Séminaire d'Algèbre Paul Dubreil: Proceedings. Paris 1976-1977 (30ème Année). (Lecture Notes in Mathematics, 641) 354008665X, 9783540086659

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continuation on page 371

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

641

Serninaire d'Algebre Paul Dubreil Proceedings, Paris 1976-1977 (30eme Annee)

Edite par M. P. Malliavin

Springer-Verlag Berlin Heidelberg New York 1978

Editor Marie-Paule Malliavin Universite Pierre et Marie Curie 10, rue Saint Louis en Pile 75004 Paris, France

AMS Subject Classifications (1970): 12H20. 13020. 13F20, 13G05. 13H20. 14K20. 16L20. 16A02, 16A26. 16A46. 16A60. 16A62, 16A66 16A72.17B20, 18H15. 20C20, 22E20

ISBN 3-540-08665-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08665-X Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting. re-use of illustrations. broadcasting, reproduction by photocopying machine or similar means. and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use. a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.

© by Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210

Liste des Auteurs G. Almkvist p. I G. Barou p. 252 - J.C. Me Connel p. 189 - F. Couchot p. 198R. Fossum p. I-G. Krause p. 209 - L. Lesieur p. 220 A. Levy-Bruhl-Laperriere p. 163 - U. Oberst p. 112 - M. Paugam p. 298 H. Popp p. 281 - G. Procesi p. 128 - J. Querre p. 358 - H. Rahbar-Rochandel p. 339 I. Reiner p. 145 E. Wexler-Kreindler p. 235.

TABLE DES MATIERES G. ALMKVIST et R. FOSSUM Decomposition of exterior and symmetric powers of indecomposable ?/pZ-modules in characteristic p and relations to invariants U. OBERST The use of representations in the invariant theory of not necessarily reductive groups C. PROCESI Les Bases de Hodge dans la theorie des invariants I. REINER Integral representations of finite groups

112 128 145

A. LEVY-BRUHL-LAPERRIERE Spectre du de Rham Hodge sur l'espace projectif complexe

163

J.C. Mc CONNEL The global dimension of rings of differential operators F. COUCHOT Sous-modules purs et modules de type cofini G. KRAUSE Some recent developments in the theory of noetherian rings L. LESIEUR Conditions noetheriennes dans l'anneau de polynomes de Ore A{X,r,cll E. WEXLER-KREINDLER Proprietes de transfert des extensions d'Ore

189 198 209 220 235

G. BAROU Cohomologie locale des algebres enveloppantes d'Algebres de Lie nilpotentes

252

H. POPP Recent developments in the classification theory of algebraic varieties

281

M. PAUGAM Sur les invariants homologiques des anneaux locaux noetheriens un cal cuI de Ja cinquieme deflection E 5 H. RAHBAR-ROCHANDEL Relations entre la serie de Betti d'un anne au local de Gorenstein et celle de I' anne au R/Socle R

298 R 339

J. QUERRE

358

Intersectionsd'anneaux integres (II)

III

DECOMPOSITION OF EXTERIOR

AND SYMMETRIC POWERS OF

INDECOMPOSABLE Z/pZ-MODULES IN CHARACTERISTIC P AND RELATIONS TO INVARIANTS

Gert Almkvist

(Lund) and

Robert Fossum (Copenhagen/Urbana).

This survey represents the extent of our

workon the decomposition

of exterior and symmetric powers of indecomposable characteristic p.

7/pm7-modules in

the relations of these decompositions to invariant

theory. relations to combinatorial theory and suggestions for future investigation.

It is a neighborhood of a lecture given by the second

author at Seminaire d'Algebre

Paul

l'Institut Henri Poincare in January.

Dubreil

at

1977. Therefore it contains

many more results and complete details of proofs. The research was started when the second author. Griffith tried to prove that the action of ring

k[Xo, •••• xnJ. with

char k=p

together with

on the power series

gave a factorial ring of invari-

ants. Because the decomposition of the graded components could be easily calculated (see chapter 111,3). the class group could be calculated and then it was shown to be zero.

Thusit was assumed

that similar techniques could the used in general. But first the decompositions, that is the structure of the homogeneous components should be calculated. The table in 111.4 was calculated by hand (over many cups of coffee in Treno's in Urbana) and discover the general

did not help to

a letter to Almkvist

Fossum pose4 the problem of decomposition.

in late 1975.

Immediately

Almkvist

solved the problem, using the fact that the representation ring is a

2

'it-ring (which it isn' t).

But enough of the techniques of

theory can be used to push

11

-ring

through the decompositions for

In

the summer of 1976 Stanley wrote to Fossum that the decompositions seemed to involve coefficients of Gaussian polynomials. This suggested further comparisons, which resulted in the general Valby Bodega theorem which allows

the change of basis in the representation ring

and thus permits the calculation of the number of components of a given dimension that appear in a decomposition from the coefficients of the Gaussian polynomials. Further months of calculations by the first author has led to the many interesting relations centered on the Hilbert series of the ring of invariants. In what follows we give an outline of the contents of these notes, chapter by chapter. Chapter I. In this chapter the basic concepts are introduced. The indecomposable

are determined (here, as always, k

is a field of characteristic R

p > 0)

and the representation ring

is defined and studied. In our mind the most usefuld result

in this chapter is the Valby Bodega theorem (Proposition 1.1.7.) that relates decompositions to Adam's operations (see also Problem VI.3.9) in the representation ring R

Also the several isomorphic

representations are given in the second section. Chapter II. This is the chapter that contains what we need from the classical theory of representations of the symmetric group. There is a meta-theorem (Proposition 11.2.3) that relates elements in the representation ring to exact sequences, and this is a key in going from characteristic zero to characteristic

p"7 O. We discuss

'A-rings

and the various families of symmetric functions. In the last section we define and give what properties are needed of the homogeneous Gaussian polynomials. Chapter III. This is the main chapter, in which we demonstrate the decompositions of the exterior and symmetric powers of the indecomposable

Z/pZ-modules. In chapter I we defined generalized

binomial coefficients of indecomposable chapter we show : !,./(V ) n

= (

v

n) r

V

2

In this

3

and

for

0,< r+n+1

p

where

V is an indecomposable k of dimension n include a table that illustrates the decompositions.

n. We

Chapter IV. In this chapter we repeat, for the reader's benefit, the calculations that show that the rings of invariants are generally not Cohen-Macaulay. This involves calculation of a principal homogeneous bundle. Chapter V.

This chapter, the longest and most difficult, is

devoted to the study of the dimension of the homogeneous components of the rings of invariants. We first define the Hilbert series, provide some examples and begin the calculations • Then the series "for large

p" are discussed. The Hilbert series for small dimensional

representations are calculated, as well as those for large dimensional representations. In one section Fourier series and integrals are used to express these Hilbert series. Results concerning counting of partitions are obtained. And counter-examples to a conjecture of Stanley concerning the Hilbert series of factorial rings are mentioned. Chapter VI. Examples and problems conclude this survey. A list of notation used precedes the list of references. However we list here those notations that are not introduced. Always p denotes a prime integer. As almost all t:heorems are true for all primes, we do not distinguish between the even and odd prime integers. We just note here that if a theorem is not The field

k

true for

p-2, it is quite

will be understood to have characteristic

Throughout the paper we denote the cyclic group with a generator notation, if

'!lIpm'lZ.

by

V m p

, and written multiplicatively. The other

not standard, is found in the list of notation.

References to a result within m, ncIN

obvious p.

a chapter are of the form

while a reference in chapter III to result

ms n

m.n

in

chapter I is written I.m.n. The bibliography or list of references is arranged alphabetically by author and then by year of publication. References are of the form (Gauss (1777D the year of publication.

they are indicated by letters 1777 a, We would

to indicate the author and

If there are two papers in the same year

1777 b, etc.

like to thank all of those who have contributed in one

3

4

way or another to this work. A. H-B. Foxby. R.

Melin, T. Claesson, P. Griffith,

Stanley, H. Diamond, C.

Curtis and I. Reiner have

given hints and suggestions along the way. Many people have listened to various versions of some of this work and have offered tions that have been helpful. Also tisk institut

sugges-

Universitets matema-

was kind enough to invite Fossum to Copenhagen for

one year at a time when the work in this are a was most active, and therefore he was able to communicate very efficiently with Almkvist We both thank our respective university for encouragement (Lund and Illinois). Fossum has been supported during the summers by the United States National Science Foundation. And a portion of visit to

Denmark

his

was supported by the Danish Statens Naturvidenska-

belige

He appreciates this support.

Finally we wish to thank Professor M.P. Malliavin who suggested this survey and kept asking for a manuscript. Thus we had to stop finding new Hilbert series and decompositions and had to writing what we

start

know.

This material is connected to many diverse areas of mathematics .... many with which we are not familiar. For example it has been suggested, and there are many indications that it might be

true,

that

there is a close connection between these decompositions and representations of the symmetric groups in characteristic apologize to those whose results

p 70. We

we have inadvertently rediscovered

But we are also interested in learning of other work that is closely connected with these results. Table of contents O.

Introduction

I.

Indecomposable

and the representation ring

I.

Indecomposable representations and the reprentation ring.

2.

Bases for representations.

II. Representations of the symmetric group in characteristic zero. I.

Partitions, representations and symmetric functions.

2.

Schur functions.

3.

;l. -operations and

11

-rings.

4. Gaussian polynomials and symmetric functions. III. Decompositions I.

The decomposition of exterior powers.

2. The decomposition of symmetric powers. 3. The decomposition of symmetric powers of

4

V m p

5

4. Tables. IV. The geometry of the group action. I. The rings S'(V + ) are usually not Cohen-Macambly. n l 2. These ring are factorial.

