Séminaire d'Algèbre Paul Dubreil Et Marie-Paule Malliavin: Proceedings. Paris 1981 (34éme Année) (Lecture Notes in Mathematics, 924) (French Edition) 3540114963, 9783540114963

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continuation on page 465

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

924

Seminaire d'Alqebre Paul Dubreil et Marie-Paule Malliavin Proceedings, Paris 1981 (34eme Annee)

Edite par M.-P. Malliavin

Springer-Verlag Berlin Heidelberg New York 1982

Editeur

Marie-Paule Malliavin Universite Pierre et Marie Curie, Mathematiques 10, rue Saint Louis en 1'lIe, 75004 Paris, France

AMS Subject Classifications (1980): 12L10, 13C15, 13H15, 14F05, 14J05, 14L05, 14M15, 14M17, 16A08, 16A12, 16A15, 16A26, 16A27, 16A33, 16A46, 16A48, 16A54, 16A55, 16A62, 16A64, 16A68, 17B1O, 17B35, 18G1O, 18G40, 20F28, 55R40, 58G07.

ISBN 3-540-11496-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-11496-3 Springer-Verlag New York Heidelberg Berlin

CIP-Kurztitelaufnahme der Deutschen Bibliothek Serninaire d'Alqebre Paul Dubreil et Marie-Paule Malliavin: Proceedings/Seminaire d'Alqebre Paul Dubreil et Marie-Paule Malliavin. Berlin; Heidelberg; New York Springer 34. Paris 1981: (34eme annee), -1982. (Lecture notes in mathematics; Vol. 924) ISBN 3-540-11496-3 (Berlin, Heidelberg, New York) ISBN 0-387-11496-3 (New York, Heidelberg, Berlin) NE: GT

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© by Springer-Verlag Berlin Heidelberg 1982 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210

Liste des auteurs L.L. Avramov p. 376 - D. Bartels p. 385 - H. Bass p. 311 - J.E. Bjork p.415 - W. Borho p , 52 - D.L. Costa p , 401 - P.. Fossum p. 261 - S. Gelfand p. I J.M. Goursaud p. 323 - T. Levasseur p. 174 - R.

MacPherson p.

- 11. P. !Iall iavin

p , 158 et p , 168 - G. llaury p , 185 - G. Mislin p. 297 - S. llontgomery p , 357 J.L. Pascaud p. 323 - J.L. Roque p. 242 - D. Salles p. 245 - P.F. Smith p. 198 J.T. Stafford p. 73 - F. Taha p. 90 - J. Valette p. 323 - E. Wexler-Kreindler p.144.

*

TABLE DES !1ATIERES

S. GELFAND et R.

MacPHERSON - Verma modules and Schubert cells : a dictionary

W. BORHO - Invariant dimension and restricted extension of noetherian rings

51

J.T. STAFFORD - Generating modules efficiently over non commutative rings

72

F. TARA - Algebres simples centrales sur les corps ultra-produits de corps p-adiques

89

M. ZAYED - Caracterisation des algebres de representation finie sur des corps algebriquement clos

129

S. YAMMINE - Ideaux primitifs dans les algebres universelles

148

M.P. lfALLIAVIN - Ultra-produits d l a l geb r es de Lie

157

M.P. lfALLIAVIN - Grade et theoreme d'intersection en algebre non commutative (Erratum)

167

IV

T. LEVASSEUR - Sur la dimension de Krull de l'algebre enveloppante d'une algebre de Lie semi-simple G. MAURY - Un theoreme de l'ideal P.F.

173

a

gauche principal dans certains anneaux

184 197

SllITH- The Artin-Rees property

J.L. ROQUE - Etude d'une classe d'algebres artiniennes locales non commutatives 241

*

R. FOSSUM - Decompositions revisited

261

G. MISLIN - Classes caracteristiques pour les representations de

groupes

discrets

296

H. BASS - Automorphismes de

et de groupes de type fini

J.M. GOURSAUD, J.L. PASCAUD et J. VALETTE - Sur les travaux de V.K. Kharchenko S.

310 322

- Trace functions and affine fixed rings for groups acting on 356

non-commutative rings L.L. AVRAMOV - Invariants d'un groupe fini, engendre par des pseudo-reflexions, operant sur

375

un a n n e a u local

D. BARTELS - On non-normality of affine quasi-homogeneous

* 400

D.L. COSTA - Semi-normality and projective modules J.E. BJORK - On the maximal number noetherian rings

elements in ideals of 413

v D. SALLES - Dimensions pures de

423

modules

E. WEXLER-KREINDLER - Anneaux filtres complets et suites spectrales associees

442

publie avec Ie concours de l'Universite Pierre et Marie Curie Ia Premiere Section de l'Ecole Pratique des Hautes Etudes

*

* *

PREVIOUS VOLUMES OF THE "SEMINAIRE PAUL DUBREIL" l"ERE PUBLISHED IN THE LECTURE 795

(1979)

NOTES, VOLUMES 586 (1976), 641

and 867 (1980).

(1977),

740

(1978),

VERMA MODULES AND SCHUBERT CELLS A DICTIONARY by Sergei GELFAND and Robert MacPHERSON

Preface

This paper was completed in June, 1980. Since that time there has been marked development in the subject. Most notably, a beautiful proof of the Kazhdan-Lusztig conjecture has been found by Beilinson and Bernstein

and by Brylinski and Kashiwara (BKJ. We have indicated

some of these further developments by adding notes marked with an asterisk

*. The proof of the Kazhdan-Lusztig conjecture has much in common

with the ideas of this paper. It proceeds by establishing an equivalence of categories between an algebraic category including Verma modules as objects, and a topological category including intersection homology sheaves of Schubert varieties as objects. However, as explained in the note on page 36", the dictionary from topology to algebra arrising from this equivalence

of categories is different from the one proposed in this paper. Never the less, we believethere is still somemerit in the point

of view we present here. Several predictions we made on the basis of this dictionary have been substantiated. One is our conjecture 2.10 which has been proved by Deligne, Gabber, Beilinson, and Bernstein (LDBBj ,[GM4J). Another is the fact that the decomposition of the coherent continuation of a projective module to the wall of a Weyl chamber parallels the decomposition of the projection of an intersection homology sheaf of a Schubert variety in

G/B

to

G/p

(See the note on page 35).

It would be very interesting to find a unification of the point of view presented here and that of [BB] and [BK].

2

Introduction In [KLI] and [KL2], Kazhdan and Lusztig made a remarkable conjecture that relates properties of certain infinite dimensional representations of a semisimple Lie algebra with those of singularities of Schubert cells in a generalized flag manifold. This paper represents an attempt to understand the source of this relation. Being a preliminary draft, this paper contains almost no proofs. Let

G be a semi-simple Lie group with Lie algebra

two rather different categories associated to the category of

There are

G. One is a subcategory of

0 modules of highest weight of Bernstein, Gelfand, and

Gelfand (see § 3.5). The other is a category of complexes of sheaves on a generalized flag manifold for

G (see § 2.1). One relation between these two

categories is the Kazhdan-Lusztig conjecture which asserts that the multiplicities of a simple module in the Jordan-Holder series of a Verma module is the dimension of the stalk homology of the complex of sheaves that gives the middle intersection homology groups of Goresky-Mac Pherson. But it appears that these categories have much more in common. This paper contains a dictionary which puts some of these common features in a more or less organized form. We should mention two things we could not do. First we do not know how to construct a complex of sheaves from a g-module or vice versa. Second (which may well be implied by the first) we cannot prove the Kazhdan-

. conjecture . * • Although we could not find a direct relat10n . between Luszt1g the category of sheaves and the category of

G-modules we have found an

indirect relation in the form of a functor from each of them to a third category (see § 2.12, § 2.18, § 3.13). This enables us to find a topological analogue (see § 2.8) for the method of "walking through the walls of a Weyl

*

The Kazhdan-Lusztig conjectures have now been proved (see preface).

3

chamber (or coherent continuation) that was extensively used in representation theory ([BG], [Sch] , [VI])' We have learned that D. Kazhdan has found several similar results.

In the other direction, we were able to refine the original KazhdanLusztig conjecture by showing a Lie­algebra interpretation of the complete Poincare polynomial of intersection homology sheaf (and not only of its value at the point

q

I , as in [KLI]).

*

§l contains notations and preliminary known results both from algebra and topology. §Z describes the topological side of the dictionary. §3 does the same for Lie algebras. We tried to arrange the material in more or less parallel form. For each result we use the notation (

§3.6) to indicate

where its counterpart may be found on the other side. We have stated some results which are fairly trivial when the parallelism with the other side was interesting. §4 contains the final comparison of the two sides of the dictionary and some remarks. Some readers may

to begin with this section.

§S contains some examples and tables.

Acknowledgements.

The authors would like to thank J. Bernstein, P. Deligne,

I.Gelfand, M. Goresky, D. Kazhdan, G. Lusztig, and D. Vogan for useful discussions on the subject of this paper. The second author would like to thank the Academy of Sciences of the U.S.S.R. and the American Academy of Sciences for support of this research through their exchange program.

*

Gabber and Joseph also found this refined conjecture (see [GJ]).

4

§l. Notations, conventions and preliminary results.

1.1.

Let

G be a complex semi-simple Lie group with Lie algebra

The

following data will be fixed throughout this paper. h

is a Cartan subalgebra of is a set of positive roots for the root system

Z

is the corresponding set of simple roots.

P

1S

of

h

in

half the sum of all the positive roots. is the nilpotent subalgebra of

(r e sp , n

generated by

root vectors corresponding to positive (resp. negative) roots. is the anti-involution (i.e. that is the identity on

h

2

1

identity. l[XY]

and transforms

to

[lY,lX]

n

is the

W is the Weyl group of corresponding reflection. £(w) generators

o

W acts from the left.

is the length function on a

, a E Z •

is the unique element of

W

o

W corresponding to the set of

W of maximal length; £(w

h* is the dual vector space to

is the bilinear form on 1S

and

non-negative integers

>

0

h*

induced from the killing form on if and only if

(X E

YE Z

ljJ- v


-

ljJ

A , not on

1.4. Schubert cells. Let root let

X

y

H be the Cartan subgroup of

For a subset G , generated by Denote by

Ac I

B and all

Hand

y

we denote by

X

peA)

the parabolic subgroup of

, a E A , so that

-a

be the dimension of

as a complex manifold

,where

combinations of roots in

6+(A)

B be

X , Y E 6+ .

P(0); B , P(I) ; G

the generalized flag manifold

card (6+'b+ (A»

h . For any

be the corresponding one-parameter subgroup. Also let

the Borel subgroup generated by

(A ') ->-

G, corresponding to

G/P(A) (d(A)

Let

d (A)

is equal to

is the set of positive roots that are linear

A). For

A' c A denote by

n(A',A)

the projection

(A)

For each C(w,A) ; B wowP(A)

w E W , define a Schubert cell

C(w,A) c

by

Summarize the properties of Schubert cells in the

following proposition :

Proposition. C(w,A) ; C(v,A)

if

-1

w

v

lies in

W(A) , and they are disjoint

otherwise. (So there is a one to one correspondence between Schubert cells in

(A») .

W' (A)

and the

7

The Schubert cells form a Whitney stratification of

¢(A) . In

particular the closure of a Schubert cell is a union of Schubert cells (axiom of the frontier). If C(w,A) c C(v,A)

Let

and

if and only if

w

ware in

TI(A',A)

to

TI(A,A')

C(w',A')

WI (A) , then

v

>

A' c A , w E Wl(A) , and

w-lw' E W(A) . Then of

v

w' E Wl(A') . Suppose that

C(w',A')

=

C(W,A) . Moreover, the restriction

is a fibration with a

2(d(A')-d(A)-2(w')+2(w»

=L

(real) dimensional open ball as a fiber. In particular, taking

A

see that

22(w')

C(w' ,A')

Suppose

is a cell in

A' c A and

¢(A')

of (real) codimension

w E Wl(A) . Then

!T(A',A)

-1---

we

C(w,A') .

C(w,A)

1.5. Sheaves. Let

X be a topological space. We denote by

of sheaves of vector spaces over the complex numbers on a sheaf is

S

=x

S

on an open set

. A sheaf

the sheaficiation Then

=

S

U c X is

SeX)

the category

X

The value of

, the stalk of

is called a constant sheaf with value

of the presheaf that takes the value

V for all connected

S

at

V if

x E X S

is

V on all open sets.

U. The constant sheaf with value

is called !(X)

We denote by

DbS(X)

the doubly bounded derived category of

(See [H] , [Vel]) . Thus an object

S·

of

DbS(X)

SeX).

is a sheaf of cochain

complexes

where i «

i (U)

.

i+l (U)

i (U)

{ •••

••. }

a complex vector space and

0 . A morphism from

S·

to

T'

in

i ( U)

DbS(X)

o

for

i» 0

and

is determined by a diagram

8

of chain-maps

t\

s' where

indicates a quasi-isomorphism, i.e.

on all homology groups of all stalks. If then the morphism is an isomorphism in

There are functors

I

'

q

induces an isomorphism

is also a quasi-isomorphism

nbs (x )

!!.i, and

f

from

S·

to

T

T

I

s (X)

I

(D)

...._--,_ _

is the complex that is

other dimensions. D

in dimension zero and zero in all

is the sheafification of the presheaf

• The translation functor

from

nbS(X)

to itself shifts the

(!

numbering of all chain complexes:

Proposition.

T

is characterized up to equivalences in

properties

!!.i (.!.(D)

The category

·e nbs (X)

if

i

if

i I- 0

by the

0

is a triangulated category

distinguished triangle denoted by

nbS(X)

([Vel] )

The

9 h

SeX) ..;;"'

_

S(Y)

f*

The functor

f*

called the pullback is defined as follows : if

5

Y is an etale map giving the sheaf

f*S

=

T

is given by the etale map

T

S

on

Y

([6], p. 110) then

X such that the following square

is a fiber square

5

The functor

f!

' called pushforward with proper supports, is defined

as follows ([Ve2], p.3) support is proper over

The functors and

Rf!

is those sections in

whose

U

f*

and

f!

determine right derived functors

Rf*

10


0 .

3) Dual support condition 2n-i

IC' (V»x f O}

dimt{x E Vi for all


0

Here, with respect to the directed system of neighborhoods x

, (Hi IC' (V»

=

x

(Hi IC' (V» x =c -

where

Hi c

lim Hi(U) ->

xEU lim Hi(U) c + xEU

denotes cohomology with compact supports.

U of

12

The sheaves of chain complexes

characterized by this

proposition are called intersection homology sheaves. They were constructed first in geometric topology in

then in algebraic geometrv in

[D]

(see [KL2]) . These constructions are proved to coincide in [GM3] . If is compact, then the hypercohomology groups of homology groups, satisfv Poincare duality: the numbers are equal.

V

IC'(V) , called intersection .

and the (2n-l)

Betti

13

§2. Topology 2.1.

Let

that

¢(A)

, the set of simple roots. Recall (§1.4)

A be a subset of

denotes the associated generalized flag manifold and

c(w) : C(w,A)4¢(A)

are the inclusions of the Schubert cells of

¢(A).

Definition. The category Chains (A) is the full subcategory of (where

¢(A)

is considered with its classical topology) whose objects

S

satisfy the following three conditions : I. Finiteness.

sional over

all

For

i

His'

is

zero if

3. Constructibility and all

1S

i E:

i

2.2.

§2.2

are all finite dimen-

is odd.

a constant sheaf on

CCw,A)

for all

;Z

Examples of objects in Chains of

HiS'

a;

2. Evenness.

w E: wI CA)

,the stalks of

(A)

are the cell sheaves

and the intersection, homology sheaves

IC'Cw,A)

of §2.5 •

Definition. For any

wE: Wi (A)

The stalk at

P

of

a;

if

o

otherwise

, the cell-sheaf i

Rc(w)!I[a;(C(w,A»].

is

H

p E: C(w,A)

is

and

i

0

Proposition.

A.

A)

The cell sheaf

lies in Chains

(A)

14

6. (

3.66)

have

the

The cell sheaves

, w E WI (A)

property that for any exact triangle in

Chains (A)

[I]

R'

I either

R'

Proof of o

v* H R'

6. -+

or

0

qi

T'

tll

0

.

Cell-sheaves have this property because by constructibility 0

is either zero or surjective.

H

from the long exact sequence in cohomology that

Then by evenness we find

R'or I

has no cohomology

and hence is quasi-isomorphic to zero,

2.3

( ...... 3.9)

of elements in

Proposition, filtration

Let

WI(A)

Suppose S = S· -0

::>

A be given. Let us choose a numbering in such a way that

S· S'

=1

J

::> ••• ::>

S· = 0 n

Let

C(w,A)

into

dew) ¢(A)

i > j

r

.

such that

S'/S' in =i =i+l k objects of the form T C(w.,A)

k

Proof.

implies

is in Chains (A). Then there exists a canonical

quasi-isomorphic to a direct sum of various

w. > W.

w1, •.. ,w

==

for

be the inclusion of the complement of the closure of , Then

satisfies the conditions of the

proposition.

15

2,4

We define the Grothendieck group of the category Chains (A) ,

K(Chains (A))

,to be the Abelian group generated by quasi-isomorphism

classes of objecm in

A subject to the relation

[R'] + [T']

[5']

whenever we have a triangle

[I]

(

We denote by

the equivalence class of

[5']

in

5'

K(A) .

I

For any Abelian group linear combinations ji E J , 7Z[q,q

-I

Ljiwi

]

J

we denote by

of elements

JW (A) I

Wi E W (A)

the group of formal

with coeficients

denotes the group of integral Laurent polynomials in

q

under addition ,

Corollary (

A.

3,8)

The Grothendieck group 2n T

rated by all sheaves

6.

K(Chains (A))

I , w E W (A) ,

There is an isomorphism k

K(Chains (A))

which takes to L P . w w 1/2 of the stalk cohomology of q

2.5.

n E 7Z

for

is the free Abelian group gene-

Let

c(w) : C(w,A) c

Schubert cell

C(w,A).

->

7Z[q,q-l]

Wi

(A)

where

s'

P is the Poincare polynomial in w at any point in C(w,A)

be the inclusion of the closure of the

16

Definition. c(w)!

( - - . . 3.66)

IC' (C(w,A»

IC'(w,A)

(w,A)

is an object in Chains (A) : finiteness is true of

A =

for any

V ; evenness was proved by Kazhdan and Lusztig

and it follows in general by applying

([GM3])

[KL2]

of §2.8 ; and

constructibility follows from the fact that homeomorphisms

is

(See §1.6)

algebraic variety when

The intersection homology sheaf

is

and the homeomorphism group of

invariant under

C(w,A)

is trans i-

tive on the Schubert cells .

2.6. v

and

The Kazhdan-Lusztig polynomial, w of

WI (A)

[IC'(w,A)]

(See §2.4) plicity of

V,w

, depending on two elements

l: P ·v v VW o ,wwo

denotes the class of

IC'(w,A)

In other words, the coeficient of 2n T C(vw ,A)

=

We say an element e=P.w+

[IC'(w,A)]

q

in n

in

in the composition series of

0

-r

The element

(q )

,is defined by

[IC'(w,A)] where

P

e

of

l W] (A)

Z>:[q,q

]

K(Chains (A» is the multi-

P v,w

=IC' (ww

0

,A)

has leading term

p·w if

l: P'v v>w v

has leading term

l·w. This follows from §1.6

property 1 .

2.7.

Proposition. (--,,3.6r). The intersection homology sheaf

IC'(w,A)

is indecomposable.

Proof.

Let

® ...

IC'(w,A) =

summands. Since

[IC'(w,A)]

also has leading term

],w

$

be a decomposition into indecomposable

has leading term Then this

R =i

]'w

for some

i

[R ]

=i satisfies all the axioms of §1.6

17

for

2.8.

c(w)! IC' (C(w,A»

Let

A and

and hence equals

A'

be two subsets of

IC'(w,A)

L:

.

Then we have a fiber square of

filtrations

(AnA')

TI(MA"/ (A')

/',AUA')

We have

Rrr(AnA',A')! Rrr(AnA' ,A)*

Definition.

Rrr(A' ,AUA')* Rrr(A,AUA')!

from

3.9) . The functor 1S

to

given by Rrr(AnA' ,A')! Rrr(AnA' ,A)*

2.9.

(see §1.5) .

Rrr(A' ,AUA') * Rrr(A,AUA')!

Properties of

A.

defines a functor

is the identity functor

Ii.

B.

Suppose

AnA' CAli c AUA'

r.

Suppose

A c A'

Chains (A')

from Chains (A) to Chains (A')

Then

,A') -F (A,A") ;

transforms cell sheaves in 1(A), to cell sheaves in Chains (A'). More precisely, if wE W .s;.(w,A)

Then

18

1 v = w(W(A) n W (A'»

where

and

n = 2(d(A) - d(A') - lew) +

. !(A,A')

does not in general transform indecomposable objects to indecomposable objects.

A.

Suppose

A

A' . Then

!(A,A')

transforms indecomposable

objects in Chains (A) to indecomposable objects in Chains (A') . In particular if

wE W1(A)

,then IC' (w,A')

!(A A') !(A,A')

never takes cell sheaves to cell sheaves unless

E. W(A')

Suppose

n WI(A)

A c A' . Let

of length

!(A,A')

=

A'

be the number of elements of

m n

d(A) - d(A') - n . Then

!(A' ,A") 0 !(A,A')

»(A o

n AI)

ej>(Am_ 1 n Am)

\/\/ \/\ ..

and hence the fiber product

*This conjecture has been proved by Deligne, Gabber, Beilinson and Bernstein [DBB] (the map f must be proper). This implies conjecture 2.10. See (GM4] for some general consequences of this result.

20 which maps to

C(w,A)

C

¢(A

m)

. This mapping denoted by

is a resolution of singularities of the Schubert variety

TT :

X({A. })

-+

C(w,A)

C(w,A) . It is

called the canonical resolution associated to the resolution data . In case all sets

AI'"

A have one element, it is the Demazure resolution [Dm]. m_ 1

We give two alternative characterizations of the sheaf

IC'(w,A)

which would follow from conjecture 2.10.

A.

( ...... 3.10 a) . IC' (w,A)

is the unique indecomposable

C (wo,A

K(Chains (A»

direct summand of ding term

whose class in

has lea-

I .w

6. R

o)

IC'(w,A)

is the unique indecomposable direct summand of

whose class in

TT!

K(Chains (A»

(In fact

I·w .

has leading term lI:(X({A.}».)

In many cases resolution data can be found for which is indecomposable. This happens when

X({A i})

R

. =a:(X({A.}» L

TT,

is a small resolution

(see [GM3]) . An example is given in §5.4.

2.12. Definition. S(¢(A»

The category Sheaves (A) is the full sub­category of

whose objects

S

I) Finiteness.

satisfy the conditions :

All stalks of

2) Constructibility all

c(w)* S

S

are finite dimensional over

is a constant sheaf on

wEWI(A)

Definition.

( . . . . . . 3.13). The total homology functor

: Chains (A)

­­+

Sheaves (A)

is defined by

C(w,A)

a:. for

21

2.13.

A.

