submodule of M. Hence the conclusion.
[)
We will next show that when ble
f E (9(8\2)
C , the curve defined by an irreduci2)f/ 19(1A &VA 2 ) is simple
, is non-singular, the module
(this is certainly well known, but we cannot find a proof to refer the reader to)
• To do this, first observe that
f-'R/R
C
R
submodule, is annihilated by J)f , and is therefore an module. However, there is an isomorphism of II:(£1f) /J3f tjJ
:
TI(o&f)
TI(fJ}) /f:fJ ; this isomorphism is obtained from
+
TI(fJ) fO
given by
=
O'f
tjJ
(0)
for
by [6, § , .6] , it follows that
0
0' E:
where
TI(.@f) R/R
TIlBf)-
k-algebras
ce
ment satisfying point is
is an
TI (Ei) /J)f-
0 EJ7is the unique ele-
Thus, as
lI(fJ)) /f[ff-".J)(C)
is a left E(C)-module. The
174
f- 1R/R
PROPOSITION 2. As a left
=
R/fR
with its natural
is
structure.
Proof. Easy.
[J
*
o
THEOREM 3. Let a curve C. If J}(A 2 ) -module-:-
C
f E
be an irreducible polynomial defining
Proof. First we show that 2 -1 (A i , f
Clearly -1
f
ff
=
-2
and
d
that
:P (A2)
y
C
for each
zero submodule of 9)(A
e- (A2)
1
2)/
&(A
&(C)
8'(A
0
completes the proof of the fact
M M
is simple, we need only show that every noncontains f- 1 Pick 0 m EM, and consider
2)
f'
*
Thus
0
* af- 1
2 E f- 1 &'(A 2) / Ct(A j
with
• Consider
is a simple
because
C
is non-sin-
f- 1 E B(A 2) .af- 1 .
[J
(1) The above proof gives a very explicit argument as to
generates
f/ &(A
af- 1
as a left J'(C)-module. As such it is isomorphic to
. However, &(C)
1
2)
. Clearly this contains an element of the form
a E f-
g(A
. f- 1 = M
Now to see that 2).m
2)f/
is a simple
-f f- 2 a (f-1) = - f y f- 2 and x' y 2)f. is non-singular, 1 E &(A 2)f + + &(A
ax (f-1)
f- 2 E J9(A 2) .f- 1
f-n
to
M = g(A
contains
• Since
Thus we obtain
f/ 8(/A 2)
is non-singular then
2)
. Later we shall show that
is a simple
JJvAh -module f- 1
tive. Hence in that case also
whenever
generates
&(A
n: C ... C is injec2)f/
. However,
our proof will not explain in such an explicit manner, why f- n
E
.f- 1 . Hence it is an interesting question (interesting for
this author, anyway) to find in some explicit cases (for example, f
=
y2 - x 3 ) operators (2)
D
n
It is clear that all the above arguments work in grea-
ter generality. That is, if
o
*
f E
B'(X)
such
X
is a non-singular variety and
an irreducible polynomial defining a hypersurface
then similar considerations (to the above) apply to I9'(X) f/ 19'(X) J) (X) -module.
Y c X, as a
175
THEOREM 4. Let a curve
0
*
f E
be an irreducible polynomial defining
n:
C. Suppose that
rJ
C
C
is injective. Then
is a simple $ (/A2) -rnodu Le ,
0'(173) f/ 8'(/A2)
Sketch of Proof. The goal is to show that each II (Ef n) -module, where
is a simple left low at once that
Rf/R
is a simple left
It is clear that for However, on
JI(£'fn) /
J3 f n
n ElN, f-nR/R
f-nR/R, R =
IJ = J3(/A2)
It will then fol-
b -module.
is a left
II (J) fn) / oVfn-module.
"" oV(R/fnR), the ring of differential operators
R/fnR, and it is easy to see that as a left d9(R/f
f-nR/R R/fnR
is isomorphic to
nR)-module,
R/fnR. Hence the aim is to show that
is a simple )3(D/f nR)-module for all
is precisely Theorem 2.3 in [6]
,
above. For
n > 1
n E
• The case
n
=
1
we must extend the results
. This is done in [7] , and here we just sketch the main steps
of the argument. There is an inclusion of algebras n R/f n R:::; R/fRQPkk[z]/(z)
such that
n
R/f R
n ) :::; \.7(C)@kk[z] "" 9'(C)QP [ z]/(z /(z n) :::;Fract(R/r kk
is of finite codimension in
,..,.,
n
Nt
) , and
the induced map on the spectra is bijective. One observes that
and this latter algebra is Morita equivalent to
J9(C) .
One therefore
can apply the same ideas as in [6 , §§2,3] to show that, if
J7( e'(C)
(t)
then .,E)(R/fnR)
[z] / (zn) , R/fnR) ...
(e- (C) 0kk [z] /
is Morita equivalent to J7(8{C) @ k l z l / (zn»
of the bijectivity of the map on the spectra , by imitating the proof of
Essen [8].
(t)
. Because
can be
, Theorem 3.4J . Then, from the Morita
equivalence it follows that R/fnR is a simple .9(R/f nR) -module. Theorem 4
(zn) ) =R/fnR
is a simple ring, and hence 0
has been obtained independently by van Doorn and van den
176
REF ERE N C E S [1]
I.N. BERNSTEIN. - Analytic continuation of generalized functions with respectto a parameter, Functional Analysis and its Applications, 6, 1972, 26-40.
[2]
I.N. BERNSTEIN, I.M. GELFAND and S.I. GELFAND. - Differential Operators on the Cubic Cone, Russian Math. Surveys, 27, 1972, 169-174.
[3] W.C. BROWN. - A note on higher derivations and ordinary points of curves, Rocky Mountain J. Math., 14, 1984, 397-402 [4] A. GROTHENDIECK. - Elements de Geometrie Algebrique IV, Inst.des Hautes Etudes Sci., Publ. Math.,n c 32, 1967. [5] A. SEIDENBERG. - Derivations and Integral Closure, Pacific J. Hath., 16, 1966, 167-173. [6]
S.P. SMITH and J.T. STAFFORD. - Differential Operators on an Affine Curve, University of Warwick, preprint, 1985.
[7]
S. P. SMITH. - The silrple i3 -mx1ule associated to the intersection horrology complex for a class of plane curves, University of Warwick, preprint, 1986.
[8]
A.van den ESSEN and R.van IXXlRN. - J) -mx1ules with support on a curve, Univern
sity of Nijmegen, preprint, 1986.