3. Related results. V.

Number of invariants and Hilbert series. I. Hilbert series and Molien'stheorem.

2. The number of invariants when

p

is large.

3. Computation of the Hilbert series for 4. Fourier series and

n=l, 2, 3,4.

definite integrals ; a formula for

Ht(S·(Vn+I)YP). 5. Symmetry of the Hilbert series

a conjecture of Stanley.

VI. Examples and problems. I. Examples in small dimensions. 2.

Bertin's example.

3. Problems. VII. Notation. VIII. References Gert Almkvist (Lund/Sverige) Robert Fossum Norges grunnlovsdag 1977 "It you can't stand your analyst,

see your local algebraist" GA 1976.

I.

AND THE REPRESENTATION RING

INDECOMPOSABLE

Z/pmZ-MODULES

1. Indecomposable representations and the representation ring. A

representation of

vector space

V

over

p

k

mover

k

is a finite dimensional

together with a group homomorphism

Ppm ---')GLk(V). This is the same as to say that generated module over the group ring

p

indecomposable if it is not the direct sum of two is irreducible if there is no proper Proposition 1.1. a) the b)

If

V:fk[T]/(T-I)n k[T] V

:= n

k [T]/

(T_I)n k [T)

V

k)7 m-modules.

k)) m-r s u bm o d u Le ,

is

p

It

p

ring kl1 m';;ik(T]/(T-l)P V

is a finitely

kv m . The representation is

m

p

k[T].

indecomposable representation,

n = dim V, k is indecomposable.

c) The indecomposable

5

V m p

, and is both free and

then

6

injective as

a

klJ m-module. p

d) isomorphism) is

only irreducible

VI

a) The group ring Define and

k [T]

kvm p

m

m-module (up to

is generated as a

k-algebra by

by extending T - - - ) Since char k .. P m p m .. I, the element (T-I)P is in the kernel. Hence there is

a surjection k

p

k,

k[T]!(T-l)pm k[TJ --)klo' m • Comparing dimensions over p

yields that it is an isomorphism. c) The ring

is a local quasi-frobenius a r>

k[T]!(T-l)pm k[T)

tin k-algebra with maximal ideal generated by the image of

T-I

Hence V m : .. kV m is free (Obvious) and injective. p p d) It is clear that k-;::k[T]!(T-l) k[Tl is irreducible as a

kV m-module. Suppose p

Then the socle of a kY m-submodule. P

Soc(V) .. (dim dim

V,

V

is a finite dimensional

by definition

If

V

Soc (V)

is irreducible, then

V

V .. Soc(V). But

p

V = 1. k b) We prove slightly more than b).

kyp m-module

kV m-module. p

Hom m (k,V), is k Vp

ask." m-mo d u 1 e s • Hence

Soc(V».v.

k Soc(V) .. dim

k

=

In fact we prove that a

decomposes into as many indecomposables as

dim Soc(V). First, since kVpm is a local ring and artinian, k any cyclic module (i.e. one of the form kY is indecomposable. p is Each ideal is of the form (T_.)n and hence each V p

indecomposable. Now As

Soc (V)---->V

Soc (V)" (T-. )n-l n

V

n

is essential whenever

injective enveloppe of

V

V

n

and is one dimensional. is of finite type, the

is determined by the injective enveloppe

Soc (k» m) c---., kY m is essential and p p is injective and indecomposable, it is seen that

of its soc Le , As kYpm

VI

Suppose there is an injection maximal essential extension of V(V.)

is a direct summand of

V. V.

with respect to the property that as to say that

wnv

l

.. 0).

VIG.-> V. Let in

V(V denote the I) V. Then the claim is that

Suppose

W

wnv(VI)"

is maximal in

o.

Then the composition

(This is the same V(VI)e.-?v-->V!W

is an injection. Furthermore it is essential, since By the maximality of

V(V.)

V

Soc(V!W)£::'V

it is a surjection. Hence

V(V.)

I• is a

direct summand. As a corollary, the module if Hence

V

is indecomposable if and only

But then there is an embedding V -->k»m" E(V I). r p V = (T_I)r kV m, But (T-I) kl.? m V m p p p -r QED. 6

7

In

the last paragraph above we have used the fact that the

Yp m-representation is isomorphic to the original representation. For as Y m-modules there is an isomorphism p

k-linear dual of a

k Vpm

Hom (kV pm • k) k

and hence Hom )1 m (V. k

Homk(V.k)

p

for each

ky m-module V.

Then it follows

p

Homk (V n' k) as

that

Vn

kv m-modules. p

The representation ring of

ky m

is defined to be the free

p

abelian group on the isomorphism classes finite type. modulo the relations V

V·

It V".

[V]

=

[V)

Denote this abel ian group by

Corollary 1.2.

of

Lv')

+ p

klJ m-modules of

[v"]

p

provided

m,

is free on the elements

Rk)1 m p

VI'" •• V pm

QED

The ring structure in

RkVpm

is induced by

'k'

So

V.W ; = V 'k W. (We omit any kind of symbols to denote the classes of a representation notation

V.W

and

confusion. Likwise

"[V]

+

[W]"

in

Proposition 1.3.

V

in

ek

V

V+W

W

p

m. And we interchange freely the

for the product. This should cause no

means

V

W

$

as modules or

RkV m) , p :

the ring

RkV m p

is generated

Vp O+ 1 • Vpl+ I' Vp2+1.· ... Vpm-I+ I A proof of this proposition depends upon obtaining the decomposition of the tensor products this paper. However in [Rally (1969)]

to demonstrate the proposition.

(The history of the decomposition of to us.

e

V This is not done in m' the multiplication table below. which is found

permits us

Ve

e

V is not clear m It seems that Littlewood knew the decomposition constants.

Also Green. Srinivasan. and Rally have discussed them. That the V ii p +

generate the representation algebra is explicity mentioned

in [Srinivasan (1964)J

• See also the papers by Re n a u d , )

7

8 MULTIPLICATION TABLE k S:7p,

If

then write

s = s

p

k

+ sl

k

with

algorithm. The following decompositions hold for each I

a)

If

k ssp ,

b)

If

s

V k

p-

I I a)

k

I 'V

k4 m-I

then =

s

(sl-I) V k P+s-sl

k

s 6 P , then

If

V k

P +

I

III V

V k

(s-I) V k

i

p +S

S

P

< (p-I) pk, then

b) I f

pk "X i

: The map

induces

'll[Xo •••.• Xm_1J - - ' )

Rj.I

p

m

induced

!?.x.

isomorphism RV m p

: It is easy to check that both rings have the same rank as modules. Hence it this follows

enough to show that the map is surjective. But

(say by induction on lots of things. for example

m)

from the formulas in the mutiplication table. QED

For future reference we need a few other relations. The first of 9

10

V

these gives the basis relation beteween the s

and the

Vi' 0

s

-s

+ Po for arbitrary o (The name is after the place where the

i .(. p •

result was proved). Proposition 1.7.

RV • In

be an element in

--

f(t) -

N

-

L:.

b. J

j=-N

b

(Valby. Bodega's Theorem). Let

j

t

7J. [t,t

P

=

b . -J

N

b

o

2:

+

where the coefficients

-I

b.

j - j (Po + Po ) =

P

1 ) The integer 2) d

Proof

=

p

P

-I

are determined as follows

IJ

=1

N+I

N

p-I

- 2:.

b. J

j =1

For

1 for --p-I

cD

2:

+ 2

expand the

2P

Then

dy

{ -,

V))

d).l

It -I]

([t])

dy

b. J

function

'l1.

j

j. T_hen

for all

J

j

b.(u + u- ) J 10 I o

polynomial

consider the

]

greatest integer in

ring

o

N

L j=1

j =1

Set

in the ---

+

and

j c j}

j =1

we have

s

-s

Po + Po p -p Po + Po

while Hence

r; r: +

P =

=

p-l

= (f o (V

p

- V

+

p_ 2)

-(p-l)

fo

V

-

2

-I (Po + Po )

)

(V

p_ 1

- V

-

p-2 -(p-2) (fo + po ) •

p_ 3)

2 (V p - V I)' p-

By induction one gets the formula 2kp+t

0·8. ) for all

fo

i

with

-(2kp+t> + Po

t

= Po

+

-t

Po

os/!.t2p.

For suppose we have this formula for -I fo + fo to get 2kp+f+1 Po

=

t+l

Po

-(2kp+f+l)

+

1'0

+

fo

-(t+l)

some k,

e.

Then multiply by

2kp+e-1 -(2kp+e-l) + Po + fo

I!-l

+ Po

10

II If

,

O(e

then the two terms on the right of each side of the

equality sign are equal, by assumption. Zkp-I

= Z (k-I) P + (Zp-I).

-(Zkp+l) + u Zp - 1 +

Zkp+1 +

fo

fo

+

-(p+s)

b

+

o

p s +

.1il-

-(p-s) Po

r

N = ZKp + L b.(p

j=1

Po

J

j

with

-j + Po ) = b

0

0, then

while the right hand side is

Zp-I + - (Zp-I) -I Po Po = Po + Po

But

fo

Now write

-(Zp-I)

fo

10

-I + Po ).

e=

If

So the left hand side is

0'

for

OJ: L 'Zp.

s

since in general,

p,

Then

-I Zp-I - (Zp-I) + b l (Po + Po )+ ••. +b Zp_ 1 (Po + Po )

0

+

o

s

k S K

(where it is assumed that

+

Zp-I

I:

£=1

K

(L k=o

b

Zkp+t

(foe

)

=

b. J

rearranging terms to get

+ u

-l

10

for

0

j'7 N). We continue by

)

Using the relations e.

fo

+

+

-t

fo

P:

P

Z(V

P

V ) p-l

-

and V

p- f+ I

-

V

p- f-l

for

Ile"p-Z

and finally Zp-l -(Zp-I) Po + Fo

Vz

we can write this as (b

0

+ Z

t

k=1

b

Zk p)

K

+ 2

L

k-o

b 2 k p+ p (V p

-

+

p-I

L e= I

Vp- I) +

K

(L k=o

b Zk p+

e.