Properties of the functor

H(A) .

takes exact triangles in Chains (A)

[I]


-

£(v,A)

->-

a

Then this extension is nontrivial if and only if

2.17.

wand

--+

Let

A be a subset of

L. The category

{a}

y(v,w)

Att(A)

of attaching schemes

is defined as follows : An object of

A is a pair of data w E WI (A)

complex vector space for each E(v)

->-

E(w)

v,w E WI (A)

for each

v

such that

V = to + t 1

A morphism for each

tn

+ ••• +

v E W'(A)

{E(v),e(v,w)}

+

E(w) =

E(v)

e(v,w)

E' (v)

e'(v,w)

*

= So

{E'(v),e' (v,w)}

+

sl

+ •.. +

is a map E(v)

sn

W

+

E'(v)

1

E' (w)

J (A) : Sheaves

and (y(w,v)t

The contravariant functor

gories between Sheaves (A)

E(w)

)

, e(v,w) = (y(w,v»t to the sheaf

is the vector space dual of

Proposition.

v

w . These data are

such that the obvious diagrams commute:

There is a contravariant functor

the data

->-

W , then

1

2.18.

is a

E(w)

is a linear map

and e(v,w)

subject to the commutativity restriction: whenever and

where

{E(w),e(v,w)}

and

J(A)

Att(A)oP.

which assigns S . (Here (S(w»

is the adjoint of

*

y(w,v).)

gives an equivalence of cate-

24

Proof.

We will construct the inverse functor

Given an attaching scheme

{E(w),e(v,w)}

J(A)-I

Att(A)

and an open set

Sheaves (A).

Uc

,

we de-

fine T(U)

{E(w) IC(w,A) nUt 0}

=

That is,

T(U)

comes equipped with maps

meets U)

such that the diagrams

E(w)

1

T(U)

(whenever

C(w,A)

E(v) e(v,w)

T(U)

E(w)

commute (whenever property. If

v

W)

U' c U then

so there is a unique map from

E(w)

and

T(U)

is the universal vector space with this

{E(w) !C(w,A) n U' t 0} c {E(w) !C(w,A) nUt 0}

T(U,U') : T(U)

--+

by the universal property of

the sheafification of the presheaf

P

T(U') commuting

T(U). Now

with the maps

J(A)-I {E(w),e(vw)}

whose value

is

is

T(U) * and

whose restriction map P(U' ,U)

2.19.

Let

A and

A'

G(A,A') Given an object

be subsets of Att(A)

{E(w), e(v,w)}

--+

L.

Att(A')

in Att(A)

G(A,A') {E(w),e(v,w)} = {E' (w),e'(v,w)}

Case 1 : A pew) = w W(A') n WI(A) coset

w WI(A').)

A' . Define (i.e. pew)

We will define a functor

, we describe

first in two special cases.

p : WI(A')

--+

WI(A)

by

is the element of minimal length in the

25

Then if

w E Wi (A'),

E' (w)

if

v

e' (v,w)

-+ w,

Case 2: pew)

w W(A)

n

E(p(w»)

{Che i.dent

Lty i f

otherwise

e(p(v),p(w»

A c A': Define similarly

p

= pew)

p(v)

WI (A) -;. WI (A' )

by

WI (A') •

Then if

wEWI(A'),

if

v -;.

E' (w)

e' (v,w)

W

E(v) $ -I vEp (w)

e(v',w') L;..I v'Ep (v) -I w'Ep (w) -+

General case

for arbitrary

G(A,A' )

2.20.

Proposition.

A,A'

,we define

C L

G(An AT ,A') G (A,An AT)

There is an evident definition of The functor

w'

G(A,A')

For any

G(A,A')

on morphisms of

satisfies formal properties similar to §2.9.

two subsets

A and

A'

diagram of categories and functors is commutative

1

1

> Sheaves

Sheaves (A)

H(A'j

H(A)

Att(A)

Att(A)

G(A,A')

>-

Att(A')

(A' )

of

L

,the following

26 In the special case of

G(A,L)

formula for the total cohomology of a sheaf

,this proposition reduces to a S

in Sheaves (A) .

(fJ

wE WI (A)

If

S

is a constant sheaf, this is the usual formula deduced by regarding

U C(w,A)

as a

C-W

decomposition of

with even dimensional cells.

27

3. Algebra Let

3. I.

n be an orbit of W

unique element of

n

'J!

--+

XE

n

an inclusion

'J!, X E n we have

X with

f('J!)

M(X)

C

= X('J!,X)

M(X+) f(X)

for a

X('J!,X) E

unique

Definition. for

be the

lying in the closure of the positive (resp. negative)

Weyl chamber. Let us fix for each Then for any

(resp. L)

. Let

'J!

--+

The characteristic elements

Y(X,'J!) E

X are defined by \ (X('J!,X»

being the anti-involution of §I.I . 3.2.

Lemma.

XI = (/)1

--+

If (/)2

XI

= 'J!I

--+

--+ ••• --+

'J!2

--+ ..• --+

(/)k = X2

and

'J!k

' then

The proof is obvious.

3.3.

Proposition.

Let

is the unique element in

Proof.

Let

'J!

--+

U(n ) -+

Then up to a constant multiple, of weight

Y('J!,X)

with the property

M(X)

and let ( , )

be

M(X)

M coincides with the kernel of the bilinear form

It is easy to see that the weight space by

X-'J!

M be the maximal proper submodule in

the Jantzen bilinear form on Then

X

f('J!) = X('J!,X) f(X) • Now for

XE

M'J!

(,)

(See [J],1.6).

is one dimensional and generated of the weight

.p-X it is clear

28 that

Xf(X)

lies in the kernel of Jantzen form if and only if

3.4.

Proposition.

16).

Suppose

l/J

X and

--+

1X(M(X)l/J)

M is an ext en-

sion

o

M(X)

--+

--+

M --+ M(l/J)

--+ 0

Y(X,l/J)(Ml/J) f {a}

Then this extension is non-trivial if and only if

U(Ec_) - free modules.

3.5.

We will consider

modules

a

VE

with the

following property (F) V is free as a

Definition.

a

Definition.

For a

3.6.

module.

U(Ec _)

is the full subcategory of

W-orbit n

*z:

in

a

we let

whose objects satisfy (F)

a en)

be

a

n

a(n)

.

Examples. X En, the Verma module

A)

If

6)

The Verma modules

either (

V

=

0

--+

or

V'

2.5)

If

--+

M(X)

--+

V'

a_en)

--+

a_en)

0

O. X En, the projective module

P(x) is not in

V

lies in

M(X) , X En, are characterized

by the property that for any exact sequence in

o

M(X)

a

is indecomposable in unless

X

P(X)

lies in

a_en)

is in the negative Weyl chamber •

a (n)

0

29 3.7.

Let

of elements in

Proposition. {O}

= Vo

C

Q in such a way that

C

Xi [> Xj

implies

XI' ... 'X n

i.:: j

V E 0_ (Q) . Then there exists a unique filtration

Let

VI

Q be given. Let us choose a numbering

•••

C

V n

sum of several copies of

= V such that V/V i_ 1 is isomorphic to a direct M(Xi) .

Proof. Define Vi as the of XI_P Xi-p . It is easy to see that these V , ... ,V

V generated by the weight spaces

V.

satisfy the conditions of the

1

proposition.

3.8. Corollary. gory

O_(Q)

K(O_(Q»

The Grothendieck group

is the free Abelian group generated by all

M(X)

of the cateX En.

for

So there is an isomorphism

where

3.9. Then

A = )K(Q)

which takes

The functors F(Q,Q')

F(Q,Q') . Let

to

Q and

is the projective functor

the following property respectively, and let W(Q)w+ • Then

M(wL)

F(Q,Q')

Let

and

w+

I.WW

o

Q'

be two orbits of

O(Q)

--+

O(Q')

W in

*

determined by

be maximal elements of

Q and

Q'

W be the (unique) minimal element in the set is the indecomposable projective functor such that

F(Q,Q') M(X+) = P(W) . (See [BG] for the definition and properties of projective functors, in particular :

Proposition. projective modules in

F(Q,Q')

O(Q')

takes projective modules in

O(Q)

to

30 3.10.

Properties of

F(n,n')

A. F(n,n') defines a functor from 6. If }f{(n) =}f{(n') B. Suppose

,then

F(n,n')

is an equivalence of categories

}f{(n) n}f{(n') c}f{(n") c}f{(n) U }f{(n')

then

F(n",n') F(n,n") = F(n,n').

r.

Suppose }f{(n) c}f{(n') • Then

to Verma modules in

O_(n')

elements in

n'

nand

F(n,n')

More precisely, if

respectively, and if

F(n,n') M(w X ) where

transforms Verma modules in

M(v

X_

and

° (n)

are minimal

wE Wi (}f{(n»

, then

) does not in general transform

v =

indecomposable objects to indecomposable objects.

A.

Suppose }f{ (n)

in

O_(n) to indecomposable objects in O_(n') . In particular, if

:::>

}f{(n') . Then

are minimal elements in F(n,n') P(w

F(n, n')

nand

n'

transforms indecomposable obj ects

respectively and

x_ and

wE W'Q+Kn», then

X )

F(n,n')

never takes Verma modules to Verma modules unless }f{(n) =}f{(n')

E.

nand

Let

W(A) c W(A') .

n

be two orbits with

Id

}f{. F(A,A')

Proof.

so that

Then F(n' ,n) F(D,n')

where

A =}f{(n) c A' =}f{(n')

[W(A')

is the identity functor in

°

W(A)] Id (n').

is an exact functor.

All of these properties (aside from the first one in

from [BG] , especially theorem 3.4.

A )

follow easily

31

We

3.lOa.

give a well-known characterization of the indecomposable projec-

tive modules

P(lji)

Proposition.

Suppose

let

in terms of the projective functors

lji E

A = )f{ (Q) , and let

weight in

Q to

lji

)f{(Qo), •.• ,)f{(Qm) weight in

Q

o

whose class in

3.11.

Let

w E WI (A)

Then

P(lji)

K(O (Q))

lji

W orbits

Qo, •.. ,Qm

C(w,A) . Let

X+

so that

be the dominant

is the unique indecomposable direct summand of

has leading term I'w . X E h*

and

. Define the vector space

V[X]

by

VX-p/U(g){ V1jJ-p} n VX-p - 1jJC>X 1jJ'h

It is easy to see that

V{X)

Lemma.

and

V E O_(Q)

of proposition 3.7 so that

3.12.

W orbit containing

be the element that takes the antidominant

is resolution data for

V[X]

dim V[Xi] = n

Q be the

. Let

• Choose a sequence of

VEO(Q)

Let

*

F(Q,Q') ( ....... 2. I IA).

may be different from zero only if

0 = V o VJV i_

1

C

VI c

.•.

C

XE Q •

V = V be the filtration n

= M(Xi) $ ... @ M(Xi)

(n

i

times) . Then

i

Proposition.

Let

characteristic element

V E O_(Q) Y(1jJ,x)

and

X, 1jJ E Q with

1jJ --+ X • Then the

defines a linear transformation

Y(1jJ,x) : V(1jJ) --+ V(X) The proposition follows easily from propositions 3.3 and 3.7.

3.13. ( ....... 2.12, 2.18) . Let and let

x_

Q be a

be the minimal element in

W orbit in Q.

*

with

A =)f{(Q)

32 Let

Att(A)

be the corresponding attaching category (See §2.17).

Proposition. wE WI (A)

The map V'IMrl" {E(w), e (v, w)}

and

e(v,w) = Y(v X_,w X_)

defines a functor

3.14.

a(Q) : V_(Q)

--+

for

where

E(w) = V[w

v,w E wI (A)

x_l for

with

v

--+

w

Att(A)

Properties of the functor

A.

a(Q)

is an exact functor.

6.

a(Q)

induces an isomorphism of Grothendieck groups.

B.

a(Q)

is not an equivalence of categories.

In particular it does not preserve indecomposability.

3.15.

Let

Q and

Q'

Proposition.

be two

W orbits with

A

>K(Q)

and

A' =>K(Q') .

The following diagram of categories and functors

is commutative :

0_ (Q)

F(Q,Q')

atOll

0_ (Q')

1

Att(A)

a(Q')

Att(A')

G(A,A') (where

G(Q,Q')

is defined in §2.19)

It is enough to prove this for two cases: A

At

and

A

A' .

In the first case it is easy. In the second case one has to use some properties of characteristic elements.

33

3.16. In [J] ,ch.5, Jantzen defined a filtration of a verma module M(X) by If-submodules

We will formulate conjectures about the relationship of this filtration with the Kazhdan - Lusztig polynomials of §2.6

Let in

Let

Definition.

be an A

W-orbit

and [KLI]

in

and

x_

be the minimal element

.

I

For

wI ' w E W (A), 2

the simple module

L(W

of the Verma module

3.17. Proposition.

I

X_)

M(w

(i)

2

let

m (wI ,w i 2)

in the quotient

be the multiplicity of

Mi/M + i 1

of Jantzen's filtration

X_) .

m i(w l,w 2)

(iii) m (w,w) o

does not depend on

=

I ; m (w,w) i

=

0

only on

for

i

>

0

The proof of the first part of this proposition is rather complicated and relies on the behavior functors

F

)

with

}f{

=

of Jantzen's filtration under projective }f{

. Parts (ii) and (iii) are easy.

3.18. Definition. (of the Jantzen polynomial) . Let of

and

be elements

WI(A) . Define

3.19. wo

wI

Let

P w

(q)

be the Kazhdan - Lusztig polynomial for

WI(A). Let

l,w2

be the unique maximal element in WI(A) (under the ordering

).

34

is a poly-

Conjecture. (improved Kazhdan - Lusztig conjecture). nomial in

q

,and

(*)

J w

Remark. For

q

=

I

w

I' 2 (*)

(q)

becomes usual Kazhdan - Lusztig conjecture,

see [K-LI] , (1.56)

3.20.

We cannot prove,of course, conjecture 3.19. The strongest evidence for

this conjecture is that it agrees with properties of Jantzen filtration from [J] , Satz5.3.

Namely, one can prove the following result. Let

Also for any

wEWI(A)

let

few)

be the set of all

WI

E WI (A)

with the properties (i)

WI

-) (ii) w




NI.

Suppose M = m +••• + 1R

(ii).

For each 1

for some positive

k, N = N n m.R is essential submodule of i

i

m.R and thus without loss of generality we can suppose that M is cyclic and M = mR.

Let E = {rEO R : mr e N}.

Then E is a right ideal of Rand

E (lIn, EI for some positive integer n. MIn

Thus N n MIn

=0

NI

and hence

= O. (ii)

=*

(iii).

Define

=' {T : T is a submodule of M and N n T = NI}. Then NI e:.. 1 and let N = m1R +••• + m n_1R.

because (i) holds.

By

induction on n we can assume 00

n

() Nl

n=l Let k be any positive integer.

Let

M= M/Nlk,

k

m = m + NI • n

n

= by (i).

= O.

(4 )

For every integer s

k

Then

-m I s n s=l n

= o

Thus

But k was arbitrarily chosen, so that

n 00

MIs

=

s=l by (4).

o

Thus (ii) holds.

In general, we pass to the ring R I• finitely generated right Rr-module.

Let M r

=

RR

1•

Then M is a

It can easily be checked that rR

r

has the intersection property and hence

This just means that (ii) holds.

This completes the proof.

A module M is an essential extension of a module N if N is an essential submodule of M. 2.5

Theorem

p.274 Theorem 2.60).

Let R be a right Noetherian

ring with Jacobson J such that R/J is an Artinian ring.

Then the following

210 statements are equivalent. (i)

J has the AR property.

n n=l MJn = 0 00

(ii)

(iii)

for every finitely generated right R-module M.

Every finitely generated essential extension of an Artinian right

R-module is Artinian. Proof.

(i)

(ii).

(ii)

(iii).

By Theorems 2.2 and 2.4, or directly. Let N be an essential submodule of the finitely

generated right R-module M and suppose N is Artinian. chain

N (\ MJ

N "i1J2

•••

must terminate and so there exists k such

that

n MJn 00

s Thus MJ

k

Then the descending

n=l

= o,

= 0 and hence 1·1 is Ar-ti.n i an ,

(iii)

(i).

Let N be an essential submodule of a finitely generated

right R-module M such that NJ = O. Artinian by (iii). integer t.

Then N is Artinian and hence M is

Thus M has finite length and MJ

t

= 0 for some positive

By Theorem 2.1 J has the AR property.

I.M. Musson (see(§, p.105») has shown that there exist right and left Noetherian domains with non-Artinian cyclic essential extensions of irreducible modules. 2.5

Lemma.

IJ

JI.

Let I and J be ideals of a ring R such that J

Suppose that J and I/J have the AR property.

I and

Then I has the

AR property. Proof.

Let M be a finitely generated right R-module and N an essential

submodule of M such that NI = O. some k

1.

Then NJ = O.

k By Theorem 2.1 MJ = 0 for

We prove by induction on k that some power of I annihilates M.

211

If k

=1

then this follows because I/J has the AR property.

and let V But MlsJ

MJl

MI 2s

i.e.

= {XE:- M : s

xJ

VIs

= o,

k-1

= a}.

=0

so that MIs

Then VIs

=0

for some s

s V and hence (Mls)I

Suppose k > 1 1 by induction.

= 0,

By Theorem 2.1 I has the AR property.

Let R be a ring and I an ideal of R.

Then I has the

if

for every submodule N of a Noetherian right R-module M there exists a positive integer k such that

equivalently, for every essential submodule N of a Noetherian right R-module M with NI

=0

there exists a positive integer k such that Ml

(see the proof of Theorem 2.1). if "AR" is replaced by "nAR".

k

=0

It is clear that Lemma 2.6 remains true The next result is due to

and

Gabriel (.;J). 2.7

Theorem.

Let I be an ideal of a ring Rand c a central element

of R such that c e 1.

Then I has the nAR property if and only if IIRc has

the nAR property. Proof.

The necessity is obvious.

the nAR property.

Let N be an essential submodule of a Noetherian right

R-module M such that NI (m

e

= O.

for some positive integer m. Mc

Then Nc

= O.

Define f: M + M by f(m)

= me

Then f is an endomorphism of the Noetherian module M and so

M).

ker f

i.e.

Conversely, suppose that liRe has

m

= O.

Thus J

n im

fm = 0

But N

= Rc has

kerf and hence im

= 0,

the nAR property (Theorem 2.1) and the

result follows by Lemma 2.6. An ideal I of a ring R is polycentral (or has a centralizing generators) provided there is a finite chain of ideals

of

212 such that for each 1

n, the ideal I./I. 1 is generated by a finite

j

J

collection of central elements of R/I. 1. J-

2.8

Corollary.

J-

The theorem gives at once:

Any polycentral ideal has the nAR property.

In

particular polycentral ideals of right Noetherian rings have the AR property. 2.9

Theorem.

For any polycentral ideal I of a ring R the following

are equivalent: (i) (ii) Proof.

I has the finite intersection property. I has the AR property. (ii)

(i).

(i) ==} (ii). radical of R. not.

See the remarks after Theorem 2.2. Suppose first that I is contained in the Jacobson

We prove that I is a Noetherian right R-module.

Suppose

Let E be a right ideal of R chosen maximal with respect to the

properties E

I and E is not finitely generated.

Then E

I.

Without

loss of generality, because I is pOlycentral, we can choose a central element c E. I with c I E.

Then E + cR is a finitely generated right ideal.

Let F = {rE. R : cr E.E}.

Then F is a right ideal of Rand E

F.

Thus

we can copy the proof of Theorem 1.8 (ii) to conclude that E is finitely generated, a contradiction.

Thus I

R

Let G be a right ideal of R.

is Noetherian.

Then G (\ I is a submodule of the

Noetherian right R-module I and hence Corollary 2.8 gives (G

n

I) (\

I. In

for some positive integer n, i.e.

It follows that I has the AR property.

(G

n I) I

213

In general, pass to the ring R (Lemma 2.3). I

Since IR

I

is poly-

central, has the finite intersection property and is contained in the Jacobson radical of R it follows that IR has the AR property. I I be a right ideal of R.

Let H

By passing to R we see that there exists a I

positive integer m such that Hn Let h e; H

n r".

r"

{rE.R : r(l­a)E. HI

for some a in I}.

Then h(1­b)E. HI for some b in I and thus h e; hb+HI = HI.

It follows that I has the AR property. Let R be the polynomial ring S[x] for some ring S.

CorollaEY.

2.10

Suppose that the ideal Rx has the finite intersection property.

Then S is

a right Noetherian ring. Proof. 3.

By Theorems 2.9 and 1.7.

Group rings Let J be a ring and G a multiplicative group.

collection of formal sums

Let JG denote the

l: a x

XE. G x

where a £J and a x x

t

0 for at most a finite collection of elements x in G.

Define l:axx = l:bxX x x

if and only if

l:a x + l:b x = l:(a + b )x, x x x x x x x (l:a x)(l:b x) = LC x x x x x x x where c

x

Then JG is a ring called a

= l: ab(xEG). yz=x y z ring.

ax = bx and

(XE

G),

214

JG + J

Define a mapping

x

Then

x

by x ) = I:a • x x

is an epimorphism with kernel

= {I:axx

I:a

x

x

x

=

O}

=

I: (x-l)JG, xe G

and g is called the augmentation ideal of JG. If J is a commutative ring then the map x to an anti-automorphism of JG.

x- 1 (x a G) of G extends

Thus JG is right Noetherian if and only if

it is left Noetherian and we say simply that JG is a Noetherian ring. A group G is polycyclic provided there exists a finite chain of subgroups (5 )

such that for each 1 is cyclic.

If

i

n, G is a normal subgroup of Gi and G i/G i_1 i-1

2S. and :J.. are group classes then an

G with a normal subgroup N such that N is an

group is a group and G/N a b.-group.

Polycyclic-by-finite groups are precisely the groups G such that there exists a chain (5) with each factor G./G. 1 cyclic or finite (1 1 1-

i

.

n).

The number of factors G./G. 1 which are infinite cyclic is an invariant 1 1of the group called the Hirsch number which we shall denote by h(G).

The

next result is due to Hall (11, Theorem 1). I'J

3.1

Theorem.

Let R be a ring which is generated by a subring S and

a polycyclic-by-finite group G such that x-ls x e: S for all s in S and x in G.

If S is right Noetherian then so is R.

Proof. (1

i

Let G have a series (5) with factors G./G. 1 cyclic or finite 1 1n).

The proof is by induction on n, the case n

=0

being clear.

215

Suppose n > 0 and let H Hand S.

Let T be the subring of R generated by

By induction on nTis a right Noetherian ring.

=m
1 and let Z denote the centre of N. AR property.

Also by the first part

If

Suppose

R has the

R has the AR property and clearly

because Z is the centre of N n R

R

=

R

s

R

=

R

R.

Thus by Lemma 2.6 g R has the AR property. 3.4 G

Let J be a ring which contains the rational field Q and

Lemma.

a polycyclic-by-finite group such that

for some a in Proof.

gJ.

Then

G

nco

gn = {r e n=l=

JG :

r(l-a) =

0

is finite-by-nilpotent.

For each positive integer n define

Then D is a normal subgroup of G and n (6)

for all n

1.

Here

[Dn .GJ denotes the subgroup generated by all

tators

with x in D , y in G. n

To see why (6) holds observe that

[x,yJ - 1 = x- 1y ­1(xy_yx)

=

- (y-l)(x-l)} g

n+l

commu-

217

provided

XE.

D yEO. G. n•

Consider the chain

Since h(G) is finite there exists a positive integer m such that Dm/Dm+1 is a torsion group. (x

-1 k )

e Dm+ l'

Let Xc D •

There exists k

m

k

Suppose x e. D l' m+

1 such that x\, D + or m 1

Then

-

1

to- g

m+1

and since x - le. gm we have k ( x-1 ) c g

Thus x - 1 E. g,m+l and

X

Dm+l'

m+l

•

It follows that D m

Let y

= Dm+1•

We can suppose

Then

D •

m

n co

gn n=l =

y - 1

and so (y-l)(l-a) = 0 for some a

E.

g'

It follows that y has finite order.