SUR L'IMAGE DE L'APPLICATION MOMENT Michel BRION Laboratoire de Mathematiques, Institut Fourier Boite Postale 74 , 38402 - SAINT-MARTIN-d'HERES CEDEX
1. Introduction Soient G un groupe algebrique reductif connexe et M I'espace d'une representation rationnelle, de dimension finie, de G (Ie corps de base etant algabriquement clos). Soit X une sous-variete fermee, irreductible et stable par G , de l'espace projectif P(M) assode aM. Dans [Kl], F. Kirwan construit une stratification de X en soua-varietes localement fermees, stables par G , qui lui permet dans certains cas de calculer les nombres de Betti du "quotient" XI/G. Lorsque k = C, cette stratification est definie a l'aide de l'''application moment", associee a. l'action sur P(M) d'un sous-groupe compact maximal de G (cf. [Kl] Part I, et [N]). Une autre definition de la stratification, valable pour un corps quelconque, utilise la theorie de l'instabilite (cf. [H], [K1] part II; pour un expose d'ensemble, voir [Bru]). Le but de ce travail est d'abord de donner une troisieme definition de X, lorsque k est de caracteristique nulle. On commence par associer a X un polytope convexe C(X); si X designs le cone affine sur X, un point de C(X) peut s'interpreter comme un covariant de X, c'est-a-dire un morphisme equivariant de X dans un G-module simple (pour une definition precise, voir 2.1). Lorsque k = C, on montre que C(X) s'identifie a l'intersection de p,(X), oii p, est I'application moment, et d'une "chambre de Weyl". On retrouve ainsi sans peine les proprietes de l'image de p, (cf. [N], Appendix; [GS], I et II; [K2]). On definit la strate de X indexee par {3, comme l'ensemble des x E X tels que {3 soit Ie point Ie plus proche de 0 dans le polytope convexe C(G.x) [ou G.x est l'adherence de la G-orbite de x dans X). Tout covariant associe au point Ie plus proche de 0 dans C(X), envoie X sur Ie cone des vecteurs primitifs d'un G-module simple; on retrouve ainsi un resultat de Bogomolov (cf. [Bo], §4). Dans la troisieme partie de ce travail, on redemontre sans trop de mal certains resultats de Kirwan (cE. [Kl], §11 a. 13) a l'aide de l'approche esquissee precedemment, et on generalise l'un d'entre eux au cas oii X n'est pas forcement lisse : la strate de X indexee par {3 est fibree sur I'espace homogene G/ P(3, oii P(3 est un sons-groupe parabolique de G , canoniquement assode a. {3. Dans la quatrieme partie, on montre comment le theorems de "structure locale" de [BLV] permet de decrire la forme du polytope convexe C(X) au voisinage d'un de ses points. On indique Ie role joue par les orbites ferrnees de G dans X, dans l'allure de C(X). Enfin, dans la cinquieme partie, on se restreint au cas ou un sous-groupe de Borel de G a une orbite ouverte dans X. Une telle G-variete X est dite "spherique".
178
Si k = C, les G-varietes spheriques sont caracterisees par Ie fait que toute fibre de l'application moment est contenue dans une orbite du sous-groupe compact maximal de G. Ce sont les analogues algebriques des "espaces sans multiplicite" de [GS], III. En outre, on peut considerer une variete spherique comme un plongement d'un espace homogene spherique au sens de [BLV]. Le polytope convexe C(X) est lie a la theorie des plongements due a Luna et Vust (cf. [LV]). Ce point de vue sera developpe ulterieurement.
Ces notes ont leur origine dans I'etude de l'ouvrage [K1] entreprise par Ie groupe de travail dalgebre et geometrie a. Grenoble pendant Ie premier semestre 1985-86. Je remercie tous les membres de ce groupe de travail, et en premier lieu D. Luna, pour leur interet et leurs encouragements. Je remercie egalement I'Univeraite d'Utrecht pour son hospit alit.e, et J. Duiatermaat, G. Heckman, T. Springer pour des conversations utiles.
2. Polytope convexe associe
a une
G-variete projective
2.1. Soit G un groupe algebrique reductif connexe sur un corps k algebriquement clos de caracteristique nulle. On choisit un sous-groupe de Borel B de G; on note U le radical unipotent de B, et T un tore maximal de B . Soit P++ l'ensemble des poids dominants de G (pour ce choix de B et T); c'est un sous-monoide du groupe X(T) des caracteres de T . Pour chaque 7r E P++, on choisit un G-module simple M", de plus grand poids 7r, avec un vecteur propre X" de B dans M". Soit C" = G· X" I'adherence de la G-orbite de X" dans M". Alors C" = G . X" U {O} est Ie cone des vecteurs primitifs de M". On note x" l'image de X" dans P(M,,); alors e" = G . x" est l'unique orbite fermee de G dans P(M,,), et Ie groupe d'isotropie P" de x" dans G , est un sons-groupe parabolique de G. Pour tout 7r E P++, on note Ie G-module dual de M", et x' Ie plus grand poids de Si Wo est l'element de plus grande longueur du groupe de Weyl W de (G,T), on a x' = -wo(7r). Pour tout x E V = X(T) C9z Q, on note encore x' l'image de x par -woo Soient M un G-module rationnel de dimension finie, et X une sous-variete ferrnee, irreductible et stable par G , de l'espace projectif P(M) associe a M. Soit X Ie cone affine de M associe a X. Pour tout x E X, on note un point de X au-dessus de x . Soient k[X] l'algebre des fonctions regulieres sur Ie cone X, et k[X]n l'espace vectoriel des elements homogenes de degre n de k[X]. Chaque k[X]n est un G-module rationnel de dimension finie, donc se decompose en somme directe de termes isomorphes a m",n copies de M", ou m",n E N. Si m",n # 0, alors tout G-morphisme non nul 'P : M" -+ k[X]n definit un morphisme Gequivariant : X -+ homogene de degre n : si x E X et m EM", on a
x
(x)(m)
= 'P(m)(x). Un
tel morphisme est appele "covariant de degre n"j par
la suite, on exlura le cas trivial n
= O.
DEFINITION. SoU C(X) Ie sous-ensemble de V = X(T) 0z Q forme des 7r In tels que m,,',n # 0 [c'est-e-dite que k[X]n contienne un sous-module isomorphe a au encore qu'il existe un covariant : X -+ M" de degre n).
179
PROPOSITION. C(X) est l'enveloppe convexe d'un nombre fini de points du Q-espace vectoriel V.
Demonstration. - Notons A I'algebre des fonctions regulieres sur X, invariantes par Ie unipotent maximal U de a . Alors A. est une sous-algebre graduee de k[X], stable par T (puisque T normalise U). De plus, les poids de T dans A sont tous dominants, done A est graduee par le monoide P++ X N, et dimA,..,n = m,..,n pour tout (1I",n) E P++ X N. II est bien eonnu (cf. par exemple [Kr], Satz 3.3.2) que A est une k-algebre integre de type fini. On peut done ehoisir un systeme de generateurs de A , tel que ehaque P, soit un vecteur propre de T (de poids 11", E P++), et soit homogene (de degre > 0). Montrons que C(X) est l'enveloppe convexe des , (1 :5 i :5 r). En effet :
n,
> 0 tel que n{3 E P++ et m n {3' ,n {:} il existe un entier n > 0 tel que n{3 E P++ et A n {3' ,n {:} il existe un entier n > 0 tel que n{3 E P ++
(3 E C(X) {} il existe un entier n
i= 0 i= 0
et un rnonome en les P, dont le degre est n, et le poids est n{3'
{:} il existe (a1 ••• a r) E N" tel que tous les {3' = a111"1 + ... + a r1l"r aln1
+ '" + arnr
{:} {3 est dans I 'enveloppe eonvexe des
1I"U ni
a,
ne sont pas nuls et
(1 :5 i :5 r).