)

e

(V + I

K

t.=1

(L k=o 11

b2kp+p+e)

-V

e_l)

(V p-e+ I - Vp- f-l )

IZ where j)=

1

d Y+ 1 =

ta

for

[Cb Zk P+ Y - bZkp+)J+Z) + p-Z

CbZCk+l)p-lJ - b z ck+ I)P-clI+z)l

and

dp

tCbCZk+I)P-I

+ ZbCZk+l)p + b(Zk+l)P+I)'

What remains is to show that these are exactly the coefficients of the

series as claimed.

L

f(t)

-I

N

L j =-N

f(t)

b. J

t

j -I

N+I

t

f(t)

t

fCt)

L:.

j=-N+l -

t-

I

b J. 1 t

J

N-I =

j

b_

L

j=-CN+l)

t

N- 1

N

b

1 (t-t )

and

f or

+ b

t

N

N+l

+ = 0 N Z

•

we

,

can write this as

N+l fCt} =

g = [N+ -zpI

L.

j=-CN+l)

J

co N+l '" LC ''"'" Cb k=-g t=-CN+I) To

j

Thus

N_ 1

Now conS1. d er,

b j+ 1 t

f Ct )

+ b Since

and hence

b.

j =-N t

Now

N

(b'_l

-

J

b

J.+ 1

j•

.

,the expans10n

e_1

-

be+l)

P t Zk +

show I} we must consider the coefficients of

in this series.

t

)

Since in general

12

e)

•

t

j

for

I

j$ p-I

13 «J

=1

Therefore

r:

N

2:.

+

b.

J

P

dim

(L.

Vd»

b

+ 2

a

follows.

+ 2

a

It extends to a map 2 Po + I = 0). Hence

P

i:

b.

J

, while

)ld V

from which the formula for

b.

J

j =!

L

j =I

L lJ=!

d V VV)

V=l

p

N

b

j ))

induced by

RlJ dim);Z

show 2) we consider the augmentation

QED

Two other relations are needed. Lemma! .8.

: a) For each

Po - Po

(Po -

n

-n

b) is

1'0

l!.

unique element

)'V n

r,s

n in

with RV

p

!$n'p, the element

fpo) .

are integers such that

For a) n

-n

Po - Po

rs ,< p ,

then there in

such that

RlJ

P

•

consider the factorization -I n-r l n-3 -(n-!) {ro - Po )(Po + Po +"'+Po ). The second term is

13

)

14 just

V For b) consider the factorization n. -rs r -r r(s-I) r(s-3) -r(s-I) Po = (Po - Po )(1'0 + Po +"'+Po ). Then we get

rs Po V

V

rs

(r(s-I) -r(s-I) r(s-3) Po + Po + Po

r

+

-r(s-3)+ ) Po ....

Hence (V

- V ) + (V r(s-I)-I ds-3)+1

ds-I)+I

QED. Remark 1.9. On

-I

(Po - Po

lized.

=

) Wn

define elements

Rp

nP-n

Po - Po

. Then for

W

n

0"-

for all n

n

we have

by

V • If

W n

n

W = - V_ The statement b) above can then be generan n , etc. So for example - V W 2V - V = 2V ' W p+ I p-2 p p+ 2 p p_ 1

s/p+2, for example, then

W

This allows us to define,

for integers

If

p

divides

s

(uniquely) m,n,

W

p+2

the generelized

binomial coefficients

in

R)). P

We now digress slighty to consider the maps on the representation

vm p

algebras induced by the homomorphisms Suppose

m z- n

jJm _ _,>vn

and

p

generated by the image of

---> RV p m

R)1 n P

is the usual surjection. We

p

get a homomorphism (surjection)

n

_:>vn. p

kv m - - - ) ky n p

whose kernel

p

1.S

(T-I)p. This yields an injection

which is just

1J. [j(.0,)l.1'···' )l.n-I ,)en""

:zz. ("o,)LI , .••

RV m p

')C.m-IJ

That is, an (indecomposable) V m-module is considered as a p

,; m-module. p

More interesting is the case Y m p

F(X)

klJ n p

m< n

=

j.J

p

Let

m, Then the k

= IF

P

and we consider the injection

j)

p

map

so that

n-m

Let

F

y m p

n,m

generate

p

n

p

is given

the Frobenius map F :

)l.p is the identity. The map on k-algebras (n-m) pn-m induced is just F ()L) = X ,the

iteration of Frobenius in the sense that

14

;,In

kV m p

=

(n-m)

k[T]/(T-I)pm

th

and

15

kLJ n p

k(S]/(S-I)P

n

and

(for example by induction on RF(n-m) is given by

RF(n-m)

n-m

= sP

F(T)

n-m)

that the induced map

Rvp m

Rvp n

(X.) = 2

It is not difficult to show

if

0

n-m-l

and

RF(n-m)()C. .) =)C.

for i-n+m V m - - ) V n is The cokernel of p

RiJ n-rm p

-_>

R\f n p

RF(n-m)

J.)

p

p

,., RV m

n-m • We get

whose composition is the augmenta-

p

dim: RV n-m -> 7l. c--> Rv m • We notice, using the

tion map

p

p

polynomials in Proposition 1.6, that (in case

2. Bases

m=n-I).

representations.

For future use we record here several different methods for writing the action of

(j on V n+ 1 a) The regular representation

(in case

n+l"* p ) ,

We know that

we can take as basis the elements

.

u.

cr·ui = u i + I' the subscript to be read modulo 0

0

0

0

0

0

0

0

U 0

V m p

oSi':'pm-J pm.

0

n+J

«(J"

(CJ" _I)e.J

e j _ 1 . Hence

(q-_I)n-j+1 = (V"-J)n-(j-I)

Homa;(W(V), W(V'» that takes isomorphisms to isomorphisms. Example l.4 a) If W = a with trivial action, then th the r symmetric pouer of V. the

r

th

b)

If

w=a:

with alternating action,

exterior power of c) If

Suppose that

W=a:S

r,

and that

of

W(T)

are among these monomials.

symmetric function.

I , ••• ,

1' ...

defined

,5

n) --;> W

WeAr

m

I' ••• '

i

n

=

T:V----')V

1- - )

Hence the eigen-

Tr(W(T»

is a

7.Z

If

n"¥m,

[lSI' ... ' -e

then mJ we get

m)' Hence there is a uniquely

corresponding to

W

21

RS

r

8 r• V

r , Clearly it is a

... ,l;m)'

:zz.[lI,., •.• , li

.

r

So

of degree

Call it

under the homomorphism W

1

1

sum of monomia ls in

W(V)

Then the eigenvalues

[s 1....

W

values

"r(V),

W(V)

the group ring itself, then

is a diagonal operator with eigenvalues T'r 6t Id are the monomials

fo

then

= Sr (V),

V.

has a basis

V

W(V)

•

22 Denote this map bS sym(W

•

I

sym RS A It is seen that r• r r: sym WI + sym W so indeed it is a group homomorphism. 2

W 2)

Suppose

1

embedding

2".,

is a partition of

,I ) r

which induces a map

)( S 1 ----" S r r

RS

I

r, There is an

)( , •• lC RS I

I

- - ' ) RS r

r

(induction from the subgroup). There are also maps for each All

I

x , •• )( A

lr

-

induced by mUltiplication -

- ,> Ar•

A calculation shows that the diagram ind

_ _ _ _ _) A

r

is commutative. If

W

is the alternating representation, then

W(V)

=

as

= a r, An easy calculation is the alternating representation on SI. ' then

we observed in example 1.4. Then sym(W) shows that,

if

WI. 1

sym (ind W .. ,)(W ) = 1X r 1 .Ir Hence sym

r

1

3

11

... a

sym is a surjection. r is a bijection.

1r

= mult (symIX ••• Xsym

1

(WI X ••• )(W I 1 r 1 r

».

By Proposition 1.1. we conclude that

Theorem 1.5.

: The map sym RS _ _') A is an isomorphism of r r: r abelian groups which preserves the products that

commutes. The set

QED RS

is also a ring,

r

structure on

A

on

A

RS

r

and

r r

so sym can be used to induce a ring r Furthermore there are inner product structures that

sym

r

preserves.

22

23 There are several other ways to set the map

sym

discussed in the next section.

2. Schur functions, Let

=

I

irreducible representations.

(II, ••• ,I

minant

forall s gn

crE:S

be a partition of

r)

I'''''!;r) is alternating

=

:

.

det

I

i,j.$r =

(In paet one can define the alternating character

r

by

Z sgn

r. The Vandermonde deter-

for

that is

Sr

These will be

r•

cr

Define the polynomial I . +j­I det (i;.r­J+I ) It is clear that V('l;I'''''!;r)

r) divides

is alternating and that

... ,J;r)

in the polynomial ring

(In fact [Mitchell (1881)] cients of

vIeS 1, ...

simple proof of Since

V

and

I"'"

Sr)

VI

r

are non negative. A

are alternating, the polynomial

Vr/V

is

r. is the (unique) function

Definition 2.1. The Schur function A

has proved that coeffi-

result appears in [Evans and Isaacs (1976B

symmetric and homogeneous of degree

in

for

given by eI(i)

=

Vr(i

r)

in

laI

Since the symmetric polynomials can be written in terms of the

a

I• n

II span Ar ,each "r J II =r Let I' denote the partition

< 0 set an = 0 and set a o I. The next result relates the Schur functions to the other functions. of

r

c o n j ugate to

L,

For each

The identities are known as the Jacobi­Trudi identities.