Thus D is torsion group and hence is finite. m

3.5

Theorem.

(ii) (iii) (iv) (v) Proof.

m

Let K be a field of ch1racteristic zero and G a poly-

cyclic-by-finite group. (i)

By (6) G/D is nilpotent.

Then the following statement are equivalent.

G is finite-by-nilpotent. g has the AR property. Every ideal of KG has the AR property.

£

is polycentral.

Every ideal of KG is polycentral. (iii)

(ii), (v)

Lemma 3.4 and Theorem 2.2. Corollaries 2.8 and 3.2.

(iv) are trivial. (v) ==}

(ii)

(L) follows by

(iii). (iv) ==9 (ii) follow by

Finally (i)

(v) by

218

Theorem 3.3 is still true if K is replaced by the ring Z of rational integers provided (i) is replaced by (i)' G is nilpotent. For a group G, and prime p a subgroup H is a p-group if every element has finite order coprime to p. normal subgroups 3.5

N

By 0 I(G) we shall mean the intersection of all

such that GIN has no non-trivial normal pi-subgroup.

Theorem.

Let K be a field of characteristic p

polycyclic-by-finite group.

(i) (ii) (iii) Proof.

p

> 0

and G a

Then the following statements are equivalent.

G/O I(G) is an extension of a nilpotent group by a finite p-group. p

g is polycentral. Every ideal of KG is polycentral. See (23).

'"

The argument of Lemma 3.4 shows that is J is a ring of characteristic p > 0 and G a finite group such that ee

n

n=i -

={re

JG

r( i-a)

=0

for some a (;; g,}

then G is an extension of a p'_group by a p-group.

A group G is called

p-nilpotent (p a prime) if every finite homomorphic image is an extension of a 3.5

by a p-group. Theorem.

Let K be a field of characteristic p > 0 and G a poly-

cyclic-by-finite group. (i) (ii)

(iii)

Then the following statements are equivalent.

G is p-nilpotent.

.! has

the AR property.

Every ideal of KG has the AR property.

219

Proof.

(L)

(H) follows by

(i)

(Hi) by (?j).

Note that for any polycyclic-by-finite group G, there exists a p-nilpotent normal subgroup N of finite index in G.

Of course. for fields

K and polycyclic-by-finite groups G. Theorem 3.1 gives that the AR and fAR properties coincide.

For other groups the situation is rather

different. Let K be a field of characteristic p

Theorem.

3. 6

Abelian group.

0 and G an

Then a necessary and sufficient condition for

to have

the AR property is that either (i)

p

=0

and G is an extension of a finitely generated group by

a torsion group, or (ii)

p

> 0

and G is an extension of a finitely generated group by a

pI_groUp. This theorem can be contrasted with Let

Theorem.

3.7

Abelian group.

K

be a field of characteristic p >,

0

and

Then a necessary and sufficient condition for

G

an

to have

the fAR property is that either (i)

p

=

(ii)

p

> 0

0, or

and for every finitely generated subgroup N of G the group

GIN has no p-elements of infinite p-height. The proofs of Theorems 3.6 and 3.7 can be found in (?7) and respectively. of p.

By a p-element we mean an element with finite order a power

An element y has infinite p-height if

n (D

yE n

where GP

n

= {x p : x e G}.

n=l

n

GP

Finally we note the following result.

220

3.8

Theorem.

Let K be a field and G any group.

Then g has the fAR

property if and only if g has the finite intersection property. =:

Proof. 4.

See (28). ""

Localization We have seen that if an ideal I of a ring R has the AR property then

R satisfies the right are condition with respect to T where T = {1-a : a a L}, Recall that C(I) is the set of elements c in R such that whenever rEo R, cr s, I or r c e I implies r

e,

1.

He are interested in conditions under whi.ch

R satisfies the right are condition with respect to e(I). 4.1

J

Let I be an ideal of a ring Rand J an ideal such that

Lemma.

I and J has the AR property.

Then R satisfies the right are condition

with respect to c(I) i f and only if R/Jn satisfies the right are condition with respect to C(I/Jn) for all positive integers n , Proof.

The necessity is clear.

Conversely, suppose that R/J

the right are condition with respect to C(I/Jn) for all n >, 1. c E C(I).

n

satisfies

Let

There exists k >, 1 such that (cR + cR)

n

Jk

(rR + cR)J.

But there exist r'£ R, c'E C(I) such that rc'- cr'E Jk and so rc'- cr' = ra + cb for some a,b E. J.

Then r(c'- a) = c(r'+ b)

and c/ - a E C( I).

It follows that R satisfies the right are condition with

respect to C(I). 1±.2

Lemma.

Let I be an ideal of a right Noetherian ring R and a a

221

central element of R such that a

I.

If R/Ra satisfies the right Ore

condition with respect to C(I/Ra) then R satisfies the right Ore condition with respect to C(I). Proof.

Let r e R, ce.C(l). rC

where J

= Ra.

R, CkE. C(l).

k

1EC(I)

I

-

cr

1

E.

such that

J

Let k be a positive integer and suppose rC

for some r

Then there exist rl .R, C

k

- cr

J

k

k

Suppose se:.R satisfies k

= sa • There exist s/E R, c/e. C(I) such that sc /-

for some t E. R.

CS'

=

ta

Then rc k c /- crk c l

k = (cs ' + t a Ia

=

and so

= where c k+1

= ckc /

.

k C(I), r k +1 = r k c / + s/a •

The result follows by

Corollary 2.8 and Lemma 4.1. 4.3

Let Q be a polycentral semiprime ideal of a right

Theorem.

Noetherian ring R.

Then R satisfies the right Ore condition with respect

to C(Q). Proof.

Let Co

= O.

c • • • • • c be a finite set of elements in Q such that 1 n

The result is proved by induction on n.

If n

=0

apply Goldie's Theorem

222 (9, Theorem 4.1).

induction R/Rc C(Q/Rc

1

Suppose n > 1.

Then c

1

is a central element and by

satisfies the right Ore condition with respect to

By Lemma 4.2 R satisfies the right Ore condition with respect

l).

to C(Q). 4.4

Let R be a right Noetherian ring such that every prime

Lemma.

ideal has the AR property. Proof.

Then every ideal of R has the AR property.

Suppose the result is false and let I be an ideal chosen

maximal with respect to not having the AR property.

Then I is not prime

and hence there exist ideals A, B, each property containing I, such that AB

I.

Let E be a right ideal of R.

By the choice of I both A and B

have the AR property.

Thus there exists n >, 1 such that

and there exists m

such that

>, 1

EA Let k

= max{m,n}.

n

m B

Then E

(EA)B = E(AB)

n

I

k

s

EI.

EI.

It follows that I has the AR property.

This contradiction proves the result. 4.5

Example.

Let K be a field and K[[x]] the ring of formal power

series in an indeterminate x.

Let R be the subring of the ring of 2 x 2

upper triangular matrices over K[[x]] consisting of all matrices of the form g(x)l

with f'(x L, g(x )

K[ [x]J •

Then R is right (but not left) Noetherian and

has only two prime ideals M > P where

g(X)J xf'(x )

: f ( x L, g(x)c;: K[[xJJ

}

223 and

Note that RIM

= K and

p2

= O.

Then every ideal of R has the AR property

but R does not satisfy the right Ore condition with respect to C(P). To check that H has the AR property let E be a right ideal of R. Without loss of generality we can suppose that E N the submodule of S

Let S = K[[xJJ and

S defined by

(g(x), h(x»E. N

i f and only if

Then N is an S-submodule of S $ S.

for some t

M.

1 (Theorem 1.1).

Lc roo

g(X

J

hf x )

e E.

But S is a Noetherian ring and hence

Then

E

n !'It +l

=

{x E R :

EM n.

Thus M has the AR property. Let

Then

CE

C(p) and P

rx E cR}.

Thus R does not satisfy the rightOre condition with respect to C(p). The above example is essentially due to A.W. Chatters

If I is

an ideal of a ring R such that I has the AR property then to check whether R satisfies the right Ore condition with respect to C(I) it can be supposed that I is nilpotent (Lemma 4.1). Question 3.

Let R be a (right and left) Noetherian ring and N the

224 maximal nilpotent ideal of R.

Does R satisfy the right are condition with

respect to C(N)? In Question 3 we can suppose without loss of generality that N2 because of the following result of Cozzens and Sandomierski 4.6

Theorem.

Theorem 2.4).

Let Q be a semiprime ideal of a right Noetherian ring R Q and R/I 2 satisfies the right are condition

and I an ideal such that I with respect to C(Q/I2). respect to C(Q/I 4.7

=a

n)

Corollary.

Then R/I n satisfies the right are condition with

for all positive integers n. Let Q be a semiprime ideal of a right Noetherian ring

R and I an ideal such that I

Q, I has the AR property and R/I 2 satisfies

the right are condition with respect to C(Q/I2).

Then R satisfies the

right are condition with respect to C(Q). Proof.

By Lemma 4.1 and Theorem 4.6. Let R be a right Noetherian ring and N the maximal nilpotent ideal of

R.

Note that N is a semiprime ideal of R Define L

= {r

R: cre.N 2

for some c in C(N)}

and K = {r Eo R : rc Then K is an ideal of R. c

1,C2,CE.C(N)

E.

N2

for some c in C(N)} •

for suppose r,r

1,r2

0

p

. (We note that most of the formulas

2 , but to avoid having to write the special

=

p = 2 , we delay the discussion of this case to the end of this

cases when paper). Let

A

=

k [[ X ]]

and set

A

l I X]]

k

'I

!(X

q)

where

'I

pe . Then

m induces a

ring homomorphism

A

A

'I

'I

III

A

'I

given by

+ J (X

III

1 +

o .

X+ X

G

so this map is well defined. In

III

A

'I

let

X+ X

X . Since char

G

X + 1, then

S

p , we get

o

X)q

G

k

x

S - 1 , and

It follows that :

A

Hence we conclude that

'I

k [Z!q Z]

is the group ring

Z!q Z

where

is written

multiplicatively with generator 0 , and the map m is just the Hop f algebra map for a group ring. Let

denote the (full) subcategory of

''I

such that

a

q

=

0 . Then

l-

consisting of those pairs (M,a)

is just the category of

A -modules of finite 'I

type. Just as before we construct the representation ring of by Re(k)

or better

(where

A

'I

and denote it

q = pe). He want to find a presentation of

Al mkv i s t r-Fo s sum ]

as a ring. This is solved essentially in

and we

review the results here. Much of the formalities is similar to the characteristic zero case.

For the next few pages we work only with k [[X]] !(X

r)

for

r

=

A -modules. Let 'I

1,2, ... '1 • The'se form a complete set of representatives

for the isomorphism classes of indeconposable group

R'l(k)

Fossum])

is free on the generators

A -oodules. Hence as an abelian q

[V I] , ... , [V] q

Green (c f c] Al.rnkv i s t-:

and others have determined the multiplication tables for these repre-

sentatives. It is seen that the virtual modules

271 [V p+ 1 J

genrate

Re(k)

as a

[V e-I p

Thus in

I - [V

+1

e R )

Re(k) (or

xo

the images in

e-I

I -I

let

and Capital letters

p

when

q = p

i

will denote variables with corresponding small letters

e R . For each

i

>0

set

[V i-I I , wi th w0

w.

i.

2

It follows from Greens formulas that

=

1

P

The results in

in

w.

i.

[Almkvist-Fossum I are given below.

Proposition 2.1

The map Z [X , ... ,X o

e-'

I] --;,. Re(k)

X.

given by

x.

has

kernel generated by the polynomials F.(X , ... ,X.) := (X. - 2 \1.) V (X.) i.

where

H.

defines the polynomials

(The polynomials

inductively.

V (X) have been defined in the previous section). p

We want to find

"better" generators and relations. Note that the polynomials

Vr(X) are even if

r

z

is odd. Hence if

(Z

is a

2

=

I), then

The isomorphism above implies that

I Re (k ) : Re-I(k) [X e_ 1 (and note that

w

e-I E

Re- I (k». Let

I(X

Y

e-]

2w) V (X ) e-I p e-I '

e-I-

= w

e-r l

X

e-I

e I as a variable over R - .

Then w 1 (Y 1 - 2) V (Y I ) • eep e-

But

w _ e 1

is a unit, so

272

Corollary 2.2.

There is a ring homomorphism

which is a surjection with

by the elements

(Y. - 2) V (Y.) l P l

Now set

W(X)

Lemma 2.3

V (X) - V p

p-

The elements

The elements

1 (X)

for

and let

u. l

w ( Yo ) w (Y) j

w. l

[V r ]

V (y ) V (Y

pop

p

i=O,I, ... ,e-1.

•••

) w( Yi-l

••• V (y

p

1)

r-

for

l'

;;;'

I .

1)

The main result of this section follows from Corollary 2.2. Theorem 2.4

(e

times).

z [Yo"'" Ye-1l / «Yo -2) Vp (Y0 ), ••. (Ye- j-2) VP (Ye- 1)'

Remark

These calculations hold for the multiplicative group law

any group law on

k , where char

k

p

and

Cj = p

e

then

m(X,Y). If F is induces a ring

F

homomorphism (denoted by F) k [[ X]] /

Hence the catorgory truct the ring

R;(k)

is also closed under the for

F

q (X )

product, so we may cons-

just as was done in the case

likely that these rings are independent of

F

=

m . It seems

F. I hope to return to this in a

later paper. See also Section 5 . (Aded in proof : These rings are independent of Later. However the A-operations depend of

F, a result that will appear

F, as is seen is Section 5).

273 §

3

Consequences

In this section we draw consequences of the main result in the last section. Theorem 3.1

Let

C be a commutative

Hom Alg

Proof

(Re(k) ,C) =

{(r 0'

This follows immediately since Hal'! Alg

Theorem 3.2

Suppose

C

HomAlg

is a reduced {R

e

Z-algebra.'l'hen ••• ,

R

r e-I)

EO

e C

1 = (R ) l8Ie

e

(r -2) V .(r.) e p 1

and

(A,C) x Hom Alg

(B,C)

Zip Z- algebra. Then

{c EO C : c 2 = 4 } e

(k) , C )

2 { ( co' ... , c e-l) : c i

Proof

We first need an expansion of the polynomials

Lemma 3.3.

V (X) r

V (X) about

The Taylor series expansion of

X

r

4 }

about 2

r-I

r )(X_2)n I t+ 2n+l

V (X)

r

Proof

Since

n=o

(l-XT+T 2 ) - 1

I

r=o 1 - XT + T

2

V

r r+ 1 (X) T

(l - T) 2 - (X - 2) T = (l - T) 2

we get

(l - (X - 2)

T ) (T-=-T) 2

Hence (l - T)

-2

(l

T

(X- 2)

n=o

I

r=o

-I

- (X - 2) - - ) (l-T/

Now expand the terms involving

I n=o

T

o }.

n

r"

- - 2 ( +1) (I-T) n

to get

(X- 2)n (2n+ r + I) Tn+r ) r

X

2.

is given by

274

and then change the order of summation, summing over

s=o

n=o

Another formula follows by setting (I

Lemma 3.4

- oT) (I -

V + ( 0+ 0

-I

r 1

0-

1

T).

Proof

Hence -I

p-I

(X - 2)2 (X+2)---Y-

-

From Lemma 3.4

(mod p).

we get ( 0 -0 0- 0

Now

X = 0 +0X

2

1

-I P

D

)

(0 -0

-I

-1 -

-I

)

. Therefore

4 = (0 +0

-

. Then

or+l _ 0 -(r+l)

) =

0- 0

V (X) p

1

X = 0 + 0-

,,-I Corollary 3.5

n+r to get

-1 2

)

- 4

( 0 -0

This congruence I first found in

-I 2 )

[Renaud J , where some of the results of this

section are suggested, in particular he proves theorem 3.2 for place of

in

N

Re(k).

Now theorem 3.2 follows from theorem 3.1

Just as in

[Renaud]

and Corollary 3.5.

it is possible to study the idempotents of

(i.e. invert 2). It should be mentioned that in have the

[V ] - [V p

I 2 (l+w

l)

and

p-

el =

1] • So in

I

2(1 -

RI (k) we get

Re(k)

I

R(2) I R(2)e l

RI(k)

RI(k)

= Z[X]/ (X-2) V (X)

z[l] we get two idempotents 2

wI)' Then letting z[

I

2]

I I R(2) e l x R(2) e 2 I I R(2/ e2 R(2)

I

=: R(2)

where, of course I R(2) e 2

P

I I R(2) / e J R(2)

we

275 (The subscript (Z) denotes the base change to

I

z[z]). So we want to consider the

rings :

z

Lemma 3.6

[X] I( (X-Z) V (X), 1 + V (X) - V I (X» p

p

The ideals

«X-Z)V(X),I+V(X)-V p

p

p-

I(X»

(X-Z)(Vo;i(X) + V _-__ L.-

z

Z

«X-Z) Vp(X), 1+ Vp_I(X) -Vp(X»

Proof

=

(X,Z) (Vp+ 1 (X) - Vp_I(X».

-z-

2

This is proved by using relations involving the elements

Corollary 3.7

-

[ZI

Z

,X]

I «X-Z) (Vp+I(X) +

-

Note that

r

n

Z

(X - Z , V 1 (X) p+ -Z-

Z

- Vp_I(X».

Z[;,X]

-Z-

Z

The ideal (X,Z)

Vr(X) ± V (X). r_ 1

(X»

-Z-

Proof

and

p-

I

[Z,X] is the whole ring. + Vp- I(X»

(X - Z ,p). Hence we get the cartesien

-Z-

square I

R(Z) e 1

I z! Z,X] I (Vp+ I (X) + V

?

t z[

Exam?les

Let p

l ]

?

Z

1

-Z-

E:=}

Zip Z

3. Then I

R(Z)

(Z

[X]/(X-2)(X+I»(Z) x 11:(2)'

Z

(X»

276

Let

p

7 . Then I

R(2) Modulo

=

(

Z[X]/(X-2)(X

3

+X

P , the elements look like

+

Vp+ I (X)

\,-1 (X)

- (X

V _ (X) p 1

-

-i-

-2-

2

- 2X-I)2 x

p-I -22)

(z[X]/(X

3

-x 2 -2X+I»2'

(nod p)

p-I

V + (X) p

2 Corollary 3.8

1

R

I

2-

(nod p) .

(X + 2)

2-

0+1

p-I

(Z/pZ [X] I (X - 2)""'2

B Z/pZ

) x

(Z/pZ

[X] I (X+2)2)

Z

We now apply this to

e, R using theorem 3.2 (A x B)

Corollary 3.9

Re(k)

B

C

=

(A

B

and the relation

C) x (B

B

B (Z/p7.) Z

0+ I

zip

z [ Xo,x l , ... 'X e

] I (X -2) o

2

p+I

- -2-

, ..• , (X -2) e-r

(This oeans (;) copies of the

Corollary 3.10

[Renaud]

.

Re(k)

rings of the form

=

z/pz

' ••• Ue-I ]

(truncated polynomial rings), where char k

B

z

5 , q

=

53. Then

U.

1.

x.1.

, (X +2) e_ r

p-I p-I -2-2, .. , (X ) e_I+2)

that follows).

is the ring product of r

Z/pz ([ U0' U I

Example

C).

Iu 0

o

r , ••• ,

± 2 •

Ue-I

e_ 1

)

2

e

local

277 §

4

Induction, Restriction and inclusion

In this section we assume A is a field of characteristic

p

>0

. As noted

nl

is the category of A [[xll -modules of finite length. Let eEN and set o e P 12. e = {M E ' 1..: x 11 = 0 } . Then /)L e is isomorphic to the category of e e+1 e p p denote the A [I X)] / (X ) -modules. Let f : A [I X II / (X )-----3'> A [[ X J] / (XI' Frobenius map given by : f

(0:

:=

(X»

0:

p (X ) .

This is an injection. It is clear that the diagram A

us n F

A

/(X

1

q)

f

>-A

pq)

I I X II / (X IF

I I x, YJ] / (Xq, y q )

[[ X,Y]] /(X

pq, pq y

)

is commutative. We get induced maps on the representation rings

and

e+1 ----?>- R F (A)

Ind :

defined by where

C

Res(M) = M considered as a module through A [[X ]l /(X

=

a B-module through

f

pq), and

A [[xl] /(X

B C

II

f

and

Ind (M)

=

C II

,

B

q)

and where

C

M is a C-module through

C

is considered to be

B

q) pq) A [[X l] / (X ----?>-A [[X II /(X

There is also a quotient map

1

the inclusion defining For

F

=

e U

e

-1t +)

and the inclusion

as a subring of

which defines

Inc: a;CA) ---7

e+ R 1. This function has been considered before.

m and the presentation given in

§

2 , the maps

Ind

and

exp lica ted. Proposition 4.1

The maps

Res

and

Ind

are given on elements by

Res

can be

278

vp (x 0)

r (x I ' ... ,x e) .

In particular Res Ind Proof

Consider

Res

p. Id.

=

first. It is clearly a ring homomorphism, so it is necessary

to determine its action on the generators with image

x

p

Suppose

The action of 1 ,x,

e

V i p -I

••

X

n-l

x through modulo

x

n

V n

A

f

It s n

is by

x

n

/(X ) , considered as an

p

0

x'e,

Then

Let

A

[I Xl)

nx

q

))/(X )

pq

/ (X

) -module

be the elements

eo,el,···,e n_ l

Apply this to the modules

e.

t.

of the algebra. Let xE A

xi

V

i

p +1

and

to get Res (V i p

+1

Res (V Res (x , ) = x _ i. i 1

Hence

V

)

P

i-I

p

V i-I -) P

)

Clearly

(p-I) V

EB

+1

V i P

(p-) )

ED

Res (V2)

while

i

so

2 VI

Res (x 0)

2 . (The res-

triction map can be made quite explicit : s = s

Let

+ sl p

0

for

Res (V,) I t fo

=

B ---7Ind V ---3'>0 r

x

B !!II A A

pq).

B to get:

A-modules. Tensor this with

o --3'>B But

A

k [[X]] !(X

pr

B • Hence

Ind V

r

vpr • Thus to establish

the formula it is enough to show that if R(k)

V r

then

in

R(k) • These

follow from a much more general formula. Lemma 4.2

[V

Suppose

[V

k

sp +r

Proof

For

r .;;; p

k

and

p

>s

;;. 1 • Then

k

sp -r

r

=

s

we have

=

[V k

[V k

P -1

P +1

x

]

k

bv definition.

Hence the formula is true for these values. Then the general result follows from Green's multiplication formulas. Suppose that [V

k

] - [V

sp +r

plr

Then

k

sp -r

by induction where

Vr!p = q(X

o'

Hence the result follows in general,

again by a mildly complication induction argument. Corollary 4. 3 Proof

Since

the theorem.

[V. k] p

Vk p

k ) . Ind (V1 ' t h i s follows by

k

applications of the result in

280

Table 4.4.

We list the polynomials giving the representations

characteristic 3 and Char k

VI

to

V 2S

VI

to

V 27

in

in characteristic 5 .