La proposition est demontree, COROLLAIRE (de la demonstration).
II existe un ouvert de Zariski Xo de X tel que pour tout z E Xo, on (ou . x designe l'adMrence de la a-orbite de x). (ii) L'ensemble des C(Y) (ou Y parcourt toutes les sous-varietes Iermees, irreductibles, stables par a, de X) est lini. (i)
ait C(X)
= C(a· x)
Demonstration. (i) Conservons les notations de la demonstration precedente. Soit X o l'ensemble des x E X tels que P,(x) i= 0 pour tout i E [1,r]. Pour tout z E X o, on dispose d'une fonction reguliere invariante par U sur . z, non nulle, homogene de degre ni, et veeteur propre de T de poids 1I"i, done '!rUni E C(a· x). D'autre part, il est clair que C(a· x) c C(X), d'ou l'egalite cherchee, (ii) resulte immediatement de (i) par recurrence sur la dimension de X .
EXEMPLES. a) Formes quedretiques. -
Soit a = SL(r, k) operant sur l'espaee vectoriel M des formes quadratiques en r variables. Soient W1 •• , W r-1 les poids fondamentaux de a , i.e. est le poids dominant du a-module A'k r • Alors M est simple,
w,
de plus grand poids 2W r-1' D'apres [V2], proposition 1, l'algebre k[M]U est engendree par des veeteurs propres de T , homogenes, de degree et poids : (1, 2wt), (2, 2W2) ... (r 1,2wr - I) , (r,O). Soit X = P(M) : alors C(X) est l'enveloppe convexe de 0 et des points 2W r - i pour 1
:5 i :5 r
1. En particulier, les sommets de C(X) ne sont pas tous
dans P++ (eontrairement remarque).
a. ee
qui se passe si
a est
un tore; d. 4.2, deuxieme
180
Soit Y une sous-variete ferrnee, stable par G, de X . II existe un entier Y soit l'ensemble des formes quadratiques de rang au plus p. On
p E [1, r] tel que
en deduit que C(Y) est I'enveloppe convexe des que Wo
= 0).
t
pour 1 :::;
i:::; p
(on convient
b) n-uples de vecteurs. On prend toujours pour G Ie groupe S L(r, k), et on fait operer G sur la somme directe M de n copies de k r [ou n est un entier positif). Soit X = P(M). Pour tout pEN, on a :
= SPM
SP(k r 0 k n )
oii G opere trivialement sur k", La decomposition de en G-modules simples s'effectue a. I'aide des "foncteurs de Schur" SA associes aux partitions X de p . Rappelons (cf. par exemple [D]) que si >.. = (al"'" a.) ou al 2:: ... 2:: a. > 0 et al + ... + a. = p, alors : • SAk r
= 0 pour s > r
• SAk r est Ie G-module simple de plus haut poids (al - a2)wI a3)w2 + ... + a.w. pour s :::; r.
+ (a2 -
De plus, SP(k r 0 k n ) est la somme directe des (SAk r ) 18> (SAk n ) pour toutes les partitions >.. de p. On en deduit facilement que C(X} est l'enveloppe convexe des points W2 Wn W2 Wr-l ( ) WI,-, ... , - (pourn
t'
X(T)QIlzR=V R
---->
U
U
ou /s.'(TKl est l'ensemble des points fixes de T K dans Ii'. On peut done considerer C(X) C comme plonge dans t', lui-meme plonge dans Ii'. PROPOSITION. -
C(X) est l'ensemble des points rationnels de
Demonstration. - Soient 11" E P++ et n un entier strictement positif, Rappelons que Cn: designe l'orbite fermee de G dans P(Mn:)' Definissons un plongement G-equivariant de X X C; dans p(snM x Mn:) [ou snM est la nieme puissance symetrique de M) par la composition X x c; P(M) x P(M,..) p(snM) X P(M,..) p(snMQIlM,..) ou i est l'inclusion; vest le plongement de Veronese P(M) -+ p(snM); e est le plongement de Segre. D'apres un resultat dfi a Mumford (cf. [N], Appendix), les conditions suivantes sont equivalentes :
(i) E p,(X) ; (ii) la sous-variete Xx C,.. de p(sn MQIlM,,) contient des points semi-stables. Notons Xn le cone affine de snM au-dessus de X c p(snM), et Y Ie cone affine de S" M QIl M,.. au-dessus de Y = X x C,... La condition (ii) equivaut au fait
qu'il existe des fonctions regulieres G-invariantes non constantes sur Y, c'est-a-dire qu'il existe un entier m > 0 tel que =1= O. Or
k[Y]m '::: k[Xn]m
QIl
k[C,..]m '::: k[X]mn
En outre, le G-module k[C,..]m est isomorphe Theorem 2). Done
d'ou
=1=
0
{:>
a
QIl
M:nlr
k[C,..]m. (cf. par exemple [PV],
k[X]mn contient un G-sous-module isornorphe a M m,.. ·
On conclut que : la proposition. COROLLAIRE
E p,(X)
{:>
il existe m
>0
tel que
(cf. [GS] I et II; [K2]i [N:, Appendix).
E C(X), d'ou
p,(X)
est
l'envelopp.e convexe d 'un nombre fini de points rationnels de VR. 2.3. scalaire
On revient au cas d'un corps quelconque. On choisit un produit VxV---+Q
(x, y)
---+
x· y
invariant par le groupe de Weyl de T , et on note I II la norme associee. Le polytope convexe C(X) possede un unique point le plus proche de l'origine (pour la norme II II); soit (J ce point. On peut trouver un entier n > 0 tel que n(J E P++ et que k[X]n contienne un sous-module isomorphe a M n[3l ; on peut donc choisir un covariant cT? : -+ M n [3, de degre n .