23

24 Proposition 2.2.

II I

If

then

= r

and 1 Yet anotner way to get these functions

'X.

If

S r ------"J

lJ!

is via the map sym

i s a character and

matrix with entries from a

'!is

det-x.(aij)

( a .. )

define

The ordinary determinant is just

J.J

a1,cr(l)

'}:.«(j) r

r ,

an

r•

r )( r

••• ar,Q'(r)'

where alt is the alt alternating character, while the so-called permanent is det where

det

triv is the trivial character.

X

Now suppose of

Sr

is a character arising from a representation

W

0

s

s2 sym

r

(W) =

I

rT

det)L

s

WI

partition

is an irreducible

111=

r ,

s

r

0

r-l

r-2

s I

r-l

representation corresponding to a

then

We can put a partial order on

RS

O:Sr-module is greater than zero,

there is a

s

r-) s

a

.

trJ.V

Then [Knutson, for example} s

If

square

ItS -module

W such that

r

r

by saying that the class of that is

X

= W.

):"0

Since

if and only if RS

r

abelian group based on the indecomposables, an element if and only if, when

X. =

are the indecomposables,

2:.

[1/ =r

WI

where

then the integers

There is an inner product on ;>1. The inverse is given by Then we get the split exact sequence. 0--)

W(n,I)-- Q:Sn+1 lD(CS

25

n

It

--)

It

- - ? O.

26 Apply this to a vector space

o

V,

to get an exact sequence

->W(n,I)(V) - 7 (ltS n+ 1 i«:S

=

But a(v)

( 0,

o

Sn(V) 8

which is split exact. The map one :

Proposition 2.3. A

!

V __) Sn+ I (V ) .i. s the o b ir o u s

relation among elements

corresponds with

r

a

split

2Z[(rl)-I]

sequence

lli

positive

Z2«r!)

-I

]Sr-modules

and conversely.

As another example, consider the relation r

j

L

o

(for

r .,. I) ,

=0

S.x S . Since a. h . sym (Ind J r-J alt l!l triv), where alt J r-J r Sr is the alternating and triv is the trivial representation respecin

Ar.

tIvely. Then we get the split exact sequence

o -_>alt

-_,>InGl

Sr_I x SI 5 r _ 2x 52 (alt@triv)-->IndS(alt(»triv)--') Sr r

----; Ind

SIX Sr_1 (alt GD triv) S r

Apply this to a free

2Z [(r!)

-->

-I

] -module

triv ---" 0 • V

to get a split exact

sequence

of

ZL[(rl)-I] -modules.

and the Define by

In this case it is possible to write the maps

splitting maps. d.

J

: /l,j(V) l!l Sr- j(V)-7 II.j - 1 l!l sr-j+I(V)

d. «vI'" J

.. 1\ V

•

J

)8w)

26

27 and define e by

L

. (u III v l " ' v .) r-J r-J 8

(u"

v =1

long, but straight forward

"» )

course

d

r+ 1

= 0 .. do

e

and

calculat ion, shows that d.

J

=

rId.

o

Hence the sequence splits whenever Further applications of this

Sr-j-I (V)

.. 'V r _ j

lJ vI"

o

(Of

(0

r- j e

Then

. : A j (V) 8 sr-j (V) _ _' ,.J+I (V) "'

r- J

r

is invertible.

principle

will be given in the next

section.

3.

-operations Suppose

R

-rings

in a commutative ring. A family

-operations on

R

A=

{'Ail

i f lN

of 0-

is a family of functions R _ _)R

satisfying the following

=

?io(x)

'AI (x ) = x

all

x R

all

xeR

z:.

i

I'l(x+y) =

'Aj(x)

'Ai-j(y)

all

x, yEO: R.

j=o In the formal power series ring

one can consider the formal

power series

. This

Let

tlo (R)

=

of

R

The three operations above are equivalent to :

[u:J]

I + t IR 1ft]

:

R

is a subgroup of the group of units

U,o (R)

is a group homomorphism. Sayan element )

=

0

x £R

for all

as sums of elements of

has j ::> O.

?! -rank

n

if

while

If we could wri te all elements of

R

-rank I and if the product of an element

27

28 of

'>.

';\ -rank 1 were again of p ( x y)

-rank 1, then we could

and

e efupu t e

We can formalize this.

In the ring

19

. . . t;m]

1) [t]

['1 ... ·•

tz

'Y\n

der the element

consi-

oD

IT

+S'Il'),t) =

(I

i.j=1

J

i.

[One takes finite products

TT 1

E;, "],

+

(I

J

nm

L.

t )

}) =0

that

[SI ... ·' l;m"

IT

The product

Thus

v

-')

11 ... ·,

11 , ....

Z

1)

Thus there is a well defined element in the limit]

--')

the elements

(S, 1 ) t

n

and notes that under the projections Z

py

I ,j

Pv

(1

S.

+

(s."])

Soo and so

is invariant under

J

are invariant and hence

pv(';'1)6A lI

A

there is a unique polynomial Z [SI . . . . 'S)). TJ . . . . . TyJ

pI/ .

such that

Pv

(5'1)

p

,;

Ca ICs), ••• ,a V(s),a I(1).···,ajJ(1»·

course we could express the symmetric functions.

in terms of the other bases for

We will not do this. as we do not need these

expressions for our purposes.] Definition 3.1.

Suppose

get) = I + bIt + b with

g

= ]

f(t)

2

+ ... 2t is defined by

z

+ a]t + azt + •••

are in

The

and

ii-product of

f

I!/}

f(t) II get)

v

L-

=0

This product is commutative.associative.with

I+t

as unit.

Furthermore (f(t) 19 (g(t).h(t» Thus The product

=

(f(t) 8 h(t».(g(t) I

h(t».

is a commutative ring with unit. 1( '

is also expandable in

TT

( ... ('

(I

-'

(3.8.) we get

L.

(3.9 )

II 1= 1>

29

e

I

in (3.6.). Then

k

I

30

The identity

L.

(S ,.." )

hp

is one form of the

e ( $' )

r

Irl

Cauchy Formula. Another form is given by

These formulas can be given another interpretation. Consider the ring

B

Define

'Ai:

S l..).

all

ted over

B

by

a

7J.

= .--m lim

by

'An(a

1 ,

=

I +Sit

(so

rk

5i =

Consider the subring of

:A

and all the

:;z [a l, ••• ,a m

consider the

'At(!;i)

? -ring.

is a

l -ring. Clearly

a A

B----+ B

Then

I for

B

genera-

-operations. Then this is again

= an so this ring contains l) and in fact is equal to A. Suppose we

8-product

co

n

(1

+

i, j =1

5.1 .lJ'I.t). /J

Then we get, on the one hand

and on the other, from

co

(L

L V

(3.10),

=0

e

111=»

I,

the expression

(5)

Hence (3.11. )

This is another form of Cauchy's Formula. Using the principle

in Proposition 2.3.,

these formulas can be

transformed to exact sequences. Proposition 3.12. there

free

:

!:..!l

E,F

be free

7Z[ S r (V )) ,. e S (V Hence we can write a

The element n

larger diagram of

k

n+ I). 0 -modules with exact columns and injections

n

on the left 0

0

1

sr-I (V

(2.3. )

Sr(V

.j, n+1

1

n+ l)

t t

sr (V ) n 0

)

N?

sr-I +p (V e?

N?

N?

n+

j

0

,

)

Sr+p(V

t

n+1

)

sr+p(V ) n

L 0

As a consequence of 2.1. and 2.2. we get that 41

Cok(N?)

is a free

42 -module for

n+l=p. Using this plus induction (decreasing) we get

Proposition 2.4. kJ.' -module of rank Proof.

» is a free n+1 for all nand r.

sr+p(V

'>

In diagram 2.3. we get an exact sequence of cokernels

o Since then

The Cok(Sr(V I) N? -n+ -I f r+p+n r+n } p l( n ) - ( n )

and

F]

F

FI _ _» F

F

_ _,>F

2

_ _:>

3

o.

are free (by induction starting at

2

n+lap)

is also free (again use that free is equivalent to

3 injective).

Q.E.D.

Corollary 2.5. Let

r-kp+r o

o is exact with

SrO(V F

with )

n+ l

a free

0,

Nk?

S r (Vn+I) _ _')F _ _,>0

Consequently the decomposition of Sr (V

use (11.2.4) base changed to For

Then

kv -module of rank

on the decomposition of

Proposition 2.6.

< p.

0

n+ k.

r < p , in

0

Sr(V

I) for

r

S (V n+ l ) • Proof. We go by induction on

R

0

) depends only n+1 r < p . For thus we can

I)

[p]

Gn+r,r(P

-I

r. By 11(2.4)

we have

,p) there is a split

exact sequence

r

r-I

A

(Vn+I)IllS

1

1

r-I

r (Vn+I)...,s (Vn+IHO

By corollary 11.4.8. there is an equality r r-V) (XY) ( 2 G (X,Y) Gn+V,v(X,Y) • O.

C (-d

v-o

Hence in

RtJ

we have

Therefore (_l)r Sr(V n+]) +

r-]

(-l)

v

.

Gn+l,r-V(r

_I

'f.) Gn+ V , I/ ( p

-I

,p)

= 0

by the induction hypothesis. But from the first formula we get l r-I v -I -I (_I)r Gn+r,r(P- ,p) + (I) Gn+l,r-P(P 'f) 'f)-O

L

JJ - 0

42

43 ) .. G (u- I u) as desired. n+1 n+r.r I " We note some unusual corollaries before going on.

Hence

Sr(V

Corollary 2.7. (a) Sr(V (b) Sr(V (1:1

I )

n+ l) n+ l)

.... ,....