3

VI V 2

V

s

Yo

X2_1

/-1

X +X o I

2 yo+(Yo-Yo-I)Y 1

I+XoX I

1+(v

0

V 3 V 4

X

0

=

U

0

Xl

2-2)v "I

V 6

2 (Yo-I)Y I

2_1) V = X X +(X 7 o I I

2 2 (Y o-2)y l+y l-1

2_I)+X V = X (X o I I 8

Yo(YI-I)+(Yo-Yo-I)YI

V 9 V 10 VIi

2

- I) (X7-] )

2 yo-yo-l U

0

x02-x 0 -1

Y 1

YI = V6-VS+V 1

"0

2

0

2 ul=yl-yl-I

V - V 7 6

2

2

=(Yo-I)(yl-l)

V9-V8

=

uou]

2 2 2 = (Yo-I)(Yj+(y l-I)Y2 + Yo(Y1-Y -I)

2-X 2-1) (X -1)(X + (Xo-i)X + (X7- 1) +X o 0 I 1 oX2

Y = u IX2 2 ou

V I2 V I3

(Xo+X I)X 2 + XoXI+1

V I4

(I +XoX )X + X +X I 2 o I

V I5

(I+X jX

V I6

(XoXI+(X7-1»X2+XO

V I7

1)+X (X o(X7I)X 2+1

V I8 V I9

2)

I) (X7- 1)X2 + X2_1 1)+X (X o(X7I)X 2 2

[The remaining relations for Y o'Y 1'Y 2

are left for masochists l

281

V ZI

(X;-I) (XIX

+

Z

V ZZ

+ (l+XoXI)X

V Z3

+ (Xo+XI)X

V Z4

Z

Z

+ X + XI o

+ XoX + I I

+ X Z) + (X;-Xo)(Xi- l) + (Xo-i)XI+XoX Z

V

zs

V = Xo(Xi- l) + XI + X Z6

z +(Xo(Xi- l)

+

V Z7

Char k

=

5

4

Z

(Y +I)Y I o-3y o

4 Z 3 (Yo - 3y + I) (yj-Zy l) o

3 V = Yo-Zy 4 o V

s

=

4

Z

4 Z 4 Z (y 0 -3y 0 + I) (y I - 3y I + I) .

Y o-3Y o+1

U

V = V + 7 3 Va

= VZ +

Vg

= VI

U

o

Yo Y I

(i-I) 0

3 + (Yo-Zy

U

0

o)

=

U

o Y1

= Vg

Z + (y I- Z)

VIZ

= Va

+ Yo (i-2)

V

I3

= V7

Z Z + (Yo-I) (yl-Z)

I4

= V6

3 + (Yo-Zy

o)

- V

4 3 3vZ Yo - Yo - • 0 + Zy 0 + I

YI

VII

V

= V

054

Z (yl-Z)

4

Z

.. Yo - 4y 0 + Z

V21

3 V I 3 + Youo(YI- 3YI)

V

4 2 VI S+Yo(YI- 4YI+2)

VIS

2 3 V12 + (y 0 -1) u o(y l-3y l)

V 23

2 4 2 V17 + (yo-I)(y l- 4y l+2)

V

3 3 VII + (Yo-2yo) u o(y l-3y l)

V 25

3 4 2 V16 + (Yo - 2y o) (YI - 4YI+ 2) .

I6

V

§

4 V19 + (y I -

3 V + u 0 (y I - 3y I ) 14

V

I7

I9

5

n

A-structures on the representation rings

In this

section the

A-structures on the representations rings are studied. It is seen

that different formal group laws give different Let

+ 2)

A be a commutative ring. Let

HI (A)

A-struc tures.

denote the ring whose underlyinf, additive

abelian group is the set of formal power series of the form

1+

I

i=1

a. Ti""A[[T]], 1

with multiplication as the operation, and formal inverse as the negative. The operation is still denoted by juxtaposition. The multiplication in the ring is too difficult to write in general. However, denoting it by

n

i=1

(l+aiT)

E

(

n

n

(I+b.T)) J

j=1

i,j

the formula:

(I+a. b. T) J

1

will uniquely define the operation. So in fact (I + T)

f (T)

E

f (T)

,

and

I +T

is the identity element.

Let

A be a commutative ring. An abelian group homomorphism (A,+)

is called a

A-structure on

A. Since for each I +

a

A-structure on

;;:. (HI (A) ,.)

I

i=1

A.(a)T

a

E

A , the element

i

1

A is given by a family of functions

A.

1

A ------'7 A such that

283 A (a) 0

A. (0) A1 (a)

0 a

=

n

I

A (a+b) n The ring

A with

::>=0

A-structure

AT

for all

a E A

for all

i > 0

for all

aEA

(b) , for all A (a) A n-p p is called a

*-ring if

a,b E A and all

AT

n

is a ring homomor-

ph i sm,

Example

Let

AT

Z

--:»1-1 1 (I:)

be given by

Then

Suppose that

k

finite length

is a field and

F

is a formal group law on

and

r

(resp. : the symmetric power homomorphic image of

(resp. : of Sr(V)

of

S;(V»

is the exterior power

V as a vector soace. Each is a

V"r = V "k':- '''k V. The vector spaces are given a

-module structure by making these surjections F-equivariant.

F

a , then r

I

i=1 r

I

i=1 If

F

m, then

V,

SF(V) , as follows:

The underlying vector space of

If

V be a

k [[xl] -module. Define the symmetric and exterior powers of

denoted by

Example

k . Let

vj ... (Xvi)···v r

k [[ X]]

284 Proposition 5.1

Let

k

be a field and

F

a formal croup law on

k

(assumed

commutative as always). Then the maps AF

T

defined by :=

are

A-structures on

a A-structure on Proof

and

n=o

RF(k).

R;(k)

char k

for each

=

p

>0

, then each of these restricts to

e? 0

It is sufficient to show that r

!I. r

F

(V

N

101)

\B

p=o

and similarly for the symmetric powers. The isomorphism is described explicitly for

r

=

2 . The general case follows by the associativity of the group law

The map

We want to show that

where

zl = vI + wI '

(using the fact that

z2 =

V

F(X,Y)

z

+

z .

W

So compare the two formulas

F(Y,X).). First,

Now

(Xz I!I. zZ) = XW !l.w + (XvI N W2 - v 2 N Xw I) + Xv I !I.v Z • I Z i j 9, (Zj !I.XZ ) and 9, (X Z I !I. X Z 2) • It is seen that the terms in Z

(This is just to say that the decompositions

Similarly for !I. 2 (w) are:

F •

285 r

A r (V

(9

W)

"'"

(9

p=o are q

x

F-equivariant). Thus the maps given are V

=0

, then

q X

ArF(V)

=0

A-structures. It is clear that if

and similarly for the symmetric powers. Hence the

coefficients of the power series defining the arguments are in In the case they are

char k

R;(k). This shows that the 0 , these

=

are each in

a;(k)

if

A-structures restrict.

A-structures have been examined in

§ I.

In fact

A-ring structures; they satisfy the identifies

(V)

So the

and

AT

and

0T

0

( V)

I.

structures are essentially the same (one is the inverse

of the other). The situation in case

char k

=p

> 0

there is an extensive study of the

is quite different. In

[Almkvist-Fossurn] on

A-structures

RI (k). Further m

papers by Akmkvist (see references) have delved more deeply into this problem for

RI(k). In particular they are not m

Problems 5.2 : Let a)

chark

= p>O and let F be a formal group law on k

For each indecomposable

V determine the decompositions :

A;CV) S;(V) b)

For each indecomposable ,\

c)

Determine

d)

Determine when

holds.

(V)

A-ring structures.

(9

5>0

=

(9

s>o

£(F ,r, s) V s m(F, r , s) V s

V determine rational function expression

.Eill. q(T)

where

q (T)

E

[T 1

•

286

The complexity of these problems is shown by the examples in the next section. In the remainder of this section, some general results, which may help in giving a partial solution to some of these problems, are given for the multiplicative formal group law

m, the problem that

I

first attacked.

Several preliminary results are recalled. (In what follows the indexing by m is usually dropped. It is almost always assumed that the group ted by an element Koszul complexes

T , is operating. The field

Let

k

Z/Q71

has characteristic

be a finite dimensional vector space over

V

p

F

or

, genera-

> 0).

k. Consider

the complexes

and

with maps given by the formulas j

e. (vI !l ... !l J

V.

J

18l

s)

\' (-1) s-I L s=1

( ' VIll

•••

,A, r. v 1l s

•••

'v) tv . J

18l

v s

s

and d.(e J

18l

vl .•• v.) J

(Note that these maps are

F-equivariant for any

F

).

These complexes are exact (they are the graded parts of the Koszul complex giving the free resolution of (p,r)

= I,

k

as an

S '(V) -module) and they are split exact in case

since: r , Id

for each Proposition 5.2

The power series

287

Proof

r T

The coefficient of r

I

in this power series is

(-I)j (Ar-j(V)

j=o

and this is zero if

sj(V» k

(p,r)

I

.

Another useful result concerns the decompositions of the induced modules. Proposition 5.3

the ideal generated by the elements

[V

Corollary 5.4.

r

If

pr]

A q(V

pr

provided

are in the ideal generated by [ V ] P

Ind (V

k

and

and the fact that

is

in

pe •

)

(q,p)

1 •

V pr

Aq - 1 (Vpr )

Res \>I) k

Aq(V

pr

)

are direct summands of

V

Dr

sq-I (V ) pr

respectively.

Another useful result relates

Res

Proposition 5.5.

the following hold

Proof

0

This follows from the relation (Ind V) a t·)

and

pr

for

p

I , then the elements

and

Proof

p

Prop. 4. I.

This follows directly from

Proof

[V ] = [V (X )]

The principal ideal eenerated by

For any

V

with the tensor functors.

Res (A q (V»

A q (Res V)

Res (S q (V) )

sq

(Res V) .

The underlying vector spaces are the same for the modules on each side of

the equalities. And the action of Corollary 5.6

The diagrams

X is eiven by

xP

in both cases.

288 AT Re+! (k)

Res

)

I (Re+ 1(k) )

)

1

°T

1

AT

-------»

Re(k)

°T

>

(Res)

Tl (Re(k»

1

commute. As an example, consider

AT(V

9

1)

in chark =3 • Then write A = L a.(T) [V. T(V4) j=1 J J

I

We conclude that :

+ +

Here the

aj(T)

have non-negative integer coefficients. Now

Res (V 4)

and (I

Since no coefficient in Hence

a

4(T)

+ 2a

S(T)

T(I+T)2

= T(I+T)2

2

2

+ [V2 I T+T ) (I +T) .

is as big as • Thus

a

4(0)

3, we get =

as(O)

=0

a

6

=

0 • Likewise

"s =0.

So our two equations

are :

T(I +T) 2 •

289 I , we get a (0) l

=

o .

and

I

Calculation with the second equation yields

where Then using these in the first equation yields

2n + m

n + Ztn

2 • Hence

n

2 •

= 0,

m

So :

as (T) 2 4 I + T + T , which implies that

Thus

=

This implies that

I + [V 4 J T +

/l2(V)=V 4 I

The similar computation for

The coefficient of

3 T

is

V 6

7V 2

[ VI

6)

Vs 1 T

2

+

a

[ V4

ffi

yields;

ffi

2 V ; This implies that 3

where There are two possible solutions to this set of equations

2(T)

1

=

0

4 3 T + T

Hence

=

I •

290 A quick calculation shows 1\3 (V6) m 1\3 (V 6) a

-

V 6

ill

V 2

V 6

(jJ

2V 7

Ell

This shows that the 0-structures induced on

V 7

Re(k)

are indeed different for these

two different formal group laws. And it raises more questions than it answers. We hope to return to these problems later. Finally we derive a result that should have great possibilities in determining decompositions. Before doing so we need some additional notation. We suppose q = pe V pq

and that

is a multiplicative generator for the group

T

have a basis

on which the generator

Z/pq Z . Let

acts as a cycle

T

permutation ; so that

T(X ) = Xi + l

for

i

and

T

We suppose also that pq i I e.1 = x - -

'

so

(X

V

pq

) = Xl


S • (V

pq-r-

=k[e,e1,···,e o pq- I ] '

1)'

291

and each has kernel generated by the linear forms (eo, ..• ,e

Let

Y. := X. X. i.

So

Y.

p-l

n

for

Tqr(X.)

1,2, .•. , q

TY.

Y

i

Sp(V

Hence the submodule of k [y1, ... ,Y ] q

Let

Proposition 5.7

pq

)

Proof

S' (V

given by

q

(0) .

are alGebraically independent and form a regular se-

pq-r-l

r+l

for

)

,,;;;

(p-l)q.

"'/pq Z-equivariant homomorphisms Zi variab les ,

q

X.

v

1+r ,,;;; (p-l) q , then

If

a)

Consider the

a

is isomorphic to

denote the subalgebra spanned by these elements.

The elements

quence on

and

q

spanned by the

n b)

Call this ideal

Note that

r,

r=o

i

r).

----? Z.

for

i

I

(0)

(p-I)q ,

and

o ,,;;; r

it is sufficient to show that

• The image of

Y.

under this map is

site

is the Frobenius which is an injection. So it is sufficient to show that Ker

a

q

,,;;; p-l

l(p_l)q .

i.

I

(p-l)q

n

. Hence the compo-

292 The one basis relation is r

I

j=o since It

for

s

>

0 •

follows that

x and hence

Ker ct

q

c

-x

r+tq

E

r

I

(p-l)q

I(p_l)q • Since both ideals are prime ideals of the same

height (by a Krull dimension argument), they must be equal. This proves (a) • Statement (b) follows immediately, since the length of :

is finite (it is isomorphic to is generated by

pq

k [ZI, .•. ,Zq] /(Zf, •••

and since this ideal

homogeneous elements, they form a regular

k [Xl" •• ,X pq ]

-sequence. But any regular sequence of homogeneous elements remains regular under is a regular sequence on

k [Xl'" .,X pq ].

There are two consequences to draw from this result. Coro llary 5.8

The algebra

k [YI, ••• ,Y

is free (but of infinite rank).

Proof

q]

This follows immediately from

Corollary 5.9

Suppose

n

regarded as a graded module over

k [Xl" •• ,Xpq ]

>q

Then

summand isomorphic to

[Bourbaki Alg. de Lie, .•• ]. Spr (V) n

contains a

'E./ pq

Z

direct

Or in other words : Sr(V) q

Ell

as a direct sum decomposition. Proof shown in

The module

Spr(V) n

[Almkvist-Fossum]

is the that

pr

th

Sr(V) q

homogeneous component of is a direct summand of

S· (V ). I t was n

Spr(V

pq

), where

293 Sr(V) q

was identified as the r

th

is graded in two ways; with deg Y i deg Y = P i

=

I

when considered as an

when considered as a subring of

o --7 I is exact as

homogeneous component of

s

n

S'(V

pq

».

Spr (V ) --7 Spr (V ) pq pq

k [Y1"."Y q ]

(which

Z/qZ-algebra and

The sequence: Spr (V ) --7 0 n S· (V ) pq

Z/pq Z-modules (where Is is the kernel of

S' (V n

».

Since

o the component

as a direct summand.

Of course the modules bed in

Sr(V) q

[Almkvist-Fossum]

have decompositions that have been explicity descri-

• I fuel that these techniques should be extremely useful,

but have not been able to exploit them fully. §

7

Characteristic

p

=

2

In this section, some of the calculations from Sections 2 and 3 that do not apply in case

char (k)

and the elements

=

2

are made. In any case the multiplications of Green apply

[V.

still generate the algebras. So we can state

the next result. Proposition 7.1

(where the

w.i.

Suppose

char k = 2 • Then

are the preimages of

)

.

The elements w. =[V .]-[V . ] are 2-unipotents so the generators Can be changed to the next result, also just as before.

and we get the

294

Proposition 7.2

In case

char (k)

2 , then

=

The next corollary also follows immediately. Corollary 7.3

Let

C be a commutative ring.

Then

Zc • }

=

If

1/2 E C , then this set is exactly the set of idempotents in

in

C, then this is just the 2-nilpotents in

Corollary 7.4

In particular if

Suppose

C

C

is a

2 = 0

Then

in

C , then

Re(k)

l8!

Z

local ring. (Compare Since

[Renaud] and his references)

(71.[X ]/«X-2)X»

Z[ l] 2

l8!

Z

Re(k)

l8!

Z

Z[ l] 2

71.[

l] 2

( 71.[

In general the square RI(k)

71. X 1--+0

x

:j:

2

\V Z

is cartesian.

\V -----'00> Z/2Z

] )

Z[ I] 2

x 2

e

2

C

71./2Z - algebra.

is a local ring "lith

C . If

,

it follows that

C

is a

0

295 References

AU1KVIST G. and

R. FOSSUI1

This seminaire. Lecture

Notes in l1athematics

nO 641, Springer. HAZEWINKEL 11.

groups and Applications London

RENAUD

J.e.

New York, San Francisco, Londm

Academic Press 1978.

The characters and structure of a class of modular representations algebras of cyclic p­groups. J. Austral. l1ath. Soc. (Ser A) 26, 410 ­ 418 (1978).

ROBERT FOSSUI1 University of Illinois Department of l1athematics 1409 W. Green St. URBANA, ILLINOIS 61801, USA l1ai 1981.

CLASSES CARACTERISTIQUES POUR LES REPRESENTATIONS DE

GROUPES DISCRETS

par Guido Mislin

Introduction Soient

K

un corps de nornbres et

C

K-representation d'un groupe fini groupes

GL. (K) , j

Les classes de Chern

c

m

(p )

E

vectoriel complexe plat sur

une

BG

sont definies comrne

a

P , qui est une fibre

(l'espace classifiant du

G). Nous designons par

borne pour l'ordre des

c

m

(P)

,

P

EK(m)

la meilleure

parcourant toutes les

representations de tous les groupes finis

K-

G. Ces bornes

permettent d'obtenir l'ordre precis des classes de

Chern universelles

=

GL(K)

G (GL(K) est la reunion des

2m H

classes de Chern du fibre associe

EK(m)

+

1 , pour les inclusions habituelles).

]

groupe discret

P : G

c

m

E

H2m

;Z)

dans le cas ou

est l'anneau des entiers dans un corps de nombres

qui n'est pas formellement reel. Les valeurs explicites donnees ici pour les nombres

EK(m)

ont ete obtenues en collaboration

avec B. Eckmann; on trouve les details concernant les sections 2 et 3 dans

.

Dans la section 4 nous considerons la con-

jecture suivante concernant la fonction zeta nornbres

K.

sK

du corps de

297

Conjecture:

EK(m)

((*): Voir la note

sK(l-m) E

a

pour tout entier

m

>

1 .

la fin de cet expose.)

Rappels sur les fibres plats

§ 1

Soient

un

X

CW-complexe

X. Alors

complexe sur

connexe et

un fibre vectoriel

est classifie par une classe d'homo-

topie

ou

j

note la dimension des fibres. Le fibre

s'il est associe

a

un fibre principal

a

est dit plat

groupe structural dis-

cret; cela peut s'exprimer par une factorisation de

(a une

homotopie pres) de la maniere suivante:

B GL ,

X

.>.

]

/

( p

G

-+

o ]n

c (p )

GL (K)

m

a

pour tout

et tout groupe fini

G}

alors

Il est facile

a

voir que la borne superieure

EK(m)

est la

meme que celle determinee par Grothendieck [6J dans un cadre plus general. De meme, la borne inferieure correspond

a

une

borne obtenue par Soule [10]. Ces deux bornes different au plus d'un facteur 2. Pour

nous avons le resultat suivant.

301

Theoreme 1: Soit

KC

un corps de nombres. La meilleure borne

pour l'ordre des classes de Chern sentations

p: G

GL(K)

+

c

K-repre-

des

(p)

est donne par si

EK(m)

est impair ou si

m

K

n'est pas formellement reel

EK(m)

1 ---

2"

EK(m)

m

si

EK(m)

K

est pair et

formellement reel.

La demonstration derive d'une analyse des des

rn

K-representations

2-groupes, en particulier des groupes cycliques, diedraux,

semi-diedraux et quaternioniens.

§ 3

a

Applications

Soient comme avant K-representation,

a

la cohornologie des groupes arithmetiques

G K

un groupe fini et

c

p'

tout

1

p $

:

G

+

GLj+l

e(K)

,

I

GL.

J

(K)

une

une representation c

. Comme

m

(p)

=

c

m

(p')

pour

est un diviseur de Icm(e)

l'ordre de la classe universelle Icm(e)

a

est equivalente

(K)

on en deduit que

D'autre part

+

est un anneau de Dedekind)

=

qui se factorise par m

G

:

un corps de nombres. 11 est facile

voir (en utilisant que

que

p

divise

c

m

EK(m)

(e)

E:

H

2m

d'apres

I ,

(GL (19"); :l)

[6] .

En utilisant

le Theoreme 1 on a:

Theoreme 2: L'ordre de

c

rn

(e)

E:

H

2m

(GL(e);:J)

, m > 0 , est s o i.t;

302

soit le

EK(m)

; il est

corps des fractions

Soit

p

si

K

de

m

est impair ou si

n'est pas formellement reel.

un nombre premier. 11 est clair que la partie

EK(m)p

est un multiple de

p-primaire

. Le lemme suivant donne une

condition suffisante pour l'egalite de ces deux nombres. On designe par

Lemme 3:

CD

Si

p

la reunion de tous les corps cyclotomiques

00

K A

CD

p

, alors la partie

00

p-primaire

EK(m)p

est donnee par

ECZ!(m)p

cas p = 2 ou si K

m

EK(m)2 =

si

est impair; et

K

est formellement reel

EK(m)2

si

m

est pair et

n'est pas formellement reel.

Ce lemme se deduit du fait que

Kp a

K A

entraine

p

Gal(K a/K)

p

p

pour tout

a > 0

On observe que si le nombre premier de

K

la condition

criminant de

p

a

K

n

p

a

=

p

ne divise le discriminant

est satisfait, car le dis-

est, au signe pres, une puissance de

p.

303

Considerons comme exemple Ie cas d'un corps quadratique L'intersection K (\

P

K

= K ,

00

K

n

p

est alors

00

K

ou

K .

Lorsque

est un sous-corps quadratique de

Rappelons que les sous-corps quadratiques de

p

00

p

00

sont donnes

par

cas

p

impair:

(corps de (_1)p-1 /2 p)

discriminant

cas

p

!D(/-l)

2

,

!D(/-2)

,

discriminants

Corollaire 4: Q(/-l)

,

Soi t

Q(M)

,

(corps de

!D(I2)

-4, -8 et 8

respectivement).

un corps quadra tique different de

K

et

Q(/2)

Q(

!(_1)p- 1/ 2 p)

,

p

etant un

nombre premier. Alors

f l2E!D(m)

si

m

est impair ou

, si

m

est pair et

On peut aussi sans peine calculer

EK(m)

K K

reel

imaginaire

pour les corps quadra-

tiques exclus dans ce corollaire. Le theoreme suivant, qui est plus precis que le r e s u Lt.a t; de Ch. Thomas dans consequence du calcul de ([5J) m de

EK(m)

m

Bm1m , ou les

Bm 1/30

, pour

m

>

0

Eq,) (m) = 2 , pour pair (le denominateur

sont les nombres de Bernoulli: etc.).

est une

pour tout corps quadratique

et des formules etablies dans [4]:

impair, et = den(B 1m)

[i i},

K

304 Theoreme 5: C K

Soient

un corps quadratique imaginaire

l'anneau des entiers de

universelles si

(a)

KC

m

c

m

E H

(& )

2m

K. L'ordre des classes de Chern

(GL (6 ) ;

est impair,

c

m

et

(6')

est alors cornrne suit: est d'ordre 2

avec les ex-

ceptions suivantes: (a) 1 :

si tout

2m m p si

(b)

m est

, p

o (p-l) ,

-

pour

pair

to, 24

c

l'ordre de

(8)

est

p-primaire de

c

l'ordre de pour

m

m

est

(&)

m = 2 , 240 pour

-

3 (4)

2pm

p

et (au

.

m

)

=

den(B 12m)

m = 4

m

etc.).