X
182
de
PROPOSITION. Pour tout choix de n et est le cone Gnp des vecteurs primitifs de Mnp.
comme ci-dessus, l'image
Demonstration. - Comme est non nul et Gnp \ {o} est une seule orbite de G, il suffit de montrer que I'image de est contenue dans Gn /3 ' Notons Y l'adherence dans Mnp de l'image de c'est un cone ferme de Mnp, stable par G, done Y :> Gnp (en effet l'image Y de Y \ {o} dans P(Mnp) contient I'unique or bite fermee Gnp de G dans P(Mp)). D'autre part, pour tout pEN, le G-module k[GnPl p est isomorphe a M;nP (cf. [VI], loc. cit.]. II suffit done de montrer que k[Y]p est isomorphe a M;nP pour tout pEN. Soit h,p) E P++ x N tel que k[Y]p contienne un sous-G-module isomorphe On a alors un covariant 1/J : Y -> MP"I de degre p, d'oii un covariant 1/J 0 : X -> MP"I de degre rip, Par suite = ; E C(X), done 11;11 ;::: 11,811 par definition de ,8.
a M;"I'
D'autre part, M;"I est isomorphe a un sons-module de k[Y]p, done aussi Par suite, SP Mnp contient un sous-module isomorphe a de k[MnP]p = SP Mp"l' On en deduit (cf. par exemple [Bou], §7, proposition 9 et exercice 18) que Ilnll :5 Ilpn,8ll, avec egalite si et seulement si "I = n,8 j de plus Ie G-module M pn/3 figure avec multipllcite 1 dans SPMnp. On peut done conclure que "I = n,8, et que krY]p est isomorphe a M;nP , d'ou Ia proposition. C OROLLAIRE (Bogomolov). Boit Z une variete affine, sur laquelle G opere regulierement avec un point fixe. On suppose que toute Eonction reguliere, invariante par G , sur Z, est constante (i.e. que Zest Iottuee de points instables). 11 existe alors un 7f E P++ \ {o) et un Grrnorphisme surjectii ie : Z -> G'!r' Demonstration. On peut choisir une G-immersion fermee i : Z -> M au M est un G-module rationnel de dimension finie, et ou 0 E i(Z). Faisons operer k" sur M par homotheties, et no tons X I'adherence de k*i(Z) dans M . La proposition precedente fournit un 7r E P++ et un G-morphisme homogene surjectif :X --> G'!r' Comme k*i(Z) est dense dans X, la restriction de a i(Z) n'est pas nulle. Enfin, comme G1r \ (o} est une seule orbite de G et que = 0, Ia restriction de a i(Z) est surjective, done 0 i convient.
3. Stratification 3.1. Redeflnissons maintenant la stratification de X. Pour tout x E X, notons G· x l'adherence de la G-orbite de x . Pour tout,8 E V+, soit Sp I'ensemble des x E X tels que ,8 soit Ie point le plus proche de 0 dans C(G· x). Si k = C, on retrouve la stratification de X definie a I'aide de I'application moment (cf. 2.2 et
[Kl] §6). Soit 8 l'ensemble des ,8 E V+ tels que Sp ne soit pas vide. D'apres le corollaire 2.1, (ii), l'ensemble 8 est fini ; en outre, il est clair que les Sp , (,8 E B) forment une partition de X. Remarquons que So est l'ensemble des points semistables de X. En effet : x E So {:} a E C(G' x) {:} le cone affine sur G· x possede une fonction reguliere, invariante par G, et non constante.
183
PROPOSITION (cf. [K1l, Lemma 12.16). -
U
8(3-8(3c
Pour tout (3 E 8, on a:
8"{.
11"111>11(311
Autrement dit, /3 est Ie point Ie plus proche de
a dans G(8(3).
Demonstration. - Soit x E S(3 - 8(3. Posons Y = G· x; soit "I Ie point Ie plus proche de l'origine dans C(Y). Choisissons un entier n > 0 tel que n"l E P++, et un covariant cI> : Y --+ M n "{ de degre n; alors cI>(x) n'est pas nul. Prolongeons cI> en un covariant cI>' : X --+ M n "{ de degre n . Comme xES(3 ,il existe y E 8(3 tel que cI>'(Y) =1= o. Done "I = E C(G· y), d'ou 11"111 11/311 car y E 8(3. Comme x 8(3, on a meme 11"111 > 11/311, d'ou la proposition. COROLLAIRE. Les 8(3 (/3 E 8) forment une partition (linie) de X en sous-vexietes localement Iettnees, stables par G.
Demonstration. -
II est clair que
8(3=S(3-(
U
8"{),
11"111>11(311 et on deduit immediatement de la proposition que UU"{II >11(311 8"{ est ferme dans X. Done chaque 8(3 est une sous-variete localement fermee de X. Les autres assertions sont evidentes. 3.2. Rappelons que pour tout tt E P++, on note G1r l'image dans P(M1r ) des vecteurs primitifs de M 1r , et P1r le groupe d'isotropie du point fixe X 1r de B dans G1r • On a G1r = G· X 1r G/ P1r • Soit /3 E V+; alors le groupe P n(3 ne depend pas de l'entier n tel que n/3 E P ++, et on peut definir P(3 = P n(3' Si on considere /3 comme un caractere virtuel de T, alors P(3 est l'ensemble des 9 E G tels que /3(t)g/3(t)-l "ait une limite quand t --+ 0" (cf. [H]).
L'enonce suivant signifie que chaque strate 8(3 est fibree sur G / P(3. PROPOSITION . Soit /3 E 8. On peut trouver un en tier n > 0 tel que n/3 E P ++, et un G-morphisme
0 tel
n/3 E P++, no tons l'ensemble des x E 8(3 tels qu'il existe un covariant cI> : X --+ M n (3 de degre n avec cI>(x) =1= o. Demonstration. -
LEMME. (i)
Cheque
est ouvert dans 8(3.
(ii) 8(3 est Ia reunion des (iii)
C
pour n E N.
pour tout (r, s) tel que r divise s.
Demonstration du lemme. -
(i) est clair.
(ii) Soit x E 8(3 : en appliquant la un covariant cI> :
X --+ M n(3 de
ala variete G . x, on trouve
degre n, envoyant G· x sur
Gn (3 ' Done
cI>(x)
=1=
o.
184
(iii) Soit x E ::f O. Posons Y
Choisissons un covariant
:
X
M rf3 tel que
G . x : alors .k[Y]r contient un vecteur propre de B , de poids r{3' (c/. 2.1). Puisque r divise 8, on peut done trouver un vecteur propre de B, de poids 8{3', dans k[Y]s; d'ou un covariant X M sf3 de degre 8, non nul sur Y. Comme Y est l'adherence de G· z , on voit que n'est pas nul, i.e. x E S(s) f3
Fin de la demonstration de la proposition. du lemme qu'on peut choisir n tel que Sf3 =
On deduit immediatement Soit E la somme des sous-
a L'inclusion E c k[X]n fournit un GE' homogene de degre n, d'ou une application rationnelle,
Gmodules de k[X]n isomorphes morphisme 1/J :
X
notee encore 1/J : X P(E'). Par construction, 1/J est definie sur Sf3. Soit G I'image dans P(E') des vecteurs primitifs de E' (i.e. des vecteurs propres d'un sousgroupe de Borel de G). Soit p la multiplicite de dans E . L'isomorphisme de Gmodules E' M n f3 ® k P (ou G opere trivialement sur kP) induit un Gisomorphisme G Gnf3 X P(k P ) . D'apres la proposition 2.3, I'image de Sf3 par 1/J cst une sousGvariete de G, sur laquelle G opere non trivialement. On prend alors pour
EXEMPLES.
a) Formes quadratiques en trois variables. - On considers G = S L{3, k) operant sur l'espace vectoriel M des formes quadratiques en trois variables, et on prend X = P(M) (cE. l'exemple 2.2 a)). Alors X est un plongement de SL(3,k)/SO(3,k), et G a une orbite fermee Y dans X ,telle que
I
o
I
I
=
2Wl'
191
b). On prend toujours G = SL(3,k). Soient M = k 3 ED (k 3 )' , et P(M). Alors X est un plongement de SL(3,k)!SL(2,k), avec deux orbites
X fermees YI = P(k 3 ) et Y2 la figure suivante :
= P(k 3 ) ' , dont
les "poids" sont
\C(X) II I II
II
I
WI
et
W2.