The following isomorphisms of I\r (V

n+ r)

n S (V r+ I )

kV -modules hold

for

n+r

for

I I

Q.E.D.

p. n+ I

s r+ I

p P

Vn+ 1 'lil Sn(V 2)

Proof : We have the following equalities

r -I S (V n+ l) • Gn+r.r(p ,p) .. Then

(b ')

Remark 2.8.

is a result of

Ar (V n+ r)

.. Gn+r,n(P

(b) applied with

: This last isomorphism

-I

n

,p) .. S (V r+ I).

ral.

Q.E.D.

is well known and very

useful in characteristic zero. To see how it is obtained, consider the group

GL(2,k)

operating on k tu,vJ • A binary n (n n-I n n-Z 2 n aou + I) alu v + (2) a v + •.• +anv • 2u of all binary n-forms is a vector space of

n-form can be written So the space Sn(V )

z

dimension of

n+l. The action of

GL(Z.k)

Mu •

I)(

U

on

Sn(V

+ 't v

and

2). Mv·

GL(Z,k)

Suppose

induces an

action

r-. u+ 8 v • Then

••• n

n

n n-I r +(I)(aoQ( o+a l

+ ••• + (a o So

V 2

and

.. ao(Olu+Yv)n + n n n-r l .(aoOl +(I)alQ{ (!+ ..

on

M·

n-i l

nn-"2/" n-rl «;( n-I 0+(1)0< ol3+')u v

n v n le vn-I n 0 + ••. + an 0 ) v + ( I) a I 0 r

n-I,.(" 1 n n n-e l M(ao, .... an) .. «aoOl.+(I)a l(;( (?>+ •• ).(a o(;( o+all"S+"')"'" a lf n- I S + ••• +a ¥ n-I). n 1

In particular take

cr (a o

.

u

n

Then we have

n-2 n n-I v+ .•• +a v ) + (n) a u v + (n) a u n l 2 2 I

(u+v)n-2 v 2 + ••• +a v n ao(ufv)n + (n) a l (u+v)n-I v + (n) a n 2 I 2 a ou

n

n-I 2 n n + (I)(ao+al)u v + (2 )(a o + \ I )

43

a

1

+ a

2)u

n-2

v

2

44 n 3 3 n-3 3 n n + (3)(a o + (I)a l + (2) a v + ••• +(a + (I)a l+· •• +a n) v 2+a 3)u o

"

, the general term being n

(aj) .. a o + a + l basis of the dual space

a ••• +a .. Suppose 2+ n J Homk(S (V with 2).k) contragredient representation then gives k L:.

I t is easy to check that

Hence So

( O.

As before we consider the surjections induced by

eo

Sr(Vpm)

> Sr(Vpm_ l)

Then we get the exact sequences

0 _ _> Sr-I (V m)

e

? o

p

47

48 Proposition 3.4. of rank

as a

k» m-module. p

Proof

In this case

Sr(V m)

r

10.1

(mod

p). the modules

S

r-l

(V p m)

p

sequence above splits. Proposition 3.5. If where

and

are both free and hence injective. by 3.2. Hence the exact

r

F is a free p

-m

r

while

S

Sr(V m) : p

0 p

[

Proof : In case

Q.E.D. (mod p). then

m-module of rank

( p m-2+r) _ (p m-r l -l+r/ p) r rip

=0

rip

(mod p), the module

Sr-I(V m)

is injective

p

(V m-l) • G. Hence again the sequence preceeding

Proposition 3.4. SPlitsPand the image of submodule of

1 Sr-l(V m), being a p

G. splits to give the result.

Q.E.D.

Proposition Sr(V

1II

p -

I) :::

denotes the of a rank that can be determined. is injective, it contains the injective ). Since Sr-l(V ) _ s(r-l)/p(V m-l) • G. pm pm p the statement of the proposition follows. Q.E.D.

Proof.

: As

envelope

of

Remark 3.7.

Sr(V m) sr-Y(V

It is possible. using these types of arguments. to

show that Sr(V m t) p -

is a free r

kl.' m-module for

t+l. t+2':",P-l

Free,for

r

(mod

rand

t

in the ranges

p), and that

Sr(V m t) p -

0

t X

is a principal homogeneous

S'(V

r

W x(X) the abelian group scheme of Witt vectors of n, over OX' By [Serre 1956 • Prop. 13] , the sequences for

the Frobenius

F F-I

Wn,X

--,>0

induces an exact sequence I

F-I

I

H (X,W m, X) - ' > H (X,W m, X)

53

54 But nr+l. Note n,r that bn,r .. br,n by 111,2.7. Define

We compute this function for

1Pr ( t ) .. lim 4>r(t),

Note that

Itlnr+I). Observe that cr,_j(n) b

= -

r,n

c .(n). r,J rn+1 )

j=1

We want to compute

c . (n) r,J

The generating function for the Gaussian polynomials

(11.4.5)

gives

.., .. L

n=o

.

"\ (L c .(n)sJtn) j r,J

L n"o co

n. c .(n)t These f .(t) S are in a Sense r,J r,J r,J f .·s we can duals to the Gaussian polynomials. Once we know the r,J compute

where

f

.t o ..

b

r,n

t

n

..

f

j_1

We are going to do induction from and f

. (t)

r,J

r

to

r+2, so we compute f l ' (t) ,J

.(t).

2,J

Lemma 2.1. The function Proof.As

(l-st)(J-s the result.

f

.(t) .. t 1, J

1-t)

Corollary 2.2. The function

j -1

•

1- s.t

L

co

s

I -s-I t ..

I

t j - I

(sL s-j) we get QED QED

Lemma 2.3. The function

and hence

66

67

Proof. As s- s

co

-I

1 v 2v+1 -(2V+O = ---t (s -s )the result 2 l-t v=o

follows.

QED

Now we get the formula for larger mula. Define

f

r

by using a recursion for-

. '=f r"-J' r,j

Proposition 2.4. The recursion formula:

Set

i -

where

Then co

2

f

(I-t ) f , n r qr+'"

t-j + t

-q')

L--

j =q+ 1

f

r-2

tj oJ

QED The usefulness of this formula is limited by the difficulties arising when Lemma 2.5. Let

i- rV r=3

and

,

v change signs.

=

Then

t

j -I

and !":q+1 )t -t 3q+1 'lt t

q-t 3q+ 2 q+3

-t

3q+3

t=2

and QED

Remark 2.6. By computing

f 3,i appear in the decomposition of

we know how many components of S3(V

means that there will never be a V 2 3 S (V + n l)

for al': n , For example n+ l) in the decomposition of

•

Lemma 2.7. Let

r-4. Then

f4,i-O

if

67

i

68 and Hence W .. (l-t+t 2)/(I_t)2(I_t 2)(I_t 3) .. (l+t 3)/(I_t)(I_t 2)2(I_t 3). 4(t) QED

we could struggle in this way to compute fS,i and f 6,i but it would be very long and boring. By summing up the "tails" we can get both 1/I and 1/1 with little further work.

s

6

Define, for all U

Note that

.(t)"'

r,J

j

I

f

Ur,-J.=U r,J+ • I

recursion formula

2.4

Proposition 2.8. (a)

.(t).

, since

. --f .• Using the r,-J r,J we get the next result. if

f

Ur,o .. Ur,1 "' Wr(t). t

I -!

UtI

vi .

U

(b)

(c) As

2, the functions

E

r- 2,

•

-rv

r-2,rv

QED

an example of the usefulness of this result we compute, once

again, 1/I3(t). Since

fl,j"'t

j -I

and 1/I1(t)-(I-t)

-I

, we have

and and (l-t

2

CD

H

3(t) .. 1/I1(t) +

2-

v-I

1= CD

- (I-t)-I + (I-t)-I

v-I

The functions WS(t) and

1/I6(t)

can be computed as well. In

order to save space and to avoid boring,completely,we reader, we list the results below.

68

69 Theorem 2.9.

The following identities hold.

3 $(t) .. I+t 4 (l_t)(I_t2)2(I_t3) l+t2+3t3+3t4+5t5+4t6+6t7+6t8+4t9+5tIO+3tll+3tI2+tI3+tl5 2)(I-t 4)(I-t 6)(I-t 8) (l-t)(I-t

QED

.. fn=o

For the coefficients in

we

have the following special cases : b

I,n

..

b 2 , n .. b

3,n

2n+3+(-I)n 4

.. 2n

2+8n+9 16

.. [n+2] 2

+ _3(_I)n + -I n) ( in +(-i) 16

[(n+2)2 8

(2. 10)

8

if

[ (n+2)2 ] + I if 8 b

4,n

..

4

In

85 + 2n 3+15n2+42n + 3 ( I ) n + 2 0. n_>. 2 n +1 ) 144 72 16 9(1->')

3

2

_ I + L 2n +15n +42n 72

where>.

The explicit expressions for

b 3,n

and

=e

2U i 3

b 4,n

(needed later)

are found by solving the difference equations gotten from

$3

and

$4' The calculations are omitted. Theorem 2.11. The function $r(t) with denominator of the Proof. We go by induction from

is a rational function over

2

TI(I-t Vi). r-2

to

69

r , the cases

r-I,2

are

70 done. So assume that

f

where r-2, qj r I + J of terms of the type taql +b/ N

nation over 7t of ( I - t Vi) , s . We want to show that

f

.

way. Then we are done since summing the

is a linear combiwhere N is a product

can be written in the same f

form for ljJr(t).

. 's r,l

yields the desired

Recall the formula from the proof of 2.4. q

00

:l-

J

0< 9.