Relations avec la fonction

Soient Si

est d'ordre 4

(V)

un nornbre premier

designe la partie

(c.a.d.

§ 4

+

2n l

n K

si

(a) 2

, c

K

K

KC

un corps de nornbres et

sK

sa fonction zeta

n'est pas totalement reel, on sait que

tout entier

n > 1

SK(I-n) = 0

pour

tier pair

n > 0

que on a pour

K =

Dans le cas n

au

impair> 1 , et

K

sK(I-n) = 0

pour

est totalement reel,

sK(I-n)

E

pour un en-

("Theoreme de Siegel"). Par un theoreme class iet

m

>

0

pair

m (-1)

2

(B

m

1m)

11 suit alors de notre description de

que

305

pour tout entier

Theoreme 6: Soit

Demonstration:

K C

11 suffit de considerer le cas ou

on trouve 2TIpa(p)

a(p)

K(2)

=

E

K(2) ,; 2}

max I o I (K a p

,; 2}

a

p

E = rrpa(p) K(2) est

r = (K: CD)

E

max {p a I (K a : K)

EK (2 ) p

K)

. Serre a montre dans

toujours un entier

• On en deduit que

E

K

est totale-

•

=

0 mod 2 r

§ 3J que

, ou

E

K(2)

et

plus claire si on rappelle le resultat de Serre K

.

et, par consequence

La relation entre le denominateur de

corps

1

>

un corps de nombres. Alors

ment reel. Dans ce cas

Si on pose

m

E K(2)

devient

(pour un

de nombres totalement reel):

Pour les autres valeurs de

la situation reste assez

mysterieuse. Rappelons tout d'abord un autre resultat de Serre (nous designons par

a(p)

les entiers

localises en

(p)

):

306

Lemma 7: m

>

0

Soit

K

est pair et si

rm 2

2-r

Corollaire 8: entier

un corps de nombres totalement reel. Si

m

>

1

r =

I:; K (l-m)

Soit

, on a

(2)

E:

KC

un corps de nombres. Alors, pour tout

,

Comme on l'a vu, on peut supposer que totalement reel et

m

r = (K:

on a

=

2Ym

2

est au

[5J)

(cf.

Si on pose

E (m) 2 K

pair. Dans ce cas

K

r

2

2

2-r

2 . Par consequence,

K (m) 2 est un multiple entier de la partie 2-primaire de 2-r rm 2 , d'ou Ie resultat, d'apres Ie Lemme 7 . E

Dans Ie cas au l'extension

K

de

est abelienne, on peut uti-

liser Ie theoreme suivant de Coates et Lichtenbaum [2J.

Theoreme 9:

Soit

K

un corps de nombres totalement reel qui

est une extension abelienne de p

Si

m

designe un nombre premier impair, alors

w(p) (K) m

I:;K(l-m)

E:

>

0

est pair et si

307

Les nombres Soit

K

(K)

sont definis comme suit ([2)):

un corps de nombres arbitraire et K . Soit

algebrique de

w(p) m

d'ordre une pUissance de On considere

w(p) (J

ments de

comme

m

E

p

K (p

K

une cloture

le group des racines de l'unite

un nombre premier arbitraire).

Gal (K/K)-module par l'action

Gal (K/K)

et

X

E

. Le nombre des ele-

w(p) fixes par cette action est note m

w (p)

m

(K)

Un calcule simple montre que II w (p) (K) m p

pour tout corps de nombre EK(m)

[5J

dans

avec celie de

En particulier, si w(p) (K) = E. (m) m

.K

K

p

p

(on compare la description de w (p) m

(K)

dans [10J).

est un nombre premier impair, on a

En utilisant le Theoreme 9 et le Corollaire 8

on en deduit:

Theorerne 10:

Si

K C

pour tout entier

m >

(jJ

,

est une extension abelienne finie de

alors

a .

308

Bibliographie

1

Charney, R.M.: Homology stability of

GL of a Deden kind domain. Bull. Amer. Math. Soc. 1(2), 428­431, 1979

2

Coates, J. et Lichtenbaum, S.: On

£­adic zeta functions.

Annals of Math. 98, 498­550 (1973) 3

Deligne, P. et Sullivan, D.: Fibres vectoriels complexes

a

groupe structural discret. C.R. Acad. Sc.

Paris, t. 281, Serie A, 4

1081­1083 (1975)

Eckmann, B. et Mislin, G.: Rational representations of finite groups and their Euler class. Math. Ann. 245, 45­54

5

(1979)

Eckmann, B. et Mislin, G.: Chern classes of representa­ tions of finite groups

6

(a paraitre)

Grothendieck, A.: Classes de Chern et representations lineaires des groupes discrets. Dans: Dix exposes sur la cohomologie des schemas. Amsterdam: North­Holland 1968

7

Milnor, J.W. et Stasheff, J.D.: Characteristic classes. Annals of Math. Studies 76

8

Serre, J.P.: Cohomologie des groupes discrets. Annals of Math. Studies 70, 77­169

9

(1974)

(1971)

Serre, J.P.: Congruence formes modulaires (d'apres H.P.F. SWinnerton­Dyer). Seminaire Bourbaki 1971/1972; Springer Lecture Notes in Math. Vol. 317, 319­338

309

10

Soule,

c.:

Classes de torsion dans la cohomologie des

groupes arithmetiques. C.R. Acad. Sc. Paris, t. 284, Serie A, 1009-1011 (1977) 11

Thomas, Ch.: Characteristic classes of respresentations over imaginary quadratic fields. Springer Lecture Notes in Math. Vol. 788, 471-481

Eidg. Techn. Hochschule Mathematikdepartement CH - 8092 ZUrich SUISSE

(*)

Le Theoreme 9 K

a

ete demontre par P. Cassou-Nogues pour

un corps de nombres totalement reel arbitraire

(voir: "Valeurs aux entiers negatifs des fonctions et fonctions

p-adiques"; Inventiones math. 51,

29-59 (1979), Theoreme 17). Il suit alors de notre Corollaire 8 que entier

m > 1 .

EK(m)

est un entier pour tout

AUTOMORPHISMES DE SCHEMAS ET DE GROUPES DE TYPE FINI par Hyman Bass *

1. INTRODUCTION.

Commen cons avec un p rob I eme auquel on peut appliquer les methodes de cr i.t.e s ici.

I

Soit

une surface compacte orientable de genre M = g

'II

0

(Home om

des classes d ' isotopie des homeomo rpb i srre s :

g. 1e groupe :

(")) l

I

-+

I

j oue un role important dans la

topologie des varietes de dimension 3 , aussi bien que dans la theorie des surfaces de Riemann (cf , [B]). Son etude est d'une surprenante d i f f i cul te . On sait que

M

g

est de presentation finie, et on a des renseignements assez precis sur ses sousgroupes finis. Recemment

E. Grossman [E.G] a montre que

M

g

est residuellement

fini, autrement dit que l'intersection de ses sous-groupes d'indice fini est triviale. Sa demonstration repose sur des arguments combinatoires assez penibles. Nous allons presenter une autre methode pour demontrer que et, en

meme

temps, que

M

g

M

g

est residuellement fini

est virtuellement sans torsion, autrement dit qu'il

possede un sous-groupe d'indice fini sans torsion.

Soit

*

rg

Ie groupe fondamental de

I.

II admet une presentation :

II s'agit d'un travail [B-1] fait en collaboration avec Alex 1ubotzky.

311 Un

homdomo rph i sme

h:

I

automorphismes interieurs de

+

f

I g

determine une c1asse d' automorphismes modulo les c ' es t-a-dire un element de :

Cet element ne depend que de la classe d'isotopie de M

g

+

h , d ' oil un homomorphisme:

rr g)

Out

D' ap r es un theoreme c1assique de Nielsen (cL [B], Th. I. 4), phisme. II nous suffit donc de demontrer que

Out

crg )

Ct

est un isomor-

est residuellement fini et

virtuellement sans torsion.

Considerons plus generalement, un groupe presentations affine

Hom (f,GLn(E))

forment, de

Rn(f) , sur laquelle opere Ie groupe

Le quotient algebrique de R (0

n

par

f

de type fini quelconque. Ses renaturelle, une variete algebrique

GLn(E)

par conjugaison. Soit

GL (D::) • C' est une var i.e t e affine ou on peut n

distinguer les classes des representations semi-simples de relle de Out (f)

Aut (f) sur

sur

Sn(f) ,

R (I') n

a

Theoreme. - Soient fini d'automorphismes de

Sn(f)

f . L'operation natu-

defini t, par passage au quotient, une operation de

laquelle on peut appliquer Ie theoreme suivant :

V une variete algebrique sur

V

Alors

E et

G un groupe de type

G est residuellement fini et virtuellement

sans torsion.

Corollaire. - Avec les notations ci-dessus, si Sn(f) • alors tout sous-groupe de type fini de

Out (f)

Out (f)

opere fidelement sur

est residuellement fini et

virtuellement sans torsion.

C'est a l'aide de ce corollaire, que nous montrons cue

M (O.Ona

g;;> 2 • Notons

la partie de

R 2(f)

injectifs et d'image un sous-groupe discret co-compact de l'image de

dans

S2(f) ,Evidemment

et

{ I}

o

I' 1

.;. , de sorte

formee des

p

qui sont

SL (R) • On note 2

et

sont invariants par

Aut (f ) g

L'application : 7

fg de

7

ad

g)

PSL2 (R) ad

ad

ou

p : r

0

ad

g)

g

7

PGL

definit une application surjective:

2(E)

est forme de tous les homomorphismes injectifs :

d'images discretes et cocompactes (cf. [P]) • Le quotient

g)

par l'action de conjugaison de

l'application: (p) r-+ (ad Riemann "marquee" de genre

0

p) • L'espace g •

APE

PGL (ffi) 2

element "general" d'ordre 2 si Si

P E

g

=

p

de

par

parametrise les surfaces de correspond

Lg

=

HI adpr ,

et l'on

g

sait que :

qui est un groupe fini d'ordre

g)

est l'image de

ad

rrg)

ad

84(g-l) (cf. [L.G,]) , En fait on sait que pour un on a

Aut(L

p)

=

{I} si

g;;> 3

et

Aut(L

p)

est

2 , il est facile de voir que: NpGL (E)(adpf) 2

=

NpGL (R)(adPf) 2

Mac beath et Singermann [M.S] ont demontre que l'indice [NpGL (R)(adpf) : adpf))] est "en general" egal a 1 pour

s > 3 et a 2 pour

g

2

=

2 • II s 'ensuit que Outer ) g

320

opere fidelement sur

pour

g

3 , et avec un noyau

g = Z . D'apres le Corollaire du theoreme du nOl , Out(f) g

g

de Grossman, disant que

est residuellement fini, pour trouver un sous-

groupe d i s t i.ngue sur

SZ(f) ; done

M d'indice f i.n i tel que M, et aussi

g = Z

est residuellement fini

et virtuellement sans torsion si Out(fZ)

3 . Pour

N d'ordre Z pour

Mn N

=

on peut invoquer le resultat

{I}. Alors

M opere f i.de l.erren t

Out(f ) , sont virtuellement sans torsion. g

L. Bers m'a signale la demonstration suivante du fait que

Out(f ) = M g

g

est

virtuellement sans torsion. D'apres un theoreme de Nielsen, tout element d'ordre fini de

M g

sous­groupe

fixe au moins un point de G d'indice fini de

M

g

ad

. 11 suffit done de produire un

qui opere librement sur

ad

g) • On prend:

G

Si

s

E

G fixe un point

(adp)

Lp

phisme de la surface de Riemann d'ordre 3 de la Jacobienne de tel

s

Lp

de

ad

,et

s

definit un automor-

opere trivialement sur les points

est l'identite.

0

H. Bass and Alex Lubotzky, Automorphisms of groups and of schemes of finite type,

[B]

s

D'apres un resultat de Serre [J.­P.S.] , un

REFERENCES [B­L]

,alors

a

paraitre.

J.So Birman, The algebraic structure of mapping class groups, in Discrete groups and automorphic functions, Ed. W.J. Harvey, Acad. Press (1977) 163­198.

[L.G.]

L. Greenberg, Fini teness theorems for Fuchsian and Kleinian groups, in Discrete groups and automorphic functions, Ed. W.J. Harvey, Acad. Press (1977) .

[E.G.]

E. Grossman,

On

the residual finiteness of certain mapping class groups,

Jour. London Math. Soc. 9 (1974) 160­164. [A.G.]

A. Grothendieck (avec la collaboration de J. Dieudonne) Elements de geometrie algebrique IV (Troisieme partie), Publ. I.HoE.S. 28 (1966)

0

321

[M.S. ]

A.M. Macbeath and D. Singerman, Spaces of subgroups and Tei.chmiil Ler space, Proc London Math. Soc. 31 (1975) 211-256.

[P.l

S.J. Patterson, On the cohomology of Fuchsian groups, Glasgow Math. Jour. 16 (1975) 123-140.

[J.-P.S. )J.-P. Serre, Rigidite du foncteur de Jacobi d'echelon

a [J .S.]

n

3 , Appendice

l'expose 17 de A. Grothendieck, Sem. H. Cartan 13(1960/61).

J. Smith, On products of profinite groups, Illinois Jour. Math. 13 (1969) 680-688 .

SUR LES TRAVAUX DE V.K. KHARCHENKO par J.M. Goursaud, J.L. Pascaud et J. Valette

Le but de cet expose est de donner une nouvelle presentation des principaux resultats obtenus par V.K. Kharchenko dans la theorie des actions de groupes sur des anneaux semi-premiers. Si fini d'automorphismes d'un anne au semi-premier

G

est un groupe

R,dans [5J V.K. Kharchenko

a

introduit la notion d'automorphisme interieur generalise l'anneau de quotients de Martindale groupe une C-algebre de type fini une trace dans Ie cas ou cas ou

R

S

de centre

au

Best semi-premiere,ce qui arrive dans Ie

Dans une premiere partie on decompose l'anneau G

C, et associe

B, lui permettant ainsi de definir

est sans IG!-torsion, et dans Ie cas ou

stables par

l'aide de

S

Rest reduit. en produit d'anneaux

sur lesquels les automorphismes interieurs generalises de

G sont des automorphismes interieurs au sens classique. Dans une seconde partie on montre que la connaissance de l'auto-injectivite de l'etude de

B ® C

B

facilite la definition de traces de

S

B

dans

et SG.

Enfin dans la derniere partie, on presente les theoremes de V.K.ronrrheruw concernant les relations entre

R

et

G R

en propos ant des demonstrations

plus concises. Sur ce sUjet on pourra egalement consulter les exposes de J. Fisher et J. Osterburg [4J , S. Montgomery [9J , A. Page [10J.

323

1.- NOTATIONS ET DEFINITIONS. 1) Pour un anne au unitaire - Z(R)

le

j(R)

R, on note de

l'ideal singulier a gauche de

- pour une partie de

R,

X

de

R,

CR(X)

R le centralisateur dans

R

X.

2) Soit

G

un groupe

- pour

a

E

d'automorphismes de l'anneau R

et

g

- pour une partie de f Ln i

E

G r

ag

X

de

R

de s Lqne l'image

de

et un sous-groupe

R. a H

par

g ,

de

G

Rest H-invariante si pour tout

x

on

t,

{x

E

X

- une partie et tout

h c H ,

- Ixi - pour

x

h

X

de

X

EX,

designe le cardinal de a

E

E

R

X,

on appelle trace de a

l' element tr a

=

Lag gEG

E

G

R

11.- L'ANNEAU DE QUOTIENTS DE MARTINDALE Dans toute la suite on supposera que R est un anneau Rappelons pour commencer la definition de l'anneau maximal de gauche de l'enveloppe injective

R, note

a

QMax(R)

gauche de l'anneau

ainsi un A-R-bimodule; on a alors la De f

Lnd, tion

. Pour cela considerons E

Hom ( E, E) AA A

R,

A

= HomR(E,E)

;

E

devient

324 Des renseignements plus precis sur

Q

Max

sont fournis par

(R)

la proposition suivante.

PROPOSITION 2.1.-

b)

a) QMax (R)

Q

Max

En fait on QMax(A)

Definition

{x

E

E , If b

E

(R)

A , Rb = 0 ==> xb = O}.

utilise aussi frequemment une autre caracterisation

necessitant la definition suivante

Un sous-R-module If

E

R

C-) QMax (R)

de

plunge dan6

D

b c A r Db

Si

de

QMax(R)

est dit dense si

o .

designe l'ensemble des ideaux

a

gauche denses de

R

on

a

THEOREME 2.2.-

Pour plus de renseignements sur

a

QMax(R)

[7] .

De meme si on considere l'ensemble R

on pourra se reporter

des ideaux bilateres de

d'annulateur nul (c'est-a-dire essentiels), on peut considerer l'anneau

de quotients de Martindale a gauche

S

=

lim ->

IE;!

Hom(I,R)

[lJ.

325

S ee {x E Q (R) Max

j

,

a un

On a

PROPOSITION 2.3.-

iJ.>omoJtplUJ.,me

U

ReS c QMax (R)

Ix c R}

I Ejt

Cet anne au possede une propriete faible d'injectivite donnee par Ie

SoU

LEMME 2.4.-

homomaJtplUJ.,me bhnadule pOM

de

tout

M

un

sc-modiu:e.

de

aioM i l

R -;

Uemel'lt

b

soit non nul,

l'annulateur dans T ill Ann ljJ(t+u)

R

=

T E$

;

(t)

Montrons que Soi t et

un

etemel'lt

s EStel.

tel que

de

I

I (b)

que:

(b)

de f Ln t s sorrs un morphisme

ljJ

et

=

S

et

Ib

-1 (R) n R de

bs

T ill Ann

tel que

R ; comme

o. R

T

Alors par

D'apres la definition de

tel que

ljJ(a) = as pour a ET ill Ann

R

coincide avec la multiplication M

et

M

T

u EAnn

de

un element de R

T

bun element de

I

a

un element de

droite par

s

S R

T

sur

M

tels que : Ib c R

I (b) CR.

Alors

'rI

i EI et

ib E cj> icj>(b)

Par consequent

=

-1

(R) n R

d'un ideal

I

et

I(cj>(b)-bs) = 0

Les elements de de

3

T

ibs .

Dorenavant on notera R

b

est nul, on a

s

MU un

M.

s i : t ET

-1 (R) n R

I E

il existe un element

un

: M ..... S

tel que

(M) non nul. Soient

S

s ,

a gauche ewte

de

On suppose (b)

de

C

C

cj>(b)

bs .

Ie centre de

S

• appele centro ide de

proviennent des homomorphismes de R-R-bimodules

dans

R.

326

De plus

a

chaque sous-R-R-bimodule

central

=

eM(x+y)

si

x

s

et

XEM

On verifie immediatement que

Pour tout

S , est associe un idempotent

c defini par

de

ve r i f Lant; : V

M de

eM

est le plus petit idempotent central

X E M

E

S

on pose

r

e

e

s

RsR

On obtient alors comme consequences

a)

PROPOSITION 2.5.-

Von

c

soiis-ci-modur».

b)

de

S

est: un C.Mp!.l

un element de

2

2

=

0

!.l,.l

ex. M

= 2

definie par

Rc

2

=c

ic(s) (l-e

s l)

E

=

jc(S) 0

et

donc

c' I

e

I

est bien definie car

2

est donne par la multiplication par un element

b) Soient

est un sous-R-R- bimodule

is (c ) c is (c) ; donc d I apr e s le lemme precedent

montre que

On verifie facilement que

es;

A

C-{O} . Alors

et l'application

(is(c )c)

nut.

S

[2J .

d) c

c

de.

jc(S)

c) c

a) Soit

[lJ .

1

qui verifie c

2

=c c ' .

C •

sic(s)

=

=

E

c'

et

. Par definition de s

=

e

l

,

o et

eIs

0 .

c) c etant regulier il suffit de montrer que pour tout ideal essentiel de C

de la forme

a

C.

Puisque par

J

=

$

Ce, l

f(e,) = e f Le . ) l

=

l

f(e

i)

i

l

l

e

i)

, tout morphisme

f : J -+ C

on peut definir une application

se prolonge

¢: eSe, -+ S l

; elle verifie les hypotheses du lemme 2.4. II existe

327 done

SES

tel que pour tout

($

ceil (xs-sx)

d)

si

Si

R

R

est premier,

e

xs

=

S

est premier done

sx

(d'apres b»

=

eixs

: s C

eisx. II en resulte

appartient

SOUS-ANNEAUX DE

anne au de

c

d'annulateur

J

•

C.

S. un anneau commutatif regulier injectif,

B

un sur

tels que

C

1)

C

a

est un corps.

est un idempotent non trivial de

J

Soient

c est eontenu dans

2) B

a

et

=

n'est pas premier, il contient un ideal bilatere

non nul

III.-

=0

XES, f(eix)

Z(B)

est un C-module de type fini engendre par

(x

i)1Sisn

et

sous-C-module singulier nul. 3) B

est semi-premier.

II est facile de voir que

LEMME 3.1.-

B

B

est un module projectif de type fini.

v.,:t un artrl.eau lti2.gU-UVt.

Montrons que pour tout ideal maximal un anneau regulier.

done un anneau artinien. Si J

(i,j)

Biro

B

de carre nul. On a alors . Comme

n'appartenant pas

C

a

de

B/m Best

C,

B/m B est un elm - espace veetoriel de dimension

finie engendre par les classes des elements

ideal

m

X.

1.

(lSiSn) .

tel que

B

est

n'est pas regulier il contient un p avec J = L C/m y i i=l

est regulier il existe un idempotent m

Biro

e

de

C

328

o

i,j ,k,l

\I

par consequent l'ideal

L Beyi i

dit notre hypothese sur

B

LEMME 3.2.­

Si

rv.,;t un ide.a1.

I

est un ideal de carr e nul, ce qui contre-

a gauche.

de.

rv.,f.,

B, I

ne

Pour demon t re r ce resultat il suffit de montrer que Soit

m

un ideal maximal de

vectoriel

I

Blm B

1

Soit

z1""

m

e

tel que la codimention

i

I+mB soit minimale. On suppose mB

m

Ell B/m B

au

(1­f)

,z.e. une base du

elm ­espace vectoriel

B(1­f)

I

ef

I

... ,ez

appartenant

a

B

Alors dans

I . Soient

B/m

1

a

B/m

e/m 1 ­ espace vectoriel et

de Donc

e

1

l

I

engendrent

il existe un element x

potent tel que

i ,

l

n'appartenant pas

a

m

en tant que e­module

.

etant essentiel dans

de

elm ­espace

B/m B (1­f)

tel que les elements et

0 •

f

e

1,

du

I ne

non nul. On a alors

11 existe alors un idempotent central non nul

ez

i

es t: un

fi

=

1

et

=

a/m a f 1

1

=

x

= x(l­f)e

un idem­

avec

et

un ideal maximal de

B (l-f)

est engendre par

B/m B f 1

est contenu dans x

=

appartient

a

I l en resulte

non nul

e

ne contenant

comme

Par definition m 1 ce qui contredit le choix

0

I

m 1

0

f

et

e

I

•

329

PROPOSITION 3.3.-

a dAaite

un

B

et

a gauche.

une

d'Azumaya. a) Montrons que

Best auto-injectif

suffit de montrer que tout morphisme I

de

B

dans

B

$

=

e a)

C

et

C

B

; la restriction de

a

D'ou

a

appartient

f(x)

etant regulier, il

a

gauche essentiel InC

es t un

f

a

Ce

$

est un homomorphisme de

a

en un endomorphisme

a

9

de

C

dans

B

sous-module singulier nul

Best C-injectif.