On obtient
192
References IA) [BLV] [Bo]
[Bou] [Br I] [Br 2]
[Bru]
[D] (GS I et II]
IGS III]
[H] [K 11 [K 2]
IKr] [LV)
[N] [Vin]
[V 1]
[V
21
[PV]
M.F. ATIYAH. Convexity and commuting Hamiltonians, Bull. London Math. Soc., 14 (1982), 1-15. M. BRION, D. LUNA, TH. VUST. Espaces homogenes spheriques, Invent. Math., 84 (1986), 617-632. Holomorphic tensors and vector bundles on projective varieties, F •A. B OGOMOLOV. Math. USSR Izv., 13 (1979), 499-556. Groupes et algebres de Lie, Chapitre VIII, C.C.L.S. Paris, 1975. N. BOURBAKI. M. BRION. Invariants d'un so us-groupe unipotent maximal d'un groupe semi-simple, Ann. Inst. Fourier (Grenoble), 33 (1983), 1-27. Quelques proprietes des espaces hornogenes spheriques, Manuscripta M. BRION. Math. , 55 (1986), 191-198. A. BRUGUIERES. Proprietes de convexise de l'application moment, Expose au seminaire Bourbaki, novembre, 1985. J. D IEUDONNE. - Schur functors and group representations, dans: Tableaux de Young et foncteurs de Schur en algebre et geornetr'ie, Asterisque, (1981), 87-88. V. G UILLEMIN et S. STERNBERG. Convexity properties of the moment mapping, Invent. Math. , 67 (1982), 491-513 et 77 (1984), 533-546. Multiplicity-free spaces, J. Differential V. GUILLEMIN et S. STERNBERG. Geom., 19 (1984), 31-56. W.H. HESSELINK. Uniform instability in reductive groups, J. Reine Angew. Math., 304 (1978), 74-96. F. KIRWAN. Cohomology of quotients in symplectic and algebraic geometry, Princeton University Press, Mathematical Notes, 31, 1984. F. KIRWAN. Convexity properties of the moment mapping III, Invent. Math., 77 (1984), 547-552. H. KRAFT. - GeometrischeMethoden in der Invariantentheorie, Vieweg(BraunschweigWiesbaden), 1985. D. LUNA, TH. VUST. Plongements d'espaces homogenes, Comment. Math. Helv., 58 (1983), 186-245. A stratification of the nullcone via the moment map, Amer. J. Math., 106 L. NESS. (1984), 1281-1330. E.B. VINBERG. Complexite d'actions de groupes reductifs, (en russel. Functional Anal. Appl., 20 (1986), 1-13. TH. VUST. Operations de groupes reductifs dans un type de cones presque hornogenes, Bull. Soc. Math. France, 102 (1974), 317-333. TH. VUST. - Sur les invariants des groupes classiques, Ann. Inst. Fourier (Grenoble), 26 (1976), 1-31. V.L. POPOV, E.B. VINBERG. On a class of quasi-homogeneous varieties, Math. USSR Isv., 6 (1972), 743-758.
MAXIMAL ORDERS OVER CURVES Jan VAN GEEL State University of Ghent. (With an appendix by P. Salberger)
In my talk for the
Dubreil-Malliavin ", I dis-
cussed some result obtained by M. Denert and myself, on maximal and hereditary orders in central simple algebras over global functionfields. It is possible to give a "geometric" interpretation of these results,
[DJ .
especially of the cancellation theorem given in [DVGl] and
This
interpretation is based on Witt's Riemann-Roch theorem for central simple algebras over functionfields. In this paper I describe this geometric approach. In section 3 it is shown how one can use Witt's result to obtain some partial results on orders in central simple algebras over functionfields. The main reason to pursue this approach is that it enables us to obtain results beyond the
case. In
an appendix Per Salberger gives a geometric result from which the above mentioned cancellation theorem follows for quaternion algebras. I am grateful to J.L
for many discussions on the sub-
ject, the proof of the stable norm theorem (originally due to Swan) given here (proposition 2.1) results from these discussions. I am also in dept to M. Denert and P. Sal berger for analogous reasons. The author wishes to thank the university of Paris-Sud (Orsay) for the hospitality and the NFWO for financial support.
194
1. SHEAVES OF ORDERS.
Let C be a complete, geometrical integral, regular curve over a perfect field k. Consider a central simple algebra A over the function field k(C). A maximal 0c - order 0A is a sheaf of maximal orders in A, i.e. f(V,OA) is a maximal f(U,Oc) - order for every affine open set V in C.
Let x be a (closed) point on C, then 0c we denote its maximal ideal with m
x'
,x is a discrete valuation ring
For 0A a max. 0c - order, 0A,x
is a max. 0C,x - order with maximal ideal denoted by M x' The degree
[0,H,X 1M x : 0c ,x 1m X ]
A,r
0c,xlmx: k]
x is the relative residue class degree of
f x is the degree of x and -
which is also relatively minimal.
Thus by the theorem and the corollary above we see that if
js an inte-
hO(Or)=l and Pc XA the k point then P lies on a smooth fibre Xz of X I Z. The
gral 0A ideal of degree 2 with to
k points of X z correspond bijectively to the integral A ideals isomorphic to
If on the other hand hO(Or)=2 for Or' the right order of which is equivalent to the fact that P lies Then by the theory of Brzezinski, cf. from X
A
then 0ApOr
on a smooth fibre of X/C.
[8J, we know that Xr is obtained
by first blowing up P and then blowing down the proper trans-
form of the fibre at XfllC containing P. But hO(Or)= 2 so X r Ie js a Chatelet bundle, i.e. there is a quadratic extension k' I k such that ->-
c
x k '
has a section. Now it is easy to see that the divisor
of 2 conjugated nonintersecting lines, providing the sections of ->-
-X K y
XA
c
is the transform of a divisor in the linear system
x k'
on X., consisting of 2 conjugated lines intersecting in P, i.e. II
to a singular fibre of XA over Z. Hence we have: Theorem A.7
0A
0c
pointA
@ 0C(-I)
m 0C(-2)
a gijgc-
ot XA/Z containing k-pointA and th2
tion ggtwggn thg ot
m 0c(-l)
l2tt 0A ideal6 ot to
ot XA/Z and k pointA ot Xf\IZ.
with hO(Or) to
2 in Q.
k -
2 lie on thg
with hO(Or) = 1 lie on
212
Finally, let us mention that the situation when r simpler. Then the linear system
1=1
and g(OA) > 2 is
-KX - Xy contains at most one effec-
tive divisor. This divisor can either be a smooth k - curve isomorphic to
or a union of 2 intersecting and conjugated lines (compare with and the k - points (resp. k - point) of this divisor corres-
pond(s) to one isomorphism class of integral A - ideals of The other isomorphism classes (i.e, those with
= 1)
-2.
consist of
single A - ideals.