L.. p

Proof. By Proposition 4.7 'IT

lim

m"'''''

J

we have

(I+cos) 'I'

gn 'I'

slnp.p

d.p

-'IT

ee

Both the Dirichlet kernels

sin (2m+l)p.p .. 1 + 2 sinp.p

L

cos 2vp.p

v -I""

_ _1_ (1+2 "" l-t 2

the Poisson kernel

involve only even cosine terms. Hence the same is true for ( ) sin(2m+1 )p.p sinp.p gn 'I' so the term inVOlving

cos.pgn(.p) integrates to zero. Thus

'IT

- lim

nr+""

As

J

gn ('I')

sin(2m+l)p.p sinp.p d.p.

-'11"

gn(O) - 8 n ( W) . we get. from Lemma 4.8 that p-I 'V

J"

4> n (t) .. -

p

L

81

and

in

82 As

it follows that a

I

p

p-I

L

(l_e\llJt)-I) \I--k

e_

where

To finish the proof we show that

'"

¢2k(t) - ¢2k(t) \I=q \I IT (I-a e ) \I--q unity different from I and p-2q+l, we get

Noting that

'"

¢2k(t) -

I

p(J-t)

is a

th

p-

root of

n+1 +

Hence (polynomial of degree at most 2(q-k As

2(q-k)

a

p - n -

(J-t P)¢2k(t) ..

».

and as

l-t:+ + polynomial of degree 1 p(J-t)

by Proposition 1.9, and as

¢2k(t)

and

'"¢2k(t)

p - n -

I

agree modulo

tP ,

it is seen that these two pQlynomials on the right of the two expres-v sions must be the same. Hence tn(t) .. tn(t) when n is even QED Already this result takes the shape of Molien's Theorem 1.4. It is possible to formulate this more precisely . Whether or not this is accidental is unknown to us - it seems unlikely that it is. Corollary 4.11.

G

be the matrix group (over

with

p

elements generated by

o

e

(n-Z)1Ti p

82

n even

83

Then

n

(e )

-

H (S'(V

t

n+ I

)vp) .. p-I

)"

gEe det(l-gt}

This follows directly from 4.10.

QED

Using the Gaussian polynomials we can get yet another formulation

of

the result. Corollary 4.12. Let

n" 2k. Then

L

aEjJ

p p-n-I

I

p(l_t)n+1

Furthermore, with I

-

(1 0

.....

+

j-o

denotin& the number of partitions

AjJ(n,j)

II I -

1.) such that J

n

and

II I II ..

u , we ge t

f

• (a I /2 a -1/2) - p A (n , j) n+J.n· v-o k.J-vP

G

c ... J

G . ( a 1/2 ,ex. -1/2) n+J.n

and

a

Proof. These results follow from 11.4. In fact

As

(a

( Cl 1/ 2 , Cl-1/2) .. G • n+J.n

.n¥-a-n.:t1 iz1--a ---.r) i +I 2 ) ••• (a h

_n

(aL-a Z) ••• (a

# I

(where

.. G (1/2 n+j.n a.

Furthermore we get

G

n+j.n

(

a.

1/2

-- I

l-t P

.0.

-1/2) .. 0

p-n-I "0

1/2

-a

-1/2

-p .0.

if a

)

it follows that

I

a

if

-1/2)

and

p-nE;;; j

n+l.

"'-.

6

I I 4 8 18

1 I 3 6 12

....-20 32 49 73 102

141 190 252 325 414

r-

32

-s-s 94 163 268

382 582 783 1082 1417 1816 2310

94 TABLE OF

v H (S'(V

t

p-s+1

)

p)=

H (S'(V

t

p-s+1

)\lp)

(l-t)S-1 I

where Ws(t) is given below

2 3

+ t2

4

+ t

5

+ 2t

6

+ 2t 2 + t

7

+ 3t

8

+ 3t 2 + 5t

9

+ 4t

2 2 + 4 t

2

2

4

+ 3t 4 + t 6 4

+ t

6

+ 8t 4 + 4t 6 + t 8

10

+ 4t 2 + 12t 4 + 5t 6 + t 8

II

+ 5t

12

+ 5t

13

+ 6t

14

+ 6t

IS

+ 7t

2 2 2

+ 18t + 24t + 33t

4 4 4

+ 18t + 32t + 58t

6 + 8 5t + tlO 6

+ 13t

8

+ tlO

l0 l2 6 + 8 + t 33t + 6t

2 + 4 10 + t l2 43t + 94t 6 + 73t 8 + 18t 2

+ 55t

4

+ 163t

6

+ 163t

8

+ 55t

lO

+ 7t

l2

+ t

l4

Example 5, 10, To see how the second table is obtained from the first we do the calculation for

4l p - 12 (t)

-

4l p _ 12 ( t )

(l-t)11 I { p (l-t)P

I --P

l-t

By the proposition } + -I-

l-t P

I:

2r+ j -II

The polynomial under the summation is denoted by

If

P >

a

(12) 2 /4 - 36. then

cients from the table for

b

n.r

n,r

a

,t 2 r 2r oJ

WI 2(t), Then

- b • so we read off the coeffin.r to get the desired result,

94

95 5.11. It is seen that p-s

is even. If

For

s-2.4.6

s>6

and

.p_s(t) p-s

is symmetric if and only if

odd. then

Ws(t)

is not symmetric.

it is easily checked directly.

considering the case

n-2.3 •••• 6

it seems that

This cannot be demonstrated by using the fact that

and then integrating and sending not commute with the operation

p

00

t -+ t-

I

•

since the operations do (Even the wrong sign

appears if this is attempted). The validity of the formula would have the following consequence : Let

00

1/1 (t) -

n

when

Lr-o

H(r)t

r

H(r) • Ar k(r,2k)

where

n · 2k. Then H(-r) - H(r-2k-l)

if

r > 2k+1

H(-r) - 0

if

r -

and

1.2 •••• 2k.

VI. Examples and problems. In this chapter we study the examples in small dimension. we prove that

depth(S'(V

blems.

\I

4)

=

4)

and consider many of the open pro-

3

1. Examples in small dimension.

In this section we study the invariants of the operation of result

v

p

S'(V

on

VI' V2, V3• Since is trivial. we obtain the immediate

VI

v

p • S'(V I) I) Slightly more interesting is the action on V2. Proposition 1.1. In degree

S'(V

2' and an invariant

v

p

there is an invariant UlaN

of degree

and

v S'(V 2) p - k (uo.uIJ

95

p

uo·x o

such that

of

96 (Consequently

uo'u,

are algebraically independent and this is a

polynomial ring). Proof. Clearly

U

o

aX

is invariant. Also

0

p-r l . N a X,(X,+X o)(X,+2X o) ... (X,+(p-')X o ) a j!odJ(X,)

+,

is invariant. Since Sr(V V for 2) t exactly one invariant in each dimension up to

we know there is r and as u is o

p-'

invariant, we see that it is the only one. Then

Sp(V ) 2

Vp (i:l V I ' So there are two invariants that are

a

linearly independent. As

u

and

l

must span the invariants in degree It is clear that

since

T P - uoT -

u

l

are linearly independent they p.

is a system of parameters and that

a

is a separable polynomial over

0

k(uo'u ) . l

The remainder of the proof is clear, either by counting dimension or by looking at Galois extensions. Proposition 1.2. In S'(V

v 3)

p

M a

ul

such that

a

xP I

- Xp-IX 0

v

p a k [uo,u"M,Nj 3) are related by one equation

uo,u"M,N

invariants I

p-I 2 N" rao n (X 2+2rX I+r X0 )

X2I - X2 X0 S'(V

there are

The elements

+ terms of the form

i+2j .. 2p).

Proof. These elements are invariant if we take the group action to be

o(X

2)

O(X I)

a

X + 2X + X 2 I o

a

XI + X o

It is clear that Consider and

P XPx 2 0

uo,ul,V

are algebraically independent. and note that those terms involving

vanish and that

divides the result.

96

97 \I

As

=

dim S'(V ) .. dim S'(V ) p 3 3 system of parameters in S'(V

3

and as

uo,u

and N form a l they are algebraically independent

3), (which it must be by our assumption that

If the ring

k[W

irreducible, the extension xP I

up-Ix 0

I

[j!!]

\l acts on V3) p 2 ] k[W , W and T - j!!2 I 2,W 3 k[uo,ul,N,M] is normal, Consider

with

I,W 2,W 3] is normal, Hence k

- M

j!!2

E

k(uo,ul,N,M), We know that

=

0

and

X2 .. X'o

-

Xo

,

X2 o

Thu.s and so

[k(X o'XI'X 2)

Hence

k(uo,ul,N,H) .. k(X o'X I,X 2)

k(uo,ul,N,M)]';; p

As

k[ Xo'XI'X2] is integral over is normal, it follows that

\l p

k[ uo,ul'N ,H]

and this last

ring

QED

The only case where it is possible to study

S'(V

4)

is treated

in the next section.