I , f(eax)

=

o.

e x g ( l ) et

a

xg(l) . [11, theoreme 5.5.7J , il suffit de verifier que pour tout

ideal maximal

m

engendre par

de

Z

(B)

(xi) 1 upp0f.>e. que.

f.>U/t

es« JteguUe.Jt -tnje.c.t-t6 et: que. le. f.,OUf.,-module. f.>-tnguUe.Jt ess: nul. MOM d

e.wte. des -tde.mpote.n;tJ., oJt:thogonaux

1 = e + ... +e

C

o

e.t

n

le. noyau de.

de.

jc(Z)

eo"'. ,en

que. pOU/t c.haque.

i

C Z

de.

(osisn)

c.anon-

e

a de.ux

r

s £

so.c: ..lnvaJUan.t pan:

te.l que.

e..t

a

fEE,

G; il

H

•

les

337 (Le. On remarque d'abord que: a) si dans E on a 2 m g5,e g g alors on a e5,e = e · 5,... 5, e (m est l'ordre de

g

GIN)

dans f

1

:5, f ,

f

donc 1

EE

=

e

e

g

b) si

tel que

f

fi

1

=

fEE (f

0

Dans l'ensemble des idempotents de nombre

k

(5,IG/NI)

supposer que

(ii) Soit

£,

,

fEE

b) il existerait

f

f

1

£

g2

f

h g2 gk , ••. ,f 1,f 1,f1 1

le definition de

1

gk

tel que

f

g2

g2

f

1

:5, £

a

g i. H

On con s i.de r-e

a

fg

1,g2, ... ,gk g EG \

k

U

a

E

d' apz e s b) il existe fEE, f 5: £

et on suppose

car •

H

1

i

st(f)

tel que

alors il existe

convient).

a

e

et dont le

deux orthogonaux est maximum,

a

H

fh 1

maximal et on peut

deux orthogonaux. tel que

hEH

f;r. fh

d'apres

et puisque

0

f

gk 1

:5, e

gk

deux orthogonaux ce qui contredirait

H

k • En consequence

i=l

Puisque

a

sont deux

seraient deux

suppose l' existence de

f-ffg

inferieurs

S' il existait

:5, f

On remarque que

(i)

, ••• ,£

fh1 < - £

5: e

E

f;r. fg

de stabilisateur

£

f:5, £

=

1

de conjugues deux

on considere un element

et

st(f)

ne sont pas congrus modulo H

et on

g. H. 1

fEE , f.:5, f,f

g2

, ••. ,f

maximum. Si on avait

et d I apre s a)

£

verifiant

gi k >

ffg

=

0 •

soient orthogonaux

i

on en dedu i r a Lt;

-1 gog f(l-f - que G

1

= n 1+ ... +n t

comme gJtoupe d' automOltphAAmv., de

..• ,n

f.>OYlt

S de

t

E

e.t teif.> que h.t Ort sn

i

'

f.>O.tt

(G) inn

un f.>OM-gJtoupe c-oYlhtitue d' automOltphAAmv., .tYlteJUeW1.J.> deMYiM par:

des e.temen;v., .tnVeM.tb.tv., de SoH

g1 = 1

g2, •.•,gn

Soient

i

un systeme de representants de

G modulo N .

les elements de ce systeme qui appartiennent k

On a

sn

C

Lee

l=1

a

Ginn

. Le lemme 3.4 entratne l'existence d'idempotents

gl

tels que

.1

t L n i i=1

o Si

est non nul, Ie lemme 3.10 montre que l'automorphisme

coincide sur Soi t g

ou

g

= hg 1

avec un automorphisme interieur.

G , tel que Sur

g

n.1.

oeO • Il existe

on a

hEN

tels que

et

g(x)

yE;,xE;,

On a donc g

gl

Puisque

g.t

Sn

y

•

est interieur au sens classique sur Sn i '

est interieur au sens classique sur

groupe d'automorphismes de

-1 -1

i,

Sn

(G) inn

i

et si on considere

G comme

est un sous-groupe constitue

d'automorphismes interieurs au sens classique.

•

342

V.- TRACES. Ce sont les elements de On definit sur

S

une structure de

B @ a-module

a

gauche en posant

C

s b.

Puisque

B(R;G)

classique, lateur

a

sG

= Cs(sG)

et puisque, lorsque

= CS(B(R;G»

droite dans

B

est interieur au sens

a

, dans ce cas on est amene o

@

G

B

de l'ideal

a

gauche

C

etudier l'annu-

B e

E

bEB

B(1

@

b-b

@

=

1)

C

0

B @ B (1 @x -x @1) qui est le noyau de l'homomorphisme g g

E

gEG

0 B e B _IJ_> B - - > 0 C

E a.

@

b . .-.....;> E a. b.

1) Dans ce paragraphe

B

injectif con tenant dans son centre C

tel que

B

designera un anne au regulier autoZ

un anneau regulier auto-injectif

soit un C-module projectif de type fini. Notons que

o

B @B C

etant auto-injectif et r(ker IJ)

de type fini,

lr(ker IJ)

= ker

et

IJ

n'est pas nul. On se propose de montrer le

THEOREME 5.1.-

r(ker IJ)

PROPOSITION 5.2.-

En

Be

r(ker IJ) Soit

libre.

ker IJ

e6t

B 03> C

a.

d!toUe.

u a galLc.he..

e6t de

M un sous-Be-module de type fini d'un Be-module

B etant regulier et

type fini,

ut un anne..au c.oh0te.J1.:t

13

Be

etant un B-module

a

a

droite

droite projectif de

M est un B-module projectif. Done pour toute suite exacte

343

o

+ M' +

(Be)m + M + 0

de

M'

Be-modules,

est un B-module de type fini

done un Be-module de type fini.

•

de

B

et

te1.J.> que

hypothe-6v.. que

En effet

et

B

r(ker 11)

du cent4e

deux

LEMME 5.3.- Soient

et:

C

CE.

(i = 1,2)

v..t Lsomonph». au pfwdui;t

r(ker III

et

est contenu dans o

est isomorphe

a

BE.

.i.

•

BE.

LEMME 5.4.- Ii -6u66U de. pftouveJ[. .e.e. theMeme. .e.OMqUe.

Le couple de

(Z,C)

on peut supposer que

3.5 et 5.3,

D' apr e s les lemmes

verifie les memes hypotheses que

ker(Z @ Z + Z)

B

est engendre par un element

v..t commu.t.a..t£6.

Best Z-libre. (B,C); l'annulateur

u . On considere alors

C o

Le morphisme naturel d I anneaux

fait que

B

@

o

B

est un

=

B e B Z

. En utilisant Le

Z @ Z-module libre, on verifie aisement que

C

ker 8

0

8 : B e B + C

C

E

B@B

Z Z

C

(l0z-z01)

et que si

a

appartient

a r(ker

11)

on a D'autre part

B

etant une Z-algebre separable, Ie noyau de

o

B @ B + B Z

est engendre par un idempotent 1-e . Soit

'1

B

@

o

B

tel que

8('1) =e

C

il en resulte (b 0 1 - 1 0 b) T 1

a

=

'1 a

ker 8

= et

T1

ua

1

. Enfin quel que soit

T 1u E: r (kezu)

b eB

•

Z

344 LEMME 5.5.-

Le.

eAt vnai. -6-

B 0

C

B --> 0 • Pour tout

droite

fEE'

on

C

pose

V

11 est immediat que les applications • Soi t o " , f = f2

un element de

Tr

s)

f

sont des traces.

f

et

B

G/H

*

( (f 0 1 ).T

SES

e

$

ef

e E E'

tel que (e 01)

T

soit de longueur minimale. On peut ecrire (e01lT =

n 1: a.0b. i=1

1: a.B = Be = 1: B b.

nul, il en resulte que pour tout

i

, f Bb

est nul et done que

i

est nul ce qui est impossible. II existe done

a EB

f e

tel que

(e01)T(10fa) '" 0 • On a alors

o '"

=

(e 0 1) T (1 0 fa)

Comme tout

sait

e'

e' E E'

$

e

f a b l::-

e . Pasons pour

s ES Tr , f ( s) e ,

=

11 est immediat que

LEMME 5.7.-

« e'

0

eX

f

1)

Tre',f

T

*

(1 0 fa)

G/H ,

s)

e

est une trace.

un eleme..u: non rtUl de. s

So.i..e..u: a

non rtul de. s. On f 1

m 1: a.0fab. i=l

que.

rB(a)

=

(1-f

1)B

du.i..gne..u: /tupe.wve.me..u: des

2

eX M un sous-modui». eX rB(M)

de.

B

=

(1-f

eX de.

2)B

OU

Z (B)

non oJt.:thogonaux. AtOM

pour: .:tout e

pOUlt .:tou..:t

I

E:

3'

o

E

E , e

0 it

tU.:t

$ 0

ef f ,.i..l 1 2

0 '" a Tr (1M) e

exist»:

.:tel que.

347 D'apres la remarque 1)

de longueur minimum

s

e'

e

et tout

k

il existe

(ef

1

e f

e'f 1

1

des definitions de

f

2

et

f

, e:"':e

tel que

o

(ef

soit

10f2)T

m

lT

l: ef a 0b 1 k kf 2 k=1

et

1

eEE'

bkf 2e'

2,

e' E E ,

ne sont pas nuls. Compte-tenu

pour tout

aTr e (x) =a[ef 1 (T*X) f 2J+a[e (T*X) J

pour tout

x E 1M

on a

:

G/H*

m e = a\:1

G/H*

n

e.

m Par definition de plus pour tout elements

o >' I: v j

v. J

g

et

m

et

H

,

f.

e

t. J

f e

de

1

'

l: Caef 1ak k=1

la somme

est directe ; de

est nul. En consequence [3.15J il existe des

g

R

tels que

o

. a ef 1a 1t. = d. = Cl e J )

= I: v j

jaef 1akt j

(1'Tre(J( (resp.

Soit

I: vjaTre(IM) = ClIMb j

etait nul, on en dedu.i r a.i.t;

(1M)

COROLLAIRE 5.8.-

e.t

done

O>'f = f

o '" Tr e (( ()/; n DR) J)

2EZ(B)

tel que pour tout il existe

G

(JUn ROll c (Jt.n D nR

tel que

J £::; tel que

on ait Tr (J e

c

o. n

G) D nR •

rB(()l;nRO) = (1-f)B Tr (J (an RO» e

IJ!-)

c f1L n RG

et

'" 0

11 existe Comme

Tr (JD) cD. e

e £: E'

Tr (x) e

•

348

VI.- LES THEOREMES.

M!.mi - pltemieJt. De. ptlL6 J.:,i

THEOREME 6.1.G R a)

pltem-a:te.uJt dans S

B

(l-f)B

ideal

de

a

=

1: Y x

g g

droi te de

B: il est essentiel

B. II existe done un ideal essentiel

tel que pour tout

a (l-f)

G•

de. R

G)

[3.2J. Si on considere des elements g

et

Rb •

{b E B ; ab E B}

dans un facteur direct dans

a

sont invariants donc nuls. II suffit d'appliquer Ie

un element de

On cons i de re

on ait

I

a

G Tre(I) C R

tel que

THEOREME 6.2. [5, theorem

e.y

G-J.:,,{mple. a1..oM

B

i C

eia(l-f)

Yg

de

EB

d'oii i l resulte I

= 1: g

e. 't , x

tels que pour tout

=

g

EB

i

(l-f) B

soit de lonm . En particulier fa i. 1: gueur minimum: fe @ 1., = 1: @ b l k k=2 k=l G Soit J E tel que l'on ait Tr (J) C R ; alors on obtient f T- 0 ; soit

tel que

e

\I x

E

J

a Tr (x) - Tr (x) a e e

o

fe 0 1 • r

349

m Or

afa

t

1

n +

E

k=2

k=m+l

il existerait

a

n +

E

appartienne

tels que

B; on aurait

0 = fal+fa2c2+ ... +famcm

ce qui est impossible. Comme a = ae '" 0

dans la demonstration du lemme 5.7, il existe on en deduf, t tion de

: e

e

el

e . En consequence

1) Supposons d'abord que Soit f

1

eo

= f

Soit

e et

un f.,otU>-R-RG-b-i.module de S • It

M

M = RaR

G

et

2

I E 1" tel que

Tr (I) e

C

G

R

et l'ensemble de ces idempotents Se

o

de l'ideal bilatere

donc nul de sorte qu' il existe E aI bk

k

C

r

B

(M)

= (1-f) B

B(a)=(I-f)B.

= 1 il existe

E v. a Tr (I) j J e

dans

r

ez

M

. On reprend les notations et la demonstration de 5.7 avec

f f

•

I = B .

e:t f = f2 E B :te.1-6 que Jf c

J E.1

tel que

ce qui contredit la defini-

e b = 0 a 1

et

THEOREME 6.3.[5,lemma 5J • Soil

ew:te

dans le cas contraire,

akc

E

k=1

e L

k

C

il existe

est cofinal dans = {a ERn Se

o

aI'f

m E ex I' Bkb k k=1

0

; 3 I

0'" a ELI n ... n L m

I' E j'

tel que

m E alb k k=1

C

tel que

M

M • Coinpte-tenu de la remarque 2)

e f

elk ERn Se

: Be f 0

a

et

E e E

L'annulateur

a Iabkc M}

I E j" m E Bb k k=1

m E I'B c r k k=1

M .

o

d'oll

tels que Le.

est

350 L'ensemble des bilatere J =

aEL verifie

etant cofinal, l'annulateur dans

3 I a E 1"

L = {a ERn sef ; RaI

L

eo

+ 1 (l-e

a

0

f)

1

(ou

a

et

1

0

e

c R)

f

Bf

L

aEM J

a

appartient

a ..1

et

Jf eM.

2) Dans Ie cas general on considere l'ideal et

de l'ideal

a I f eM} est nul. En consequence

E l'

0

sef

, J f a 5:: RaRG a

E.1'

direct

Bf

existe un ideal essentiel

B

(a.l = (l-f )B a

Cet ideal est essentiel dans un facteur

On verifie aisement que

o

avec r

a

de

Ceo

$

f = f

. De plus d'apres [3.2J , il

o

tel que pour tout

C

i

on ait

i

i

on peut associer

en effet il existe

alors si

Ej'

J'

a

n

de

verifie J' a

J.e.f 5:: L J' Sn f 5:: a a n n

L J

a

f n

13 cJ n

a

L J. e.

B

a

n

c

n

J. E:t

tel que

J. e . f c M

tels que e. f = 13 f + .•. +13 f 1 a n i an. 1

et si J.=J' a

1

n ... n J' a

on obtient n.

I1 est, clair que l'on peut supposer

M

n

E l'

et verifie

Jf eM.

•

On dispose d'un theoreme analogue pour les sous-RG-R-bimodules de

THEOREME 6.4. [S,theorem

7J

S .

le

on on a

a) POUlt tou.:t

I E$,I

b)

I 1E.1'1

POUlt

c.) (R.:f )

tou.:t G

es«

1

a '5 1

G In R

u. ewte a (RG)5

J E

1

j" tel que

J c RI

1

351

a) Soit que

o

I-f

J E j" b) r

f2 EZ(B)G

f

(II) = (I-f) B

c) Soit

avec

fEZ (B)

J E.1

,

PROPOSITION 6.5.G R

que

cG

=

est nul G

(R ) -r 0'1

de f i n.i

SJ..

G R

et

t,

I

est un produit de corps et

un ideal de

B

I

a

RI

3'

E

a

0 •

(I-f) I c R

tel que I c RI 1 . gauche

Comrne

1

et

RI

(Rj') G

1

•

e;.,t G -

est regulier, Ie theoreme 6.4 montre

I

est egal

C . Comrne

a

C . Par suite

est un anne au semi-simple. si ct

est

•

SJ.. G.

es« C.OYL6:tUue. d'acdomoJtph.J...6me;., de.Mn..L6 paJ!.

:Lnn

S

GIG.

:Lnn

G. R

tel que

=

et donc

des e.ie.me.nt6 J..nveMJ..bie;., de. :Lnn

E.:t

(1-f)

est engendre par un idempotent invariant

qui est central dans

PROPOSITION 6.6.-

e:,>

G) (I nR (1-f)

un ideal essentiel de

B

invariant,

et

()(, l'ideal

un element de

cG = c G/ N

est un corps. Soit

I

se prolonge

ess: pftemJ..eJt, B

est premier et

eEE'

et il existe

est non nul,

C

(l-f)B • Il suffit de montrer

• Il existe

est nul et par suite que

contient

Comrne

I-f

un element de

at

B(RI 1)

= Tre(JI) (1-f) c

O;tTre(JI (1-f»

Le lemrne 5.7 montre que

que

r

Dans Le cas contraire il existe

tel que

B

tel que

•

Supposons qu'il existe supposer que

G

g

;t

1

E

GIG.

:Lnn

est engendre par Ginn

x-exterieur sur et

G. R :Lnn . On peut

g . II existe donc

o ;t

G.

XES a nn o

352 tel que l'on ait : G.

\I s E S a nn

En particulier

G. S lnn

de f

o

Soit

f

EI

I E.1'

g

Puisque

g

ae;t

0

m

G. S lnn

G. S lnn f

est trivial sur

tel que la longueur

f

dans

de

done au centre

associe

a

x

o

. En consequence si S

et

(e' f I8l 1) . T

e'

c(f)

un

soi t minimale

XEI

est X-exterieur sur tel que

a I b

1

=

R

et que

fa

1'

E k;t1

il existe

•

ce qui est impossible.

0

(cf. question 15 de [4J).

at un

PROPOSITION 6.7. [5, lemma 11J .

v...6entie1. daM

R

v...6 entie1. dans

RG

Soit

D

un ideal

idempotent de

B

,

a

at. n

R

G

gauche

v..,t

un ideal

;t 0

tel que

fB

'r

de E

(e.). d'idempotents de C tels que l lEI

\I i

Soit

J

i

J

G. (e'f I8l 1).T*IcR lnn, on obtient

veri fie \I

=

l'idempotent central de

designe la couverture centrale de

element de si

o

G. CS(S lnn)

est invariant et

c(f)

a

x

sx

E:J

tel que

J.e. l

l

E

R •

a

a

(a

qauch»: (flv..p. it

tel que dtn D = 0 • Soit r

B

f

(M n RD) . 11 existe une famille

un

353 Soit

E JieiM i

N

effet soit

a E

I l existe

J.e.a r,

et soit

x

E Jieia-{o}

ax = a x e. E M.

B

(N n RD)

B

= fB

En effet

n RD) e fB

(N

d'apres la definition de

x EN n RD

Pour tout

a

En

.i,

r

r

i

()U

J.e. ax e J.e.M.

On a alors

et si

I l existe

()(,

a E R

a

est un sous-module essentiel de

N

N'

-
,0

si

R est de Cohen-Macauly, Ie premier coefficient non-nul apparait pour

i ; dim R, et sa valeur est appelee Ie type

t(R)

de

R, la condition

t(R)

caracterisant les anneaux de Gorenstein (iv) la serie de Poincare de

i

R:

0

qui est un polynome si et seulement si

Rest regulier

(v) la serie de Hilbert-Samuel de ;

.L.

R

-R /PR p P - P P

R /pR

est separable.

Pest ramifi§, et on sait que l'ensemble des G ramifies sur R est Ie ferme de Spec R, egal au

support du module des differentielles de Kahler Jt / RG. En particulier, un ideal R definissant cet ensemble est la differente definie comme etant l'ideal de Fitting des mineurs maximaux d'une presentation de JL / RG . (Pour une R exposition detail lee de la theorie locale de la ramification nous renvoyons au

[IIJ).

cours de Scheja et Storch

Dans ce nume r o on se contente d ' enoncer Le principal r e su l tat de [2]. On a besoin d'un lemme, qui dans Ie cas complet est une consequence d'un resultat plus precis, demontre dans [10, pp.8-9J. Lemme (2.1)

[2, (12)J

caracteristique de Alors

h

elements

Soi t

h

un automorphisme de

k ; on suppose en plus que

h(x)-x, quand

x

parcourt

premier 11 la caracteristique de sur l'anneau local dans

R, d ' ordre fini premier 11 la

induit l'identite sur

est une pseudo-reflexion si et seulement si l'ideal

R. Soit

9

H engendre par des pseudo-reflexsions,

k,

H

G operant generiquement sans inertie

l'ensemble des pseudo-reflexions contenues dans

l'ensemble des ideaux de

R, qui sont H-stables et ne sont pas contenus

","HR. Alors on ales egalites : G) Q.(R/R =

k.

engendre par les

R, est principal et non-nul.

Theoreme (2.2) ((2, (4)J) On suppose

G,

h

n

E.e

b

380

Remarque (2.3). En suivant

la demonstration donnee dans [21, on voit que

l'egalite de gauche reste vraie sans supposer que

H

est engendre par des

pseudo-reflexions, si les conditions suivantes sont satisfaites : (i) premier a la caracteristique de inertie

(ii) Rest

k, et

G opere sur

RG-libre de rang

R

IHI

est

generiquement sans

Jel ; et (iii)

est d'intersection

complete.

,w

En reprennant un exemple de [9], soit 6-ieme de l'unite (6 f

une racine g(X)

=

Rest

lJJ X, g(Y)

Y, H

k), g

( g ) . Alors

= G

le k-automorphisme defini par 6 2] G R k[X , y est r egu Li e r , done

RG-libre de rang 6, et

est d'intersection

complete. On voit, par la remarque precedente OU directement, que G) = (XSY)R. D'autre part, comme est donne dans la base evidente 2 par la matrice diag(lJJ ,w, 1) , la seule pseudo-reflexion de G est

D(R/R

l'element

h

=

g

3

et

(XY)R, done l'egalite de droite n'est plus assuree

=

par ces conditions. 3. ANNEAUX DE BUCHSBAUM. 11 est bien connu que la propriete d'etre de Cohen-Macaulay descend de H

R , pour tout groupe fini

dont l'ordre est inversible dans

H

k

R

[6J. Nous

allons retrouver ce resultat, dans le cadre plus general des anneaux de Buchsbaum, introduits dans [14] par la propriete que pour tout systeme de parametres xl"" ,x k de R, la difference de la longueur et de la multiplicite reste constante ; cette valeur independante de x est appelee l'invariant

i(R)

de

R, et on a

i(R)90

avec l'egalite caracterisant les

anneaux de Cohen-Macaulay. Proposition (3.1) Soit (i) Si (ii) Si

R H

G un groupe fini d'automorphismes de l'anneau local

soit premier a la caracteristique de G est un anneau de Buchsbaum, R l'est aussi, et

tel que l'ordre de

H

=

est engendre par des pseudo-reflexions, et

G

R,

k. i(R)

opere sur

R

generiquement sans inertie, on a i(R) Demonstration. (i) Soit

xl""

D' ap r e s [12J, pour voir que

groupes d 'homologie sont annu l e s par /WI

H. (K)

H

1

H

IHli(R

=

un systeme arbitraire de parametres de

,x d

R

G).

est de Buchsbaum, il suffit de montrer que les • H R sur Xl"" ,x

du complexe de Koszul de

pour tout

sur un systeme de prametres de

i

1. Comme

R, on a deja

K IlJ H R -

d

est un complexe de Koszul

R H. (K IlJ H R) = 0 1 R

pour

I.