References.
[0]
M. Denert, Orders, in non Eichler algebras over global
functionfields, haVing the cancellation property, To appear. [DVGl]
M. Denert, J. Van Geel, Cancellation property for orders in non Eichler division algebras over global functionfields,
J. retne angew. Math. 368 (1986), 165-171. [DVG2]
M. Denert, J. Van Geel, The Classnumber of hereditary orders in non Eichler algebras over global functionfields, To appear.
[H]
R.
Hartshorne, Algebraic Geometry, Springer Verlag, Berlin -
Heidelberg
1977.
LR]
I.
[Sa]
P. Salberger, K-theory of orders and their Brauer-Severi
Reiner, Maximal Orders, London 1975.
schemes, Thesis Goteborg 1985. [Swl]
R. Swan, Strong approximation and locally free modules, in
Ring theory and Algebra III, ed. B. McDonald, Marcel Dekker, New York 1980 [Sw2 ]
R. Swan, Projective modules over binary polyhedral groups,
J. reine angew. Math. 342 (1983), 66-172.
[v]
M. Van Den Bergh, Algebraic subfields and splitting fields of division algebras over function fields, Thesis Antwerp 1985.
[VV]
M. Van Den Bergh, J. Van Geel, Algebraic elements in division algebras over function fields of curves, Israel J. of Math.,
52,no 1-2,1985, 33-45.
lid
E. Witt, Riemann-Rochser satz and Z-functionen in Hypercomplexen, Math. Ann. 110, 1934, 12-28.
213
References added in the appendix.
M. Artin, Left Ideals in Maximal Orders, Springer LN 917, 1982, 194-210.
[BJ
J.
Brzezinski, Brauer-Severi schemes of orders,Springer LN 1142,
1985, 18-49.
nt,
V. Iskovskih, Birational properties of surfaces of degree 4 in Mat. Sb.(N.S.),88 (1972), 31-37.(=Math. USSR-Sb. Vo1.17 (1972) no 1,30-36.). V. Iskovskih, Minimal models of rational surfaces over arbitrary fields, Izv. Akad. Nauk SSSR ser Mat.43 (1979), 19-43.(=Math. USSR-Izv. 14 (1980), 17-39.),
Ghent,Octobre 1986. State University of Ghent Seminar of Algebra and Functional analysis. Galglaan, 2 B-9000 Belgium.
GHENT
LINEAR COHEN-MACAULAY MODULES ON INTEGRAL QUADRICS J.HERZl. Thus a linear MCM-module over S has the projective A-resolution
o It
follows
-+
Zl(M)0A(-l) R
from
this
-+
ZO(M)0A R
resolution
-+
M
that
-+
0 M
is
in
fact
a
cohen-Macaulay module. The main goal of this paper is to prove the following
Theorem 1: Let R be the ring of integers of an algebraic number Field. Then For each integer number
of
isomorphism
classes
of
there exists only a Finite linear
MCM-modules
of
rank
m
over S.
This Theorem implies a result on matrix factorizations which in fact is equivalent to Theorem 1 if R is factorial. Let Em denote the unit matrix of size m, Eisenbud
[5]
According to D.
one calls an equation QE where a and (3 are m=a(3, square matrices with linear forms (elements of A as coeffil)
216
cients, a matrix factorization (of size m) of Q. Two such factorizations QE
are called equivalent, if there exist matri-
m
ces S,TEGI(miR) such that a'=SaT ization
-1
-1
, and
.
• A matr1x factor-
is called decomposable if there exist matrix factor
izations
and
g2]i
such that
is equivalent to
otherwise it is called indecomposable.
Now Theorem 1 implies
Theorem 1': Let Q be a quadratic the
ring of
integer
form with coefficients in
integers of an algebraic number
field.
Then for any
there exist only a finite number of equivalence clas-
ses of matrix factorizations of size m of the quadratic form Q.
We will prove these Theorems by applying the methods introduced in (2], where we proved similar (but stronger)
results for
quadratic forms with coefficients in a field. On the one hand we consider the category
of linear MCM-mo-
dules. The linear MCM-modules are naturally graded with all generators in degree o. Thus we define the morphisms in
to be the
linear homogeneous maps of degree O. On the other hand we consider the even part Co (Q)
of the Clifford algebra C(Q).
that Co(Q) is a (free) R-order of R-rank 2 category
of
Co (Q) -modules which
generated as R-modules.
are
n- 1.
It turns out
We denote by
R-torsionfree
and
the
finitely
217
Theorem 2: The category of Linear MCM-moduLes
and the cate-
gory A of CO(Q)-moduLes (which are finiteLy generated over Rand O R-torsionfree) are equivaLent.
Theorem 1 is an immediate consequence of Theorem 2 and the Jordan-Zassenhaus Theorem ([10] and [4], Theorem (79.1». We shall which establishes
see in Proposition 3 that the functor the
equivalence
of
the
categories,
has
the
property
that
for all So it suffices to show that there exist only finitely many isomorphism classes of Co(Q)-modules of a given R-rank.
But this
follows from the Jordan-Zassenhaus Theorem, since, as Q is non-degenerate,
the tensorproduct K0C (Q) O
is a
semi-simple K-algebra,
R
see [6].
Before we define the functor
::£-+"«0 we show
(following the
ideas of D.Eisenbud in [5]) how the linear MCM-modules are relatad to matrix factorizations.
proposition 1: a) compLex 0
-+
homogeneous
Linear MCM-moduLe with
Z1 (M)0A(-1) S ZO(M)0A -+ M homomorphism
Q.id Z (M)0A=a{3.
o
Let M be a
Moreover,
-+
{3:Z0(M)0A(-2) (Coker{3) (1)
-+
o.
Then
there
Z1 (M)0A(-1)
is again a
Linear
exists a such
Linear
that
MCM-mo-
duLe.
b) Given (finiteLy generated) torsionfree R-moduLes NO and N and I, homogeneous homomorphisms N 10A(-1) R
S N00A R
and N 00A(-2) R
N10A(-1) R
218
such that Q.id
then Cokera is a linear MCM-module.
No
Proof: a) Since M is an S-module we get a commutative diagram
where lemma
0
ZO(M)@A(-2)
0
Zl (M)0A(-1)
S a
Zo(M)@A !id
ZO(M)@S !E. M
ZO(M)0A
is the natural induced homomorphism. we
see
that
and
using
0 0
Applying the snake b)
it
follows
that
is a linear MCM-module. N1@A(-1) S NO@A
b) Let M=Cokera, then clearly 0 the linear complex of M. In fact, a equation Q' idN @A
o
M
0 is
is injective because of the
•
Thus it remains to be shown that M is R-torsion-free. equation
Q' idN o
implies
where
Q'id N
and
'fI
(i)
1
denotes the i -th twist of the homogeneous map exchange the role of a
The
'fl.