2, Bertin's Example. In this section we study what we call Bertin's example [Bertin (1967)], Let

k

be a field with

V be the regular representation of 4 factor V Suppose 3,

char(k) '" 2

and let

2/42, with indecomposable

and

The action of a generator 0 of

is, as usual,

Proposition 2,1, The following elements are S'(V

4)

and

S'(V

3)

respectively,

97

invariants in

98

v\

:-X\

v 2:-X2(X2+X\)

- - - -2 - v 3:=X 3X1 (X 3+X\)+X2(X2+X\)

u3:=X3Xo+X2X\+X2(X2+Xo) y\ 2

2

Y2:=X3Xo+X2X\+X2X\(X2+Xo) u4:=N:-X3(X3+X2)(X3+X\)(X3+X2+X\+Xo)

v

4:=X3(X3+X2)

(X 3+X\) (X 3+X2+X\)

Y3:=X3(X3+X\)X\(X\+Xo)+X2(X2+Xo)u3 22 2 2 224 y 4 :=X 3{ X3Xo +X2X\X o (X 2+X\+Xo)+X\ (X\+X o)+X 3X\ (X 2Xo +Xt+Xo)+X o } 2 4 +X2X\(X2+X\)(X2Xt+Xo)+X\(X\+Xo) Proof, A direct calculation shows that each of the elements is invariant,

QED

Let the homomorphism S'(V -+ S'(V be denoted by 4) 3) Its kernel is the principal ideal generated by the invariant

Xo

It then follows that

x

o

S'(V )

Proposition 2,2. The ring one relation v

4

n

S'(V )Z/47J _ X S'(V )71/4Z

404

S'(V ) 'Z / 4Z . k[v ,v ' V ' V ] 1 2 3 4 3

with the

2 2 3 + (v\v • 0 + (v\v 3 2)v 3 4+v 2)

Proof. It is clear, almost by observation, that the elements v\,v

form a system of parameters in k[X\,X 2,X3]· S'(V 3) 2,v 4 Hence the algebra k[v\,v has Krull dimension 3, In fact 2,v 4] k[X\,x is integral over k[v\,v with 2,X3] 2,v 4]

.

v\ Xl -2 + X v\X 2 + v 2 = 0 2 2 -2 + -4 X + (v\+v (v\v 2)X3 + v 4 2)X 3 3

and

another proof that k[v\,v

.

0

k[v\,v has Krull dimension 3, Therefore 2,v 4) is isomorphic to a ring of polynomials in the three

2,v 4} variables v\,v

2,v4,

The relation

98

99

T

2

2

+ (v\V

3

+ (V\V a f(T) is irreducible by Eisenstein's 4+V2) 2)T criterion applied to the prime ideal (v It is also easy to 2,v 4). check that the ring k[v\,v

is normal. As

2,v 3,v4]

'" a

k[v\,v

a

k(X\,X

2,v4][T]/(f(T»

and

we see that Hence k(v\,v 2,v 3,v 4) and as

k[v),v

k[v

l,v 2,v 3,v 4)

We now write the images of

.. k[X

I

-

- ) ?/./4:r

2,X 3

k[X),x , X3] 2

is normal with

2,v3,V 4]

it follows that

-

,X2,X3)7J/47J

u \ " " " 4 ' in

these algebra generators.

'1

a

integral over it

S'(V 3)

QED

in terms of

3 vI

)'2 .. v lv 2

2 )'3 .. v\v 3 + v 2 2 5 '14 a v\v 3 + vI

U .. v 4 4 Thus the image of the algebra S'(V

3)7J/4r

2

k[ u l,u 2,u 3,u 4,y) "2"3'Y4]

2 v), v 2' vI' v lv 2' v lv 3,v lv 3

is generated by

3

in and

Let D denote this algebra. Also let

Then B [v Since

3]

dimB" 3, the ring

.. k[vl'v B

2,v 3,v 4]

is also a complete intersection, but

is not normal. Proposition 2.3. The ring of invariants ] S ' ( V ) 7J / 4 7J .. k[ u l ,u 2,u 3,u 4'YI 'Y2'Y3'Y4 4

and depth

.. 3.

99

100

Proof. The ring

k[v

"D + DV + Dv We want to show l.v 2.v 3.v 4] I 3. f is in the ring generated by u ••••• y4. We l is homogeneous and that it is of least degree not

that each invariant can suppose in

f

k!u l •••.• y4]

element

XI

Consider its image

in

S'(V

has image

4)

I

vI

in in

k[v l.v 2.v

S'(V

2

3)

The

3.v 4] while

in S·(V Thus there are X3(X3+X I)X\ + XZ(XZ+X\) has image v 3). 3 homogeneous elements do' d d in k[ u I' ••.• y 4] and a homogel• 3 neous

g

in

S'(V

f

Since term

"

d

such that the invariant

4

o

deg g < deg f. Xog

we can assume that

g E k[u ••••• y4] l a to f

can be omitted. Now apply the generator

2

• so the to get

2

o " a(f)-f " (dl+dZ(X3+X3Xo+X\+X2Xo»Xo Hence

" O.

dl +

Apply cr

" O.

dl + and then s ub s t r a c t

Hence

d

Z"

to get

to get

,

d]" O. Therefore

and so

0

k]

U

I···· 'Y4]

We now have the following data : a) depth

S· (V )71/4'1 4

=

I + depth D.

= B c S(V 3 )Z/4Z = B[ v 3] c) depthBB " depth B[ v 3] = 3 .

b) D

c D[ VI]

d) depthDB " depthBB " 3 Consider the long exact sequence of local cohomology at the irrelevant maximal ideal

o Since

.....

..... depthDB

=

of

D

Hm(B/D)

3 ..... ••• ..... HitCB/D) .....

.....

3. we get that

0 for

100

i,&3. Hence

o.

101

But B

B • D + Dv in

D

l•

BID·

hence

en

=

where

=

{f : fBCD}

is the conductor of

{fED: fv lED} 2

So we calculate Clearly the elements vI' v 2• 2 v and v are in Ct. The Ld e a.I g e n e r a t e d by lv 3 lv 3 elements is contained in the ideal vlB + v of B. As (v l.v 2.v 4) 2B is a regular sequence in B. no power of v is contained in 4 vlB + v 2B and hence no power of v is contained in the ideal gene4 rated by these elements in D. Now we show that no power of v is 4 contained in For suppose vrvI E Now deg(vrvI) = 4r + I. A general monomial in D of the form

Ct.

_e 2 _f _e _f 2 3 3 "z u 3 YI Y2 u 4

_f 4 _eS

Y3

Y4

has degree

2(e + 3(e + 4(e ' Therefore any 4+f 4)+Se S 2+f 2) 3+f 3) expression homogeneous of degree 4r+1 would have at least one of

e 3.f different from zero. But then vI would divide this 3.eS r expression in B and would imply that v E vlB + v 2B. 4 We now continue by showing that

v

4

w in

pose that there is a homogeneous

is regular on D

such that

Write

v

DIG:.. Supwv E D. 4 I

2 3 2 t w· terms in (vI.v2vI.vlv2.vlv3.vlv3)D + constant v 4

which we can do since

2

(v\ ••••• v

ideal. Then v

implies Hence Therefore

4

wv

I

E D.

E

4)

generated the irrelevant maximal

D

a contradiction,

depthDD/GL = depthDB/D > I. depthDD> 2. Since D. s·(v depth S'(V ) 2 / 4 Z >3 4

=

But

4)a/4a /(X o

) we get

(1+2t 3+t 4)/(I_t)l_t 2)2(I_t

4)

by Proposition V.I.9. and as mentioned. by [Stanley (to appear)], the ring

S·(V4)7/4Z.Which is factorial,

cannot be Gorenstein. and

hence not Cohen-Macaulay. So depth S'(V

4)Z/4Z

< dim S'(V 4) - I 101

=3

102 Remark 2.4. This method for calculating the depth of this ring clearly cannot be used by sane humans. The number of invariants of degree 5 in the Bertin example is 14. The 7 invariant generators of degree at most 4 give. 14 monomials of degree 5. But there is one relation:

u l Y3 + u 2 Y2 + u YI - O. This relation can be used to 3

show that

is not Cohen-Macaulay - for

(u!'u

2,u 3,u 4) is a maximal l,u 2,u4) regular sequence. In any case. there must then be at least one "new" is clearly a system of parameters. While

(u

invariant of degree 5. This one was found by many hours of hand calculations. Many more hours have been spent trying to find the ideal of relations. Computations can proceed as follows, + t + 3t 2 + 5t 3 + IOt 4 + 14t 5 + 22t 6 + 30t 7 +

-

+ 43t

8

+ 55t 9 + 73t

lO

+

«I_t)(I_t 2)2(I_t 3)2(I_t 4)2(I_t 5»-1 • I + t + 3t 2 + 5t 3 + IOt 4 + l0 + ••. + 15t 5 + 26t 6 + 38t 7 + 60t 8 + 85t 9 + 125t So the (number of monomials in the generators) (number of invariants) lO + ••• t 5 + 4t 6 + 8t 7 + 17t 8 + 30t 9 + 52t There is one relation of degree 5 which gets repeated in the higher degrees, so at least one takes these away by multiplying

h

t

by

t

to get excess of monomials over

3t 6 + 5t 7 + 12t 8 + ZOt 9 + 38t

invariants - relations

10

+ •••

generated by one of degree 5 Hence there are 3 relations of degree 6. There are 2 2 2 Y2+UtU3Y2+uZu3+ulu4

0 Z

4

3

2 2

=

0

+ 5t 9 + 8t

lO

YtY2+ulu3Y2+uIY4+uluZY2+u2u3+ulu3+UtUZYI+UtYt+ulu2 2

3

2 Z

3

YI+UtUZYI+UtYZ+UtU3+u2

=

0

Take these away to get the series

2t 7 + 3t

8

There are two relations of degree 7. These start with

102

+ •..

5

103 Then we get the series so there is one more relation of degree 8, and that should be enough But we have completed the main part of the computations - sufficient to show that

depth{Bertin)· 3, and we don't care to do any more.

Remark 2.5. By Corollary{2.7) of [ Fossum,Foxby, Gri ff i th and Reiten (1975») (which is due really to Hartshorne and Ogus) we conclude that there is a prime ideal P in S'{V such that ht{P) = 3 4 and for localization, • 2 This holds since

S'{V)a/4a 4

cannot be Gorenstein and Serre

condition (S3) + Factorial, would imply the hypotheses of Corollary (2.7) of [Fossum, Foxby, Griffith and Reiten (1975»)

3. Problems. In this section we list problems that appear naturally. Problem 3.1. What are the ..c!ecompositions of for the indecomposable

n>p ?