381

H R __:>R

1

L. g(x) s'etend a une gEH section de l'inclusion canonique K '--) K H R, qui commute aux differentielles, H. car elles sont donnees par des matrices a co:fficients dans R On obtient donc f:

Or l'operateur de Reynolds

que la suite exacte de

(f(x) = IHI-

RH-modules :

donne en homologie de Koszul les isomorphismes de RH-modules H.(K 1

donc

H

R

fll

1

H.(K 1

H T).

R

H D'une part on voit que annule l'homologie de Koszul de R et de H R est un anneau de Buchsbaum et T est un RH-module de Buchsbaum.

T,

D'autre part, comme on a d

(_l)i-l t

i=l

H. (K 1

H R) , R

= R/hn montre que la longueur d'un H,:: R voit que

et comme l'egalite etre.calculee sur (3.3)

R

i(R) = i(R

H)

R module peut

+ i(T)

ce qui demontre (i) pour G H. Pour passer au cas general il suffit, puisque H R G H R est RG-libre de rang IG/RI et AlI = IM R (e g , (2,(7)1), d'appliquer i

Ie lemme suivant :

---+ (B,1L)

(3.4). Soit est A-plat et et dans ce cas

Alors

B A K = K

A

B

commutatif :

'

A est de Buchsbaum si et seulement si

B

B l'est,

i(A) = i(B). A K

Demonstration. Soit generateurs de

un homomorphisme local, pour lequel

Ie complexe de Koszul de A sur un systeme minimal de . A A 1(K) , et posons H =.H HomA(K ,A). D'apres nos hypotheses i A) B) A) Hi(K = Hi(K B = H1(K et on a un diagramme

AIm- -

A

H. (K

A)

Hi (K

A)

1 (A)

(A)

B!:! Hi (K

B B

B)

1 (B)

ou l'isomorphisme des modules de cohomologie locale vient de la platitude

382 i

i

Ext (A) M B = lim Ext (A/rttl n A) M B B A A -;> , A -----, A que la s ur j e c t i v i t e de lp est equivalente

i B, B)

a

celle

de

i H (B). On voit 1t epB (descente

fidelement plate), et cette propriete caracterise les anneaux de Buchsbaum (13J. D'autre part, en calculant i(A) comme dans

(3.2), on obtient

(ii). D'apres Ie lemme on peut supposer

G

i(A)

H, et comme

=

i(B).

Rest d'apres

(1.2) RH-libre de rang IHI, on a Ie resultat par (3.3).

Exemple (3.5). Soit

R

l'anneau gradue

3 4 4 3 k[x , X Y, Xy , y ] du cone sur la

quartique gauche. II n'existe pas de groupe

H engendre par des pseudo-reflexions,

operant par

R, tel

dans

k-automorphismes homogenes sur

k. En effet,

R

que

est un domaine de Buchsbaum avec

lHI soit inversible i(R)

=

I, et on conclut

par (3.1 i i )

Bibliographie [1]

L.L. AVRAMOV, Homology of local flat extensions and complete intersection defects, Math. Ann.

(2)

(1977), 27-37.

L.L. AVRAMOV, Pseudo-reflectilln group actions on local rings, Nagoya Math. J. (to appear)

(3)

N. BOURBAKI, Groupes et algebres de Lie, chapitre V, §.5, Hermann, Paris,

1968. [4]

C. CHEVALLEY, Invariants of finite groups generated by pseudo-reflections, Amer. J. Math. 67 (1955), 778-782.

(5]

H.-B. FOXBY and A. THORUP, Minimal injective resolutions

under flat base

change, Proc. Amer. Math. Soc. 67 (1977), 27-31. [6]

M. HOCHSTER and J.A. EAGON,

Cohen-Macaulay rings, invariant theory, and the

generic perfection of determinantal loci, Amer. J. Math. 93 (1971), 1020-1056 [7]

C. LECH, Inequalities related to certain couples of local rings, Acta. Math. 112 (1964), 69-89.

383 [8J

£9]

H. MATSUMURA, Commutative Algebra, Benjamin-Cummings, New-York, 1980 E. PLATTE und U. STORCH, Invariante ragu La r e Differential- formen auf Gorenstein Algebren, Math. Z. 157 (1977), 1-11

[10]

J.-P. SERRE, Groupes finis d'automorphismes d'anneaux locaux reguliers, Colloque d'Algebre E.N.S.J.F., 1967.

[11]

G. SCHEJA und U. STORCH, Lokale Venzweigungstheorie, Schriftenreiche Hath. Inst. Univ. Fribourg/Suisse, nOS, 1974.

[12]

P. SCHENZEL, Applications of dualizing complexes to Buchsbaum rings, Adv. in Math. (to appear).

[13]

J. STlJCKRAD, Kohomologische Char ak t e r i s ierung von Buchsbaum-Hodules, Math. Nach r ,

[I 4]

265-272.

J. STUCKRAD und V. VOGEL, Eine Verallgemeinerung der Cohen-Macaulay Ringe und Anwendung

auf ein Problem der Multiplizitatstheorie, J. Math.

Kyoto Univ. 13 (1973), 513-528.

ON NONNORMALITY OF AFFINE QUASI-HOMOGENEOUS SL(2,a)-VARIETIES Dina Bartels

Introduction: By an affine quasi-homogeneous Sl(2,cr)-variety we will mean an affine algebraic variety with regular Sl(2,cr)-action containing a dense orbit. The normal affine quasi-homogeneous Sl(2,cr)varieties have been classified up to Sl(2,cr)-isomorphism by Popov [11]

. We were interested to know, whether orbit-closures

in simple Sl(2,cr)-modules, in particular those orbit-closures containing zero, are normal. For instance, Kraft and procesi [8] have shown that for the adjoint module "of

all orbit-

closures are normal. It is known that in simple Sl(2,cr)-modules all orbit-closures of dimension less than three are normal. Special examples of non-normal three-dimensional orbit-closures were given by Popov [11J

, Kraft and Hesselink [4] , using

various types of arguments. The aim of this paper is to introduce a new method, which is sufficient to show in general that for simple Sl(2,cr)-modules all three-dimensional orbit-closures containing zero are not normal. Let us mention furthermore that Luna and Vust [9] have classified up to Sl(2,cr)-isomorphism all normal quasi-homogeneous, but not necessarily affine, Sl(2,cr)-varieties. I would like to thank W.Borho for several discussions and hints.

385

O.

Notations

We denote by

G

the group

such that

[ ac db]

ad-bc=1

of complex 2x2 matrices . Throughout the paper, we use the

following symbols to denote some special subgroups of B

is the Borel-subgroup of matrices

B

is the Borel-subgroup of matrices

u

is the unipotent subgroup of matrices

U

is the unipotent subgroup of matrices

T

is the subgroup of diagonal matrices

N(T) is the normalizer of matrices 'em

T

[g

n

G, consisting of

o

[6

is the cyclic subgroup of matrices

Rn

[6 ]

[t 011

is the semidirect product

By

in

G

i;m: 1 .

such that

em ·u

we denote the complex vector-space of all binary n-forms

with complex coefficients. For a particular such form

f , we

use the notation a

as polynomial in

X

and

Y . The group

v

E«: G

acts on

Rn

by

386

1.

Orbit-closures in finite-dimensional G-modules

1.1

By an (affine) G-variety we will mean an affine algebraic

variety on which space on which

G G

acts regularly. A G-module is a vector-

acts linearly. The following fact is easy

and well-known [10J.

Theorem:

For every (affine) G-variety there exists a closed

G-equivariant embedding into some finite-dimensional G-module.

From representation theory of

G

it is well-known that

1)

every finite-dimensional G-module is semisimple,

2)

every simple G-module is isomorphic to some

Definition:

R

n

A G-variety is called quasi-homogeneous, if it

contains a dense orbit.

The subject of this paper is to study quasi-homogeneous G-varieties. As a consequence of the facts listed above, it is equivalent to study closures of G-orbits ina G-module V=Rn

1.2

$ ... $

1

Rn

for any numbers n , ... ,n r . 1

r

Let us recall the structure of closures of G-orbits in

finite-dimensional G-modules as described by Popov in Let

Gf

denote an orbit, generated by

f , and let

[12j . Gf

denote

its closure. We will distinguish six types, according to the dimension of

Gf

and the structure of the boundary

Gf'Gf

These are listed in the following table. In addition, the stabilizer

G f

of the generator

f

(up to G-isomorphism)

387

is given in the last column, where we use the notations introduced in section O.

dim Gf

-Gf ....Gf

0

¢

G

I'

a)

2

¢

T or N(T)

b)

2

3

a)

3

b)

3

type 2a)

c)

3

type 2b)

type 1)

I

1.3

Gf

type 1)={0}

U

m

for some m

any finite subgroup of G

¢

em

em

for some m for some m

For our subsequent considerations orbit-closures of

type 3c) will be of special interest. Let us make some suitable choice for the generator

f

of the dense orbit in this

case.

As a consequence of the Hilbert-Mumford-criterion,

f

can be

chosen such that

as explained in [llJ. Moreover, cyclic stabilizer

f

can be chosen such that the of order

is the cyclic group

m

R $ ... $ R , say f=f ... $f these special r n n 1$ l r for each i=l, .. ,r assumptions on f amount to choosing f.ER l n

For an

f

in

i

as follows: n , (i) n. n.-v v f.= Zl a (l)X l Y l V v· v=O (i) with a =0 i f m)'n.-2v v

l

or i f

388

For each nonzero v

(i)

such that

OSk < i

n.+1] [

(Remark: zero for

v and IT

l

k. l

the maximal number

is

. Note that by the above assumptions

that

k.>O

The number

for at least one

k.

l

max

,where

l

n.

.i,

l

k. l

n,

is defined to be

l

l

Gf

in the

[llJ ).

Regular functions on orbits and their closures

For an algebraic variety

Gum

7f let

denote its ring of

regular functions. An affine algebraic variety pletely determined by

closure

?J is com-

SO, taking the algebraic point for an orbit-

of view here, we will consider the ring

2.1

i

f.={O}, is the "height" of the variety

terminology of

2.

we denote by

f.

Gf.

First we will state some general results concerning the

regular functions on G-varieties.

(All these results are valid

for any reductive linear algebraic group).

Let

?J be any G-variety, then

g(F(v)):=F(g

Proposition:

-1

v)

If

for every

G

acts on

vE 'l.t, F E(JHV)

7.J is a G-variety, then

(5)

Schanuel's example.

(5)

Suppose a£A\A is chosen so that the conductor

(4)

C =(A : A [a])

of A in A [a]

is not a radical

ideal of A Ca]

. Since

C is the largest common ideal of A and A [a] we may choose b"I]""""'l;\A. Sayb

n"

C.

Since C is an ideal of A raJ

'

b

m

for all m

406 Let

n

o

be the

smallest integer such that

n-I

and d = b

'A,

0

I f B is a

(4)"9(3).

suppose

but d 2 ,

A is not

for

d 3 E A contradicting

A

subring of

seminormal and

Let B = A

bme,\

m". no· Then

no'

(5).

containing A,

then +BA=+7\Ar'lB.

let aE. +XA'-A. B,

• Then +BA

so for

each prime

ideal P of A

there is a unique prime ideal Q of B lying over P and the map k(P) _

C

k(Q)

(A:B)

F

C

is an isomorphism.

is not a

radical

0 since B is a

We shall now show that

ideal

in B,

contradicting

finite A-module,

and C

Let P be a prime ideal of A minimal over C, of B lying over P.

ideal

f

in B and since CA P P Now if C is a of PB

Q,

CAp

Q

phismk(P) -+ k(Q), which

Ap '

radical

= QB

Then CA Ap

ideal,

also.

t

(A

p Bp

CAp

PAp

p

Q.

: Bp)

is

and

(I,).

A since A

#

B.

let Q be the prime the conductor

of Ap

also have that Bp = B • Q since QB

Q

is

the radical

We now have from

an isomorphism,

the

isomor-

and hence Ap

= Bp '

is false. (After Gilmer-Heitmann)

A[X I , ••• ,

Xn]-module,

polynomials. ring

and

But we

Thus PAp = QB

Ap/PA p -+ BQ/PA

p

f

the conductor

Let

{C

Since

B is

a

malization

C

t

} be the finite

+'BB.

Ais seminormal.

Let A

o

rank one projective

set

of coefficients occup-

let B be the subring of A generated by c

finitely generated

finite in tegral closure

a

then P is determined by an idempotent matrix

I , ... ,

in the matrix and

.If P i s

13,

2-algebra,

and hence

= +'!lB.

It

it

from A

and A o

and

o

,... ,c

t'

is noetherian with

the same is

true of the seminor-

is possible to see that Aoc;;,A,since

The matrix which defines P also defines a

projective Po

r ark

i

satisfies the hypotheses

therefore P is extended

of theorem I,

from A.

soP

o

is of

is extended

407 III - Seminormality and projective modules over polynomial rings. Theorem 6 allied with the methodology of Lequain-Simis now permits

to extend the Serre conjecture a bit farther,

US

and in a way

which brings seminormality into play. Theorem 7. [7, (i) (ii)

A

Thm.2J Let A be an integral domain such that

is a Prufer domain,

Spec

(A ) p

and

is finite for all P6 Spec(A).

Then finitely generated projective A [Xl"'" extended,

if and only if A is seminormal.

Sketch of proof. are extended,

modules converse tegral

-modules are

If finitely generated projective A [X]", then A is seminormal by theorem 6. To

show the

we apply the Lequain-Simis methodology to the class

domains

is clear from

A

satisfying

(ii)

that

is less easy,

(LS

(i),(ii)

.0)

but follows

and

of in-

(iii) A is seminormal.

l;. .

holds in

-

(LS.l)

It

is easy to verify.

from well-known facts

about Prufer

domains. Let us now verify (LS.3)

if Af.

e.

is quasi-local,

by generated projective A [Xl-modules are free. finite Serre

and as

By

(ii)

then finite-

Spec

(A)

is

in the proof of theorem 5 we have by the theorem of

that every projective A [X]-module has

an ideal and F free.

But A is seminormal,

the form IEDF

so by theorem 6 I

with I is exten-

ded and hence is principal. The following immediate corollary to theorem 7 seems to have been known to several authors. Corollary. finitely

If A is a one-dimensional Noetherian domain,

generated p r o j ective A [Xl""

if and only if A is seminormal.

then

XJ -modules are extended for

408 This result applies to algebraic curves. (For a precise interpretation of seminormality in algebraic curves the reader should consult Davis's ever,

enlightening paper

US}

since Pedrini

domain

A = K

isomorphism,

IV To

so A is

result.

+ Z2

-

x2

is limited

to curves,

it

YZ)

has

but NKo(A)

Pic(A

10.

(NKo(A)

Swan completes

is

refer

to Traverso's

seminormality as T-seminor-

Gilmer and Heitmann gave further

for any reduced ring A,

for Pic(A)

weaker

-+

Pic(A [XI"'"

Xn] )

to be an isomorphism.

and hence T-seminormal, with Pic(A) Thus for

reduced rings

In

to its own total quo-+

Pic(A

[xl, ..

,X

n])

T-seminormality is formally

than the condition on the Picard groups. Swan gives

the following elegant definition of seminormality.

A c c mmu t e t i.v e ring A is seminormal if whenever b, there

They

T-seminormality is a necessary

they constructed a reduced ring A equal

not surjective.

in

by redefining seminormality.

showed

ring,

exists a

A such that a 2=b and a 3=

Let us first pose be A and b 2=O.

>

the waters more

on the relation between T-seminormality and Pic.

tient

the c o k e r

with a refinement of Traverso's

US

information

addition

an

This situation has recently been resolved by

In addition to theorem 6,

condition

[xll

the picture.

is a result which muddies

in a remarkable way

that

how-

the two-dimensional affine

But with the apparently disparate circumstances

Let us henceforth ma l i ty ,

(y3

seminormal,

theorem 6 holds,

[19J

has shown that

theorem 6 provides

than it clearsthem. Swan

The result

Seminormality and Pic

be sure,

original which

[x,y,z]/

[9].)

note that A seminormal Then b 2= b 3=

0 and

so

c Eo A satisfy b 3= c 2

c. implies A reduced. there

For sup-

is an aE A with a 2=b,

409 Then which implies that It is fairly easy to see that if Q(A)

is a product of fields,

then A is seminormal if and only if it is T-seminormal.

agree in all situations hypothesized in theorem 6,

definitions

all situations of consequence vis-a-vis following

theorem of Swan

Traverso's

Thus the two

theorem. (A

Theorem 8.

r ed

[19J

the Picard group.

i;e. in

And so the

provides the ultimate version of n i Lr a d i c a l ) •

denotes A modulo its

Let A be a commutative ring.

The following are equi-

valent. I)

pic (A) ->- Pic (A

[XI'"

2)

Pic (A) ->- Pic (A [XI])

,X n])

is an isomorphism for

I

is an isomorphism.

3) Ar e d is seminormal.

v -

Related results.

Here we shall compile a list of results related to those above. Anderson tegral

[IJ

has studied the question of when,

domain

A

I

for a g r: aded in-

$ ... , the natural monomorphism Pic(A o ) + Pic(A)

is an isomorphism. He has obtained the following result. Theorem 9. For

a graded integral domain,

A is seminormal

if and only if A

Pic(A ) ->- Pic (A) O

o

is seminormal and

is an isomorphism.

This has the interesting corollary that for A seminormal with Pic(A

if and only if Pic(A) o)

->-Pic(A)

O.

an isomorphism,

a field, A is

Anderson also gives an example but neither A nor A

By using the theory of divisorial ideals, ved the next

o

o

seminormal.

[16J

has pro-

theorem.

Theorem 10. Let A be an integral domain.

Then A is integrally

closed if and only if every rank one reflexive A [X] -module is extended

410

This puts theorem 6 in greater perspective. It is an inxeresting formal is s em i n o r ma L,

consequence of theorem 8

was given in the case when Q(A)

[81

In

then so is A [x]

that

A

a direct proof of this

is absolutely flat.

That proof

result s t emmed

from the following quite general result. Theorem II. A [XJ

is

be rings.

Let A£ B

If A is

(2,3)-closed in B,

then

( 2 • 3 ) - c1 0 sed in B [X]

(In Swan's formulation of seminormality. A(2,3)-closed equivalent to

A seminormal in B [19,

Thm

so theorem

2.5J

in B is

II

shows

that "A seminormal in BI! extends to polynomial rings.) Proof. closed

in B,

and suppose also

Weshall refer butf¢A

[X]

are commutative rings with A (2,3)-

Suppose that A £ B

that A [x]

to any polynomial f(X)

is not

B [X]

(2,3)-closed

such that f2,

in B [X].

f3'-A

[x]

as a counterexample.

A minimal counter example is a counter-example f=a + .•. +a X o n such that (ii)

(i)

f

has minimal degree n

aMong all counter examples and

among all counter examples of degree n,

string of coefficients in A (i.e. every counterexample at By our assumption

counter

o

.... ,

a

f

has

is not

(2,3)-closed

initial

Note that

i_ 16A,

least has a EA since A is o

that A [x]

the longest

(2,3)-closed in B. in B [x]

, minimal

examp I e s ex i st.

Let f=a and a i

a

n

¢ A.

If

o+'

.. +anx

n

be a minimal counterexample with a

r f. A and ra

i

E A,

then rf E A [x]

o"

.,ai_IEA

. For otherwise rf

would again be a counterexample and would violate the minimality of f. The coefficient of Xi in f2 2aoai

A.

likewise,

It follows

is 2a a.+ terms o

in A and hence

from the preceding p a ra r r a p h that 2a

the coefficientof xi in f3

is 3a 2 a. o

+ terms

of

A [x]

in A and so

411

3a; f G A[X] . Now consider the_ polynomial. g(X)£B(X] defined by 2 f (X) 2 - 2a f (X) + a2 6 A[X] and x (X) f(X) - a . X g (X) 2 o 0 0 g 3 2f(X) 3 X g(X)3 f(X)3 -3a f(X)2 + 3a - a t A [X] and hence g2 o 0 0 g3e A [X] f

.

But deg g(X)< deg f(X),

+ Xg(X) : A

=

We say

[X]) a

so g(X)E. A [X]

In much the same way

show that A n r o o t

closed

r

This was done in

[8J

.

in B if A

that

theorem 11

in B implies A [X]

Recently

Hatkins

Band b £ B,

but

He

has

shown that

that for

implies

the

[21J has

same for

[12J

then

A

for

and As a n u n a [X]

for all primes p.

in B [X]. the stabili-

power

series

A n r o o t; r

or

(2,3)-closed

in B [[X]]

remark that

p-seminormality,

of Hamann ring,

we

A [[X]]

one can

studied formal

bnE A

is not n-root closed in II [[X]]

with A absolutely flat,

Finally, called

[[X]]

7

was

n-root closed

ty of such closure conditions under formation of rings.

Then

contradiction.

that A is n-root closed

implies bE A.

.

i

n

each [2J

[19J

Swan also defined a notion

integer p .

He

by proving that

t

asn

generalized result

if A is

any reduced

is A-invariant if and only if A is p-seminormal The reader

is referred to

for

a discussion

this r esu It. References

(1]

D.F. T.

Dl

Anderson,

Asanuma,

Seminormal graded rings,

to appear.

D-algebras which are D-stably equivalent to D [X],

preprint. H.

Bass,

Torsion free and

projective modules,

A.M.S.

102(1962),319_327. H.

Bass,

Algebraic K-Theory,

H.

Bass and M.P.

Murthy,

Benjamin,

N.Y.,

1968.

Grothendieck groups and Picard groups of

412

abelian group rings, Ann. of Math. 86 (1967), 16-73. [6] J. Brewer and D. Costa, Projective modules over polynomial rings,

J.

Brewer and D.

pol yn om i a I [8J

J.

sure

Costa,

J.

r in g s,

Brewer,

D.

Pure App.

J.

Alg.

13

(1978),

some non-Notherian 157-163.

Seminormality and projective modules over

AI g e bra 58

Costa and K.

in polynomial rings

(1 979),

Mc Crimmon,

208 - 21 6 • Seminormality and root

and algebraic curves,

J.

c Lo

>

Algebra 58 (1979\

217-226.

[9]

E.

Davis,

A.M.S.

[IOJ

S.

68

Endo,

Japan 15

VI]

R.

On the geometric

(1963),

16

Heitmann,

(1980),

[12J E.

Hamann,

OJ)

Lequain and A.

Lindel,

C.

Pedrini,

On a On

J. Qu e r r e , 64 (1980),

On pic R

J.

Pure App.

the K

o

for R seminormal,

R

Alg.

18

(1980),

Spinger Lecture Notes nO 342,

Seminormality,

[19] R.G.

Swan,

On seminormality,

1973,

Pure

,X

n]

preprint. in

92-108.

over polynomial rings,

J.Algebra

Inv.

Math.

to appear. p r e p r i n t. •

Seminormality and Picard group,

(1970),

J.

165-172.

Ld e a ux divisoriels d'un anneau de p o l y n Sm e s , 270-284.

Traverso,

Soc.

(X] , J. Algebra 35 (1975) ,1-16.

of certain polynomial extensions,

Rush,

24

[X]

conjecture of Quillen and Suslin,

[18] D.E.

Pisa,

Math.

Proj ective modules over R [Xl'"

D. Quillen, Projective modules 36 (1976), 167-171.

C.

J.

251-264.

Simis,

Prufer domain,

H.

over polynomial rings,

The R-invariance of

K-theory II,

[161

Proc.

339-352.