Hence we may
and show instead that
is
R-torsionfree. In the above commutative diagram, if we replace the Zi(M) by the Ni we obtain as before Suppose we know already that S is R-torsionfree, then, as R is a Dedekind ring and No is R-torsionfree,
N and hence also 00S
is R-torsionfree. Now we show that S is R-torsionfree. Let s S, r R, r#O, such that rs=O.
If a A is a representative for s then ra=Qh for some
polynomial h A. Let
be a prime ideal of R. We denote by v
ation, and for f A, seen that
V
v
XV we set v
the
valu-
(f)=min{v (a )}. It is easily v v
for all f,g A.
219
Applying this rule to the equation ra=Qh, we get for
all
since
Q
is
primitive.
But
this
implies that h=rh' for some h' A, and therefore a=Qh', and s=o.
Corollary:
The
isomorphism
cl.asses
of
l.inear
MeM-modul.es
M
over S=A/Q, for which the homogeneous components M are free R-mo-
i
dul.es, correspond bijectivel.y to the equival.ence cl.asses of matrix factorizations of Q.
The Corollary shows in parti.cular that,
in fact,
implies Theorem l' and is equivalent to Theorem l'
Theorem 1
if R is fac-
torial.
Proof
of
the
MCM-module M for
Corollary:
The
linear
complex
of
a
linear
which the homogeneous
components M are free i R-modules is an A-free resolution of M. Having chosen bases for ZO(M) and Zl(M), the equation Q·id z
o
yields a matrix fac-
torization of Q. Conversely, Proposition 1),b) shows that a matrix factorization of Q yields a linear MCM-module. It is clear that this correspondance establishes a bijection between the
isomorphism classes
of
linear MCM-modules
and
the
equivalence classes of matrix factorizations of Q.
We now briefly describe the Clifford algebra C(Q) associated
. Q. Let F=A* be the R-dual of the free R-module A w1th . dual w1th 1 l . ei=x * for 1=1, . bas1s ... ,n, and let T=T(F) be the tensor algebra of i
220
F.
The Clifford aLgebra C(Q)
of Q is the quotient algebra TjI,
where I is the two-sided ideal of T which is generated by the elements X=
x F.
Here
we
have
set
n 2: x e .• i=l l l
Q(x):=Q(x1, ... ,xn)
for
i
Proposition 2:
a)
C(Q)
is a
7l.j2Z-graded aLgebra,
decomposes as R-moduLe into a direct for
c.L
L i,j {O,l}
one
has
that
is,
sum CO(Q)lDC 1 (Q), such that where k {O,l} and
k=i+jmod2. b)
CO(Q) and C1(Q) are free R-moduLes with bases j even} and j odd} respectiveLy.
c) The composition of the naturaL maps and C(Q)
is an injection,
is generated by the image of F in C1(Q)
(which we identi-
fy with F).
Proof: a) and c) follow immediately from b). It thus suffices to prove b). elements
The defining ideal of C(Q)
2
ei-a
for i=l, ... ,n, and ii Using these relations we see that
certainly contains the
e.e.+e.e.-a.. l J
J l
lJ
for
is a system of genera-
tors for the R-module C(Q). We will write Q* if we consider Q as a quadratic form with coefficients in the quotient field K of R. Then
and under this isomorphism
is mapped onto a
R
basis of C(Q*), as one knows for Clifford algebras over fields, see [6]. But then
must have been an R-basis for C(Q) as well.
221
We denote by A the category of Z/2Z-graded C(Q)-modules which as R-modules are finitely generated and torsionfree. Our next aim is to define functors Definition
and
of
Let
then
we
set
and
To explain the action of C(Q) on ZO(M)$Zl(M) it suffices to define the multiplication by elements of F, since F generates C (Q) . Given two R-modules MO,M
1
and a
free R-module G,
there are
natural isomorphisms
C*)
R
Thus to the homogeneous components Zl(M) Zl(M)@A1 of a and
P
oZO(M)@A
ZoCM) and F@ZoCM)
R
Zl(M),
R
which define the multiplication on
x
*)
and 1 correspond R-modules homomorphisms
F@ZlCM)
It follows 2m=x(xm)=Q(x)m
a
R
easily
from
by the elements of F.
the
identities
for all x F and
Q-id=ap=pa,
so that in fact
that has a
C(Q)-module structure. Finally, maps 'PO:ZO(M)
a morphism 'P:M
M'
in
ZO(M) and 'P1:Z1(M)
induces unique R-linear Zl(M), and we set
It is easily seen that the so defined
A is a
functor. Definition of
Let N=No@N1 A. The multiplications with ele-
ments of F define maps F@N
R O
phisms (*), we obtain maps NO
N and F@N 1 R 1 N1@A
R 1
and N 1
NO- Using the isomorNO@A
R 1,
which can be
uniquely extended to homogeneous A-linear maps NO@A(-l)
R
N1@A and
R
222 a
N -+ N such that Q-id O (ii)
Let
M
be a maximal ideal of
be a maximal we have
r.
in inj.
is an Artinian regular in
RI
dim
[X
M
Suppose that
+ . . + xnR
1R
(quasi-Frobenius)
R/l x
1R
now that
+ ... +XnR
R.
Now R
Thus
nR] is
n+ 1 k';; n
is a maximal k the ideal MI [ xl R + ... +XkR] which is regular in
RI
X
X
+ ... + XkR
1R .;; k
(R/l)
Is
1R
+ ... + xkR J contains Since
where pd.
denotes the
we have a short exact sequen-
R
n
Thus
k
n
Thus
.;; rank
Let
S
be a
is integral over a say.
x
(M)
Ext
n
R
(S,R) M
0
for each
n
and since
and
is minimal (or by 6.1),
is
By assumption, M
R
1R+
the image of
by the generalised principal ideal theorem
Thus (i)
...+X
[X
Extn(R/l,R) .... Extn(S,R) .... R R ExtRn(R/l,R) o since pd. R/l k < n
n+1 (Y,R). Now R n+1 Ext (Y,R) 0 since r. inj. dim. R which is a contradiction. Therefore over
[1,2.2]
RI
simple right R-module and let central subring,
contains a
By [1,2.2J ,
r.
inj.
of length dim.
RI
annRS
M is a maximal ideal [X
1R
n
+ ... + xnR
o
So R/[ xl R + ... + x R is a simple is a Quasi-Frobenius ring. Now S n RI [ xl R + . . . + x n R ] module so we may assume S c RI [ X 1R + ... + x n R] Therefore Now 0 where R* HomR*(S,R*) RI X 1 R + ... + x n R] by 2.1
"
Thus
R
k
This gives
.... Ext
maximal ideal of
, . . . ,x
" " ,x
I
(R/l)
1,x 2
Then by
contradiction.