Gilmer and R.

R a

seminormality,

(1978),1-5. Projective modules

Ap p , Alg.

Y.

interpretation of

Ann.

585-595.

[2l] J. Watkins, Root and integral closure for

R [[xn

'

preprint.

Scuola Norm. Sup.

On the maximal number of

elements in

ideals of noetherian rings By Jan-Erik Bjork

Introduction Let

R be a commutative noetherian ring and let ot be an ideal of

R. In

[I) G. Valla introduced the following concept: 0.1 Definition. A subset fat" .ad every h:mogeneous form in coefficients in I/,. •

R[x ... l

of

o:

is called on-independent if

vanishing at

(a ... a l

k)

has all its

Thi s leads to: 0.2 Definition. Put

sup(/Il.)

sup [k

0

3 a k-tuple of al-independent

elements} Following a recent work by N.V. Trung we are going to determine for each given ideal ell.

of a commutative noetherian ring

sup(tt)

R. So except for some

minor modifications the proof of the Main Theorem below is contained in [5]. Before it can be announced we need some notations. They are introduced in Section

2 below, while the proof of the Main Theorem is carried out in the

subsequent sections. 0.3 Remark. In [3, p.35] it was proved that the following inequalities hold for every ideal tT. : grade (Dz..) the ideal

sup(tl.)

1:

ht(t1l.)

where

ht

denotes the height of

or equivalently the Krull dimension of the R-module

414

I. Statement of the Main Theorem Let us first observe that

f

general, if ring

decreases under localisations. In

is a prime ideal of

R

which contains (Jl. R f

Ot generates the ideal

where

Ot then we get the local

and wi th these notations we

have. sup (OLR" )

I. I Lemma. sup (DZ..) Proof. Let

fa) .. .

r sDlxO(

akl

R" [xI"

in

be

It is then sufficient to prove that

IJI.R" -independent. To show this we consider some hanogeneous form

they are also

lilt 1= 0( I + ... + 0( k =

.xk]- so here ()(= (o() •• 'O(k) are multi-indices and m for some fixed integer m.

t

Suppose now that

l)(

a

so that all the coefficients Since

sOle Ol

and then

lit.

[a) ... a

R,.

tsc:(

are

(1f.

k} for all

ex

e

R, .

in the ring

= 0

and in addition

R

Then we can find

r

Ol

(ts",)a

-independent it follows that

= 0

tso( e

(Jl.

in for all

elL

are

which proves that

-independent. 1.2 The

cot1!pletion

know that that rJi. R

of the local ring

Rp

is faithfully flat over

f -independent elements are

R

rt

can also be introduced. We

and using this fact it is easily seen

Rp

!L

-independent which together with

Lemma I.) gives.

1.3 Lemma.

hold for all f2dl.

SUP(t'tRp

This inequality will ve used to prove the Main Theorem. Before it is announced we shall need another definition. 1.4 The ideals

define the sets

). Given an ideal

U.

U. (£)

1-

1-

for each

t z I

Of course, if

then

I,

of

some noetherian ring

S

we

by :

0

appears as an ideal of

S

and considered

as an S-module it has a Krull dimension. This exp lains the definition of the sets

U i (.« ) • I t is easily seen that

they increase. Finally, if

U. (I/. ) 1-

Kr.dim(S) =0

1.5 Main Theorem. For a given ideal

sup (dt)

=

inf [inf

f

Ui

(bL

are ideals of

bl. of a ring

)f.bL: fJ

S

and of course

Us (J(,)

is finite then R

= S

holds.

we have

f: Ass

The proof requires several steps. In Section 2 we prove the easy part, namely

415

the inequality

, the opposite inequality is more involved and to prove it we

need several preliminary results in Section 3, while the actual construction of tt-independent elements is carried out in Section 4.

2. An upper bound of sup

)

Given an ideal

x

sup (dl. Rf»

for all

in a ring

f

in

R

we have already seen that

Ass

Theorem follows from the result below- applied to the ideals

2.1 Proposition. Let

S

sup (i7/.)

Therefore the inequali ty

be a local ring and let

I,

c

in the Main

PtR;'

S

be an ideal. Then

sup(£H:inf{i20: Proof. Let

kz O

and assume that

. We must prove that

sup(l.)

is

To do this we begin with some preliminary observations.

Uk (I:. )

Proof of Sub lemma I. Firs t, the ideal xI ••.

Now we can choose

It follows easily that Kr.dim(£wUk(A))L), and then all pairs ht(f / f)

d. We shall assume that

d

2

(pcl

f

with

e Ass (R/ IJL )

here.

For the construction we shall need Sub lemma . To each pair

ht(f/tp)

=

I

and

1

contained in

Uf :

r« roo

contains

f

that

and then

as above there exis ts a prime

and hence there exis t

r

0

i

n

so that

1

shows that

1

is not

which is outs ide

f

of the ideal

f'" so

I. I t remains only to see that we can choose

also holds. This follows because

shows that there exist infinitely many primes fESpec(R)\V(tt)

a

f"

'

we find a minimal prime divisor

ht(f /tp)

f e Spec (R) \ V(d!.)

(Jer o

and; ¢Uf:

so the defini tion of

F' 0

Given such an element

(tp, 0 f

x Ass (R »

422 dim(R)

for all

t? 1.

References

[1]

Valla, G., Elementi independenti rispetto ad un ideale. Rend. Sem. Mat. Univ. Padova 44 (1970), 339-354.

[2]

Eisenbud, D., Herrmann, M., Vogel W., Remarks on regular sequences, Nagoya Math. Journ. 67 (1977), 117-180.

[3]

Barshay, J., Generalised analytic independence, Proc. AMS

(4)

Zariski, 0., Samuel P., Commutative Algebra, vol.2

(5]

Trung, N.V. Generalised analytic independence. To appear. Preprint from

(1976), 32-36.

institute of Math. 208 D Dei Can, Hanoi.

[6]

Valla, G., Remarks on generalised analytic independence. Math. Proc. Cambridge Phil. Soc.

DIMENSIONS PURES DE

MODULES

par Danielle SALLES

INTRODUCTION. foncteur

Depuis la definition en 1961, par J.E. Roos des derives du

Fm, de nombreux auteurs (en particulier Barbara Osofsky (I) et

Christian Jensen (2))

se sont attaches a

ou les anneaux pour lesquels les derives de

les systemes projectifs 1im

s'annulent pour des entiers in-

ferieurs ou egaux a un en tier n , Un article recent de C.U. Jensen ("Dimensions cohomologiques reliees aux foncteur

lim i" a paraftre aux Proceedings du Se mi.na i-

3) ?

n

On sait

(8)

dimension pure globale. s'ils

existent ce seront donc des anneaux de grande dimension glob ale et de petite dimension faible.

(9) qu'on peut construire des anneaux

Barbara Osofsky a

de valuation ayant une aussi grande dimension globale que l'on veut. Ces anneaux

a

sont de dimension faible

1 ; ce sont donc de bons exemples d'anneaux ayant

une grande dimension pure globale. ci­dessous ces resultats : dim

P gL dim

Absol. plat

weak dim. n

n

n

n

>n

Valuation

PROPOSITION 6. Soient

quelconque et

C

A

K

P

o

Hom (F, GJ

­+

F

>

n­r l

uri anne au, G un A-module injectif, F un A-module

-+

0

une suite 8xacte Hom (P, OJ

Hom (K, GJ

Hom (P,G)

Horn (K,G)

la suite 0

est une suite sci.ndee de modules Montrons que la suite

CD

o

-+

Hom (F,G)

­+

0

428 est pure. Elle est exacte car

G

est injectif. Elle est pure si et seulement si

la suite Hom (P,G) 0 R -> Horn (K,G) 0 R -> 0

0-> Horn (F,G) 0R est exacte pour tout

A-module

R

de presentation finie (car les limites induc-

tives commutent aux suites exactes et aux produits tensoriels.). La suite

o est exacte car

-> Horn (R,K) -> Hom (R,P) -> Horn (R,F) -> 0

R

etant de presentation finie est pur projectif. La suite

o ->-Hom(Hom(R,F) est exacte car

G

,G) -+Hom(Hom(R,P) ,G) ->-Hom(Hom(R,K) ,G) ->- 0

est injectif.

On sait que

G

etant injectif et

Horn (Horn (R,F) ,G)

m

Les suites

et

G etant injectif, tion 1)5 Horn (F,G)

CD

pour tout

sont isomorphes, donc la suite

Horn (F,G)

et

Horn (P,G)

F.

CD

est pure.

sont pur-injectifs (proposi-

est pur-injectif et sous-module pur d'un pur injectif, il en

PROPOSITION 7. Soient

Hom (E, F)

etant de presentation finie, on a

Hom (F,G) 0 R

est donc facteur direct et la suite

al ox-e

R

est un

A

CD

est s c i nde e .

un anneau coherent, F

A module plat pour tout module

un E

A module injectif sous module pur d 'un

injectif· Preuve:

SaitO -> E - , P -> R

injectif. Montrons tout d'abord que

a

qu'il est plat vis 0-+ D->-

etant coherent,

-+

une sui te exacte pure au

Horn (P,F)

P

es t

est plat. II suffit de montrer

vis des modules de presentation finie. Soit donc

une presentation finie/ou D

0

Best projectif/de

C. L'anneau

A

est de presentation finie. II nous suffit de montrer que la

suite

o

-+

Horn (P,F) 0 D

est exacte, soit encore, puisque

Horn (P,F) 0 B -+ Hom (P,F) 0 C ->- n

D, B et C

sont de presentation finie et que

F

429 est injeetif, que la suite isomorphe :

o --

Hom (Hom(D,P) ,F)

Hom (Hom (B,P) ,F) -- Hom (Hom (C,P) ,F) -- 0

--+

est exaete. Le module

o

F

etant injeetif, il suffit de montrer que

--+

Hom (C,P)

Hom (B,P)

--+

est exaete;ee qui est verifie ear Nous avons vu (prop. 4)

Hom (A,P)

--+

0

--+

E

--+

0

Hom (P,F)

Pest injeetif que si

--+

P

--+

R

--+

0

--+

0

est done plat. est une suite

pure, la suite

o --+ est seindee, done

Hom (R,F)

Hom (E,F)

--+

et

Rappelons qu'un module Ext! (M,F)

=

0

F

Fest dit

A

FP

sont plats. FP

injeetif s'il verifie

M de presentation finie.

un anne au coherent,

son enve l oppe injective, al.o re

Preuve : Les modules

Hom (E,F)

--+

Hom (R,F)

pour tout module

COROLLAIRE 8. Soient tif,

Hom (P,F)

Hom (E, F)

E

un

A-module

FP injec-

est plat.

injeetifs sont purs dans taus les modules qui

les eontiennent.

PROPOSITION 9. Soient

A

un anne au,

F

un

A-module pur-injectif,

G uri

A-module injectif, al.ove :

e

E}

module pur de type fini

E

V

(Hom (F, G)

2) Si,

isomorphe

a

est isomorphe

a

Hom (Hom (E,F},G)

pour tout sous

d'un module de presentation finie.

de plus, I 'anneau

A

est coherent, al ore

Torn (Hom (F, G),E)

est

Hom (Extn(E,F),G).

Preuve : I) Rappelons que eet isomorphisme est toujours verifie quand

eonque et

E

de presentation finie (6). Soit

0

--+

E

P

--+

F

R--+o

est quelune

430 suite exacte pure nie. Puisque

E

OU

est l'injection canonique et

est de type fini,

P

est de presentation fi-

R

est de presentation finie.

-->

Hom (P,F) -+ Hom (E,F) -+ 0

Fest

pur injectif, la suite

o est exacte et

G

-+ Hom (R,F)

etant injectif, 1a suite:

O-+Hom(Hom(E,F) ,G) est exacte et isomorphe

o

--+

a

Hom(Hom(P,F) ,G)

Hom(Hom(R,F) ,G)

0

la suite exacte :

Hom (F, G) 0 E --+ Hom (F, G) 0 P

Hom (F, G) 0 R

0

car les deux derniers termes des deux suites sont isomorphes. On a donc : Hom (Hom (E,F),G)

Hom (F,G) 0 E.

2) On termine comme dans (4) en prenant une resolution projective de et en calculant l'homo1ogie des complexes induits. En effet tion tinie

E

est de presenta-

comme sous-module d'un module de presentation finie car

A

est co-

herent.

PROPOSITION 10.

Soient

A

anne au,

W1

A-module pur projectit; alors pour tout n n n Pext (E g F,G) et Pext (E, Hom (F,G))

Preuve

.

E et: G deux A-modules, F un

JV

les

A-modules

sont isomorphes.

Soient

une resolution pure injective du module ...

--+ L

P

...

G,

et

-+

une resolution pure projective du module E. Hom (L 0F,Q ) isomorphe P q

Considerons Ie bi-complexe

a

Hom (L ,Hom (F,Q )). P q

Calcul des suites spectra1es convergentes associees 1) Cal cuI de l'homo1ogie quand Pext P (E, Hom (F,

q

Qq ))

est fixe

a

ce bi-complexe on obtient

E

431

or

Q q done:

etant pur injectif,

est pur-injectif (prop. I) il reste

Hom (F,Q ) q

Hom (E, Hom (F,Q ))

Hom (E 0 F, Q ) q

q

dont l'homologie en

q

est

Pext

q

(E @ F,G).

2) Cal cui de l'homologie quand

Pext or

L

p

q

(L

p

0 F,G)

P

etant pur-projectif ainsi que

(prop. 2)

F,

L

dont l'homologie en au bi-complexe

p

pest

0 F,G)

@F

p

donc on obtient Hom (L

on obtient

est fixe

est un module pur-projectif

Hom (L , Hom (F,G)) P

Pext P (E, Hom (F,G)). Les suites spectrales associees

Hom (L

@ F,Qq) degenerent donc en les isomorphismes p n n Pext (E 0 F,G) Pext (E, Hom (F,G)).

COROLLAIRE 10. Bis. Soient

F un

/i-modul.e pur projectif et

G un

A-

moduZe aZors p, i dim Hom (F, G)

THEOREME 11. Soient

A

pure-projective egale

p. i

un anneau,

a s,

E

dim G.

G

un

un A-module,

un

F

/c-modul.e

A-module plat de dimension pure

projective finie r. Alors et 2) pp. dim E Preuve

@

F

,< r-re,

Par recurrence sur

une resolution pure injective de -+L

G

s

a) Faisons

s

=

I.

et

P

une resolution projective de E (elle est pure projective car (prop. 3).

Soient

E

est plat)

de

432 F

etant de dimension pure projective egale

a

il existe une suite

exacte pure :

CD sont pur-projectifs. Appliquons

a

cette suite exacte Ie foncteur

F

o

et F Pour

L

p

il vient I

0-+ Hom(F,G) -+ Hom(Fo,G) car

Hom (.,G)

Hom(FlG) -+ Pext (F,G) -+ 0

sont pur-projectifs.

l p

fixe, appliquons a cette suite exacte Ie foncteur

Hom(L ,.). p

etant projectif on obtient la suite exacte : 0-+ Hom(L ,Hom(F,G»

-+ Hom(L ,Horn(F ,G»

p

p

0

-+ .• ,

1

Hom(L ,Hom(Fj,G»-+ Hom(L ,Pext (F,G»-+ 0 p

p

CD

Reprenons la suite exacte On

et appliquons lui Ie foncteur (L

obtient :

P

@.) .

0-+ L @ F) -+ L e F -+ L @ F -+ O. p 0 P P Appliquons

a

cette suite exacte pure Ie foncteur

Hom (. ,G)

on obtient

la suite exacte :

o

(3)

-+

Hom (L

-+ Hom (L car

L

P

e F

0

et

P

p

@ F,G) @

-+

Hom (L

P

@ F ,G) -+ 0

I

Fl,G) -+Pext (L @F,G) - - 0 p

sont pur-projectifs.

Les isomorphismes des 3 premiers termes des suites exactes prolongent aux derives ; on a Hom (L , Pext

l

(F,G»

p

Pext ) (L

P

"" F, G) .

'I

car

i

(L

F

p

@ F,G) = 0

Pext P (E, Hom (F,G»

est

p

0 F,G) -

et

Hom (L , Pextl(F,G» p

pest:

Recapitulation

l

(F,G».

Les termes de la suite spectrale de ler terme

q

des que

>

1

et on a :

"E Pq

Pext P (E, Pextq(F,G»

"E Pq

0

2

2

si

quand

est de dimension

est fixe sont donc

Pext P (E, Pext

sont reduits a 0

(L

@ F,G)

P

pextl(L dont l'homologie en

1

I.

Les termes restants quand

dont l'homologie en

Pext

(F,G»

=> Pextn(E @ F,G)

434 Remarquons que puisque

E

est plat, il revient au meme d'ecrire

"E Pq = Ext P (E, Pext

q

2

En particulier, si a lorsque

E

(F,G))

Pext

n

(E 0 F,G).

est de dimension pure projective egale a

r

on

n = r+2 Pllxt

r+1

(E, Pext

1

pext r+ 2 (E, Horn PExt Done si

Pext F

r+2

r

(E, Pext

o

(E 0 F,G)

(F,G))

}

0

(F,G))

0

2(F,G))

0

on a aussi pour tout

i r+2-i Pllxt (E,Pext (F,G))=O

pour tout G. On obtient ainsi et

est un module de dimension pure projective egale a

module plat de dimension pure projective finie pure projective au plus egale a

r+1

p.p. dim E 0 F

i,

r

alors

E 0 F

E

un

est de dimension

c'est-a-dire p.p. dim E+I.

La suite spectrale precedente mon tre de plus que Pext

r

(E, Pext

1

(il suffit de remarquer que lorsque p = r q

et est egale

(F ,G)) n = r+l

b) Supposons qu'elle est uraie en

Pext

Soit

F

r

(E, Pext

s- 1

s-I

(H,G))

o

n'est non nulle qu'en

z:

(s>I), alors : Pext

r+ s- I

-+ F

alors

-+F

s

H

(E 0 H,G)

®

s-l.

un module de dimension pure projective

une resolution pure projective de F. Appelons -+ F -+ 0

(E 0 F ,G)

s = r,

H de dimension pure-projective

o

F

r+l

EPq 2

a

La proposition est done vraie en

pour tout module

Pext

o

s

et soit

-+F-+O

Ie noyau du morphisme

H est de dimension pure projective (s-I) et verifie

("0

435 F

o

est pur projectif donc la suite exacte pure O-+H-+F

montre que

Pexts(F,G)

Pext

o

s- I

-+F-+O

(H,G),

O-+H@E-+F

est exacte donc

Pext

r+s

d'autre part la suite

@E.-+F@£-+O

o

(E @ F,G)

Pext

r+s-I

(E @ H,G), l'isomorphisme

devient : Pext

r

(E, Pext

S

Pext r+ s (E 0 F, G)

(F,G))

ce qui termine Ia recurrence. La technique de recurrence est la meme pour montrer que pp. dim E @ F ( pp. dim E + pp. dim F.

PROPOSITION 12. Soient A un anneau (Ga) ae./N un systeme proiject i.]: de modules plats et pUI'-injectifs dont les morphismes inteI'mediaiI'es sont sUI'jectifs.

AloI's la dimension pUI'e-injective de

Preuve

lim G a a,,-W

est infeI'ieuI'e ou egale

a 1.

Soient

une resolution pure projective de

o -+

lim G -+

-

455

gr ILx

gr d

o

X

gr

1

LX o

d:t

n+1

)

La filtration du complexe filtre separe et complet

etant decroissante, on

peut lui associer une suite spectrale de cohomologie [4, ch. XI, §.8) de la maniere X suivante. A IL nous associons le A-module a droite gradue,

(tous les elements de

A

etant de degre

0), filtre par la filtration exhaustive

et separee Ell

r

'o'p&lN

n61N

, defini par

et muni du f-morphisme differentiel pour tout cohomologie

'fE.Lx n H (.;t )

et pour tout

(A

n EIN.

2

=

0).

A(lf) = d*(lf} , n a droite de

Le A-module

du A-module differentiel gradue f i.Lt.re

:t

est muni de la

filtration quotient F

et le gradue associe

(1)

P

H (,;t)

a

gr(H(':&))

(F

P

Ker Ii +

rm t:.)

cette filtration est F

Ell

p6/N

qui est un gr A-module

=

a

P

FP +

a

Ker Ii + 1m 1 Kera + Imli

droite gradue et un A-module

a

droite bigradue.

Posons le resultat suivant evident.

Lemme 3.1 Les A-modules gradues et bigradues filtres suivants sont isomorphes H('&)::!

Ell

Ext;(M,I)

gr

H(.,l;)::!

nc/N

Ell

n f

gr Ext (M, I)

nc.1N

le deuxieme isomorphisme etant aussi un isomorphisme de

r

gr A-modules

a

droite

gradues.

Notation:

;. -p), on pose, en omettant parfois d'ecrire

les indices pour

q = n-p ,ou d*:

(2)

On aura la decomposition directe

457 zp,q

et de

k

q=-p des cocycles

(Zp , q ) k

k (B

merre

pour (Zp)

k&IN'

respectivement. Les suites

et

r

sont decroissantes et celles des cobornes

k k6IN

croissantes. On a les inclusions

p,q \6IN'

et les egalites BP,q

X p+q

IX>

U

BP,q k

la derniere, en vertu de l'egalite

BP,q

(3)

tion de

("lIm d

OD

k=o

p+l pour

k

Puisque la filtra-

est separee, on deduit enfin

r;

zp,q

(4)

co

Posons pour tout

k'lN

zp,q k

k=o et tout

p E f'I, n = p+q

0,

zp,q + F P+ 1 LX

c:

p+q

k

BP,q + F P + 1L X k p+q

k

E{

p=o

On aura alors /I)

, \>'k lOIN, 'fp SIN.

$

q=-p De meme, pour tout

p, tout

p-l zk+l

k

et tout

q, on a

ZP+l,q-l k+l p+k,q-k+l

Bk + 1

BP = 4 F P + 1.t . On pose o Preuve de b) On a par la proposition 3.2

Preuve de a)

: Evident, puisque

Ker gr A rm gr A

H(gd)

et, d'autre part,

X

(Ker gr d l)/Im gr p p+q+ P

co

Ker gr

$

n=o

1m

gr

a 0

grA.

458 ce qui prouve bJ et l'egalite

6k

Defini tion de x

E

x

,b )

° °

x E.Zp,q

: Pour tout

classe de

H(E

1

k

'

on pose

modulo

classe de d;+q+l (x)

modulo

+

J p,q

Alors

0

est bien de f i n.i , rp+k,q-k+l r p,q 0 et est une k ok k k k application differentielle de Eko Les arguments standards de modularite (2] et

0"

les proprietes enoncees au debut conduisent aux isomorphismes zp,q + F p+l L x p+q Ker $p,q '" k+l k - BP,q + p+l x F L k p+q I

Sp-k,q+k-l :! m k

Preuve de eJ

+ F P + 1 L;, p,q BP,q + F p+l L x k p,q k+l

On va ealculer le module limite EQDo Pour tout k, E k FP ;;t Alors pour p et q, p+q n z, o , done de

°

est un sous-faeteur de

F

est un sous-faeteuY de

x

p

F

Lp + q

p+l

x

° °

On considere 1a suite de sous-modules

L

p+q

BP,q.,.. BP,q,. BP,qr 1 ... 2 '" 3

Zp,qczp,qc FP =. . ....r zp,qc 3 - 2 - 1 -

dont l'image dans

gr

p

LX

p+q

LX

p+q

est la suite croissante

LX

ou