3.1,
Let
n
and so
However,
which is a
and let x
>
k
0
in M, in the ring RI [ xl R + ... + x k R 1 does not contain the image of an element of a
R
is injectively homogenous.
o
240 4.2.
PROPOSITI,ON. Let
R
be a right Noetherian ring integral over a central subring Z
Let
be a
in
If
R
homogeneous t.h e n so is the ring R*= R/(Zk
1.=1
r.
inj.
dim.
inj.
dim.
x R)
Moreover
i
R-k,
Clearly i t is enough to check this for
Proof r.
R* = r.
inj.
dim R = n
right
r.
Then
R*-module.
Then by
inj.
2. 1 Ext
D
result follows.
is right injectively
R
dim R*
n
since by
[1 , 1.4] ,
242
r.
inj.
dim.
R(:";;
n
we have r.
[ 11 , 2 . 7 ]
. Thus as in
i
1+.,,+i p
= n
1
O'i
O'i
2
P
(
[ et cela pour tout couple d' elements Pour une haute 0 -deri vation
.L
0' (a) Lsps n p si pn , et
a
et
b
de
{J
n,p
(b)
n, avec KJ.
ou
{J
n
i >0 , r
)
(on) de K, 11 faut "adapter" la formule cr-
0
dessus, en y apportant une legere modification (modification due au « decalage » d'indices : 0 = 0' 1 ' n n+
pour tout
3.2. Pour tout automorphisme 0 yation de K
K toute suite
telle que, pour tout
«$»
et
pour tous 6
pour tout 3 . 3.
n = i
0
(0) n n
n
':': ,+r i+l=
En [7), I I ,
(
0 d'applications additives de
on (ab) = 0
n-j
0
0
r 1 r2
avec
r
s
Ce qui donne ceci :
K , on peut appeller
1
aE" K, bE" K,
i , U1
de
n , nE" N).
0
> :.
K dans
au ($$)
r i+ 1 )
nous avions considere des suites serielles de longueur ... ,(fk)'
Mais, iei, plutot que de parler de hautes finie ({fl' {f2' derivations "de longueur finie", on parlera de "segments de hautes bis derivations", comme en (5 ], § 0: une suite finie (0 0 ... , Ok) 0, 1, d'applieations additives de K dans K (au sera dite segment de 00-deriyation verifiee pour tout
si, et seulement si, l'egalite n
«$»
de (3.2) est
tel que
3.3.1. Toutes les notions ou definitions concernant les hautes se transferent de
evidente aux segments de hautes rr-derivations. Par
exemple : -- un segment (0 ) de haute {J-derivation est trivial si n
0 = 0 pour n
285
-- (on)O(n(k tel que
est un segment de haute
I
fondamentale s'1l est
:= 0 pour tout n non multiple de ll:. Et on detin1t l'indice n numerique, et Ie premier terme non trivial d 'un seEf11!lBDt non trivial de haute fondamentale, exactement comme on l'a fait pour une
S
haute
fondamentale non tr1v1ale.
a
Pu1sque l'on s'interessera 1ci uniquement fondamentales (y
n
)
= (0
-1
nx
des hautes 00-derivations
(0), il sera sou vent naturel de traiter avec la suite n ). Et il est utile d'ecr1re la formule expr1mant ¥ (ab), n
b1en que cette formule ne s01t pas plus simple que celIe donnant 0 (ab): n
3.4. S01t = 0 un automorph1sme periodique de K , de periode IT • 0 -1 Soit (S ) une haute SO-derivation fondamentale, et not one Y = S n ml: n pour tout n I n 0 Alors Y = id , et, pour tout n , nn 0 K «&JJ pour taus Lie: K , bE: K , (ab) = Yi Yn ou
r n0
et
Yn
pour tout
rn = i
i , D1
3.4.1 Et si
L
>
Y
(
avec
r
e
... 11¥
r2
ire+1
I
rill:
(&&)
r i 11:+ 1
)
est un segment de haute 00-derivation fondamentale
tel que
alors l'application
pour tout
r1
r 1+... +r i 11:+ 1 : n-i
n
tel que
1
nlt: ( k
3.4.2. Reciproquement, si
Y n
S
-1
possede la propriete «&))
nll:
au
(resp.
famille d'applications additives de
K dans
est une
K telle que
YO
que la condition «&JJ soit verifiee pour tout
n,
alors la famille
dafin1e en posant :
[resp.
(om)O(m(kre J
resp.
= 1d K
Snre = Y pour tout n, n s O [resp. n { S = 0 lorsque m n'est pas multiple de re m
est une haute
[resp. un segment de haute]
et ],
et
l1-derivation fondamentale.
Dans ce qui suit, on utilise souvent les suites (Y
et il est done n), commode de disposer d'une terminologie specifique designant ces suites: 3.4.3. DEFINITION. On appelle d'applicat1ons addit1ves de
l1-differen1ielle de K dans
K
I
K toute suite
telle que
et que la condition «&» de (3.4) soit satisfaite pour tout Et on dira qu'une suite finie dans
K est un segment de suite
¥O
= idK
n. n;l.
d'appl1cations additives de si
YO = id
K
K
et s1 la
286
condition «&» de (3.4) est verifiee pour tout
n
tel que
3.4.4. La suite (¥ ) definie en (3.4) a partir d'une haute n
fondamentale (on) sera appelee la suite
associee
De
la suite (0 ) definie en (3.4.2) m rentielle sera appelee la haute
a
la suite
Et on emploira Ie
a
a
(on)
partir d'une suite fondamentale associee
vocabulaire pour la correspondan-
ce indiquee en (3.4.1) et (3.4.2) entre segments de hautes fondamentales et segments de suites Conventions de notations n
toute famille si
k
ecrite simplement
est un entier fixe (k>O), toute famille n
sera notee simplement 3.4.5. Comme en en (3.2),
n
[5
bi s]
, la famille
des applications definies
cs s », sera dite famille des tteres de 1a haute 00derivation n
De
la famille (r i)O\Hn definie en (3.4), (&&) , sera dite fa111ille des tteres de la suite . n La formule de definition de [ resp. de r montre que : i
n
,
ne depend que de 0 , 0 1, 0 fn ne depend que de [ resp. i et qu'en particul1er : 3.5.
3.6.
n n
n+1
on+1 = 0
[ resp.
r nn =
sni
( Cf , alJss 1' ( 5bis J ,
, Y n i
Yl'
(f
ml:
(O,S))
= id K
(cf .
rs bi s ]
I
(0.6»
3.7. Etant donne un segment de haute 00derivation (on)O\n\k ' 1a formuIe
($$)
n
de (3.2) permet de definir la famille des iteres de ce segment, soit (cf. (3.5»
. De
on peut M,finir la famille des iteres
([n) d'un segment de suite i O\i\n\k+i
•
3.8. Soit (on) une haute 00derivation de