Problem 3.2. The representation rings and

and

Ar (V ) n

v p -modules Rv m p

Rv

have

A-operations,

is close enough to being a A-ring so that the decompop sitions of Ar{v ) can be accomplished. What properties, short n

of being a A-ring, but stronger than admitting RV p m

A-operations, does

enjoy? [ This was suggested by Rentschler) •

is the relation between decompositions of Prob 1e m 3. 3. Ar{V Sr{V + and repreSentations of the symmetric n), n 1) characteristic p? Problem 3.4. Compute Problem 3.5. Are the completions -

S'{V

V m

n+ I

) p

factorial? It was

this question that started us on our investigation of the decompositions. As seen in Chapter IV, the decomposition of

m) allow m p the computation of the divisor class group Cl{S'{V pm) p ) • O. It was hoped that the decomposition would be of use for the other indecomposables. As yet this hasn't helped.

103

v

104 Problem 3.6. Sr(V

n+ l)

Is there a formal relation between decompositions of

and semi-invariants of Schur [Schur. Satz 2.21] 1

Problem 3.7. Does the Hilbert series of a graded algebra give any information about its depth 1 (Partial answer - probably not because a (Hilbert) series can be the Hilbert series for a Cohen-Macaulay ring and a non-Cohen-Macau1ay ring). Problem 3.8. What is the generalization to

RV pm

of the Va1by

Bodega Theorem 1 Problem 3.9. Work out the Adam's operations for the representation ring.

(We started. but they did not fit directly into the subject

matter. The elements

+

are Adam's operations. for example).

Problem 3.10. What are the combinatorial properties of the triangles of numbers in

III.4? V m

Problem 3.11. Show that the Hilbert series Ht(S'{V ) p ) n+ l m-I +2. symmetric provided n is odd and n ;;. p Problem 3.12.

•

(See V.5.11).!.!

is not

1

v m

S'(V +1) P Cohen-Macaulay 1 In particular n v m Cohen-Macaulay 1 and S'{V m 2) p

p +

Problem 3.14. Is there a factorial local ring Cohen-Macaulay. dim A

=

5

A

with

and which satisfies Serre's

A S3

not condi-

tion 1 Problem 3.15. It is shown in Chapter III that '"• Free for r+n· p-l. Show that S r (V n+ l)

4. Final remarks.

S r (V for

) '" • Free(BV s n+ 1 r+n· p-2.

(July 1977). After the handwritten version of this

paper was completed we found that Sylvester and Franklin. a century ago. computed

for

n=I.2 .... 10

and

12

Sylvester (1973)

There it is the "counting function" of the covariants tiants") of a binary form of degree

n

•

(or "differen-

(in characteristic zero).

This (remarkable 1) coincidence will be the subject of a forthcoming paper.

104

105

Problem 3.12 has been solved by R.P. Stanley (private communication) but the result was used by Sylvester in his computations, so certainly known to him.

VII. Notation. In this chapter is listed, in order of appearance, the notation used in the manuscript, with chapter and section references : Standard notation 'Z

Integers.

N

Positive integers.

No

Non­negative integers. Field of rational,

IP (V)

real and complex numbers.

nV :

Projective variety of lines through space V. Direct sum of n copies of V.

v0 n :

Tensor product of

vn

V

An tpn

n­space over

Projective

k

th n­­

An(V) n

S (V)

a

copies of

in the vector

V.

Sn

Affine

k

n

o

k.

n­space over

k.

exterior power of V. n

Sym (V)

th : n­­

Symmetric power of

V.

T,U,V,X,Y : Indeterminates (sometimes multi­indexed). Chapter O. Vp m

=

a

'Z/prn'Z : The cyclic group of order

p

rn •

Generator of

Chapter I. Rk V p rn : Representation ring of

v p rn '

Chap ter II. Sr

The symmetric group acting on

r

letters (11,1) (II, I)

1I I

=

I I + 1 2 + •• + In

(II, I)

105

106

(II. 4) Symmetric polynomials of degree

Ar a

r

hr kr s 7

th r-

elementary symmetric polynomial

th r-

complete symmetric function.

(II.5)

th r-

monomial symmetric function.

(II.5)

(II. 4)

(II. 6)

r f

(II.4)

r

l = 2[f- ]

(11.7)

w{V) :

(II. B)

sym

{II. 9)

r:

eI

Ith

I

Partition conjugate to

I

Schur function.

(11.11) (11.11)

I

A-ring : Section 3 A-operations : Section 3 Gn,r (X,Y): Homogeneous Gaussian Polynomial.

(11.25)

Chapter IV. 'm{B) : Groups of units of

B.

(IV.7)

Chapter V. H. (A)

The Hilbert function

Ht{A)

The Hilbert series

B ---'t B G k (tJ

i=o

A

k

B is given by a

is

(it)-1 di(b)ti , tP=O •

In this situation the following assertions are equivalent (i)

« operates reductively on B

(ii)

There is a

z B

with

d(z)

If (i) and (ii) are satisfied, the element

zP

is invariant and the map

C (ZJ/

G is linearly reductive. If

G is too if and only if

X is linearly

X/G is affine ••

A special case of the second half of the preceding corollary is due to BialynickiBirula (BBJ. This corollary can also be proven by applying Serre's criterion for affinity to

X/G

since the existence of

X/G as algebraic k-scheme is known a

priori.

119

-9II - Proofs I only give outlines of the proofs. The details will appear in a forth-coming paper in the J. of Alg. The situation and notations are those from part

I.

The following two lemmas are used for reduction purposes. (2.1) Lemma: Let

kC.t be a field extension. Then

if and only if

G operates reductively on

The preceding lemma is applied with Let

N :g G be a normal k-subgroup of

(2.2) Lemma: The group operate reductively on

G B

.t

B

G • Then

operates on

B

k. fiB

if and only if

N resp. G/N

N

B.4

In characteristic zero every k-group is smooth, in positive characteristic every k-group is an extension of a smooth by a finite k-group LD-GrJ , Exp. XVII, Prop. 3.1, p. 625. Since both theorems

A and B are known for finite k-groups

(see e.g. (D-GaJ, III. 2, 6.1) the preceding lemmas permit the reduction of the proofs to the essential case of a smooth k-group over an algebraically closed field.

(2.3) Proof of theorem A : (iii) ====+ (ii)

This is a special case of the theorem

of faithfully flat descent (see e.g. CD-GaJ, 111.4 , 6.3). One has only to notice that

a

(G,B)-module

V gives rise to the G-scheme

Sp(B

$

V)

over

X. The

details are due to Voigt (VoJ. (ii) ===f. (iii)

Since the functors

are adjoint to each other and hence quasi-inverse equivalences under the assumption (ii), in particular faithfully flat. The map Xx G

B 0: (-) is faithfully exact, and hence CeB C (inj, A) : B I&C B B 0: A , induced from

X x X : (x,s)r----t (x,xs), is bijective since

120

B

.n

an algebraic closure of

operates reductively on resp.

G operates reductively on

is

-10-

G(_)

is bijective and

and

B 0 A are considered

(G,B)- modules in a suitable way.

as

(i)

G(_) in

B

is an equivalence. Here

---+ (ii), (iii) This is the difficult part of the proof. Since

is an equivalence if and only if (G,B)

But

B is a projective generator of finite type

B is obviously of finite type and by (i) projective. Hence

one need only show that for every non-zero G

V = HomG,B (B,V)

assume that

(G,B)-module

V the C-module

is non-zero too. After the reductions mentioned above I may

G is smooth and

k

is algebraically closed. By indirect proof and

noetherian induotion I may further assume that the statements (ii) and (iii) are valid for all pairs of

B, but not for

where (G,B)

runs over the non-zero G-invariant ideals

itseld. However since

an open dense G-invariant subset

G operates free on

U' of X and a morphism

braic, but not necessarily affine k-schemes such that pal G-bundle [D-GroJ, Exp. V, Th. 8.1. Since Let then Then

V be a non-empty, open,

U

¢

X one has

U

is G-invariant and non-zero since

Gv

2V

f

X • Then the ideal U

f ¢.

This implies

=

GV

f

0

=

is a princiU' and V'

p,-1 (V).

X this implies

G-invariant ideal and

rX

GV = GeV/(O :

by induction hypothesis

B (w.l.o.g.) 0

since

G(_)

C = GB over

'I?"( (0 : V))

and thus

V)V) f

V is a non-zero

The finite 0eneration of

= (X-U)red

h with

since

is a

is exact on

then

¢ = Supp(V/SV) = Supp(V) () '\?"(b) U

V' and U :

V , then by induction hypothesis

(G,

(G,B)-Mod. If, on the contrary, V

Since

V'

V be a non-zero (G,B)-module, of finite type over

0 . If

non-zero

there is

V is a principal G-bundle of affine k-schemes. By construction resp.

assumption on

Let then

p' :

X is non-empty, so are

subset of

B

p' : U' ---t V' of alge-

0

(0: V)

since

1\

(X-U)

(tr= zero set)

0 • But then again

(0: V)

is a non-zero

(G,B/(O: V)) - module. k

will be dealt with in the proof of

theorem B .Il

121

-11-

(2.4) Proof of theorem B : I shall only indicate the proof for the finite generation of C. The proof of the fact that

Y is the quotient of

X by G in the cate-

gory of k­schemes and of the other properties is inspired by that of the corresponding result for linearly reductive groups and proceeds along the same lines (see (MumJ, Th. 1.1). Assume first that theorem

G operates freely on

A are satisfied. The k­algebra

B

so that the equivalent conditions of

A is of finite presentation, the same

holds for the B­algebra

B '3: B B '3: A . Since C C B is faithfully flat this C B is a C­Algebra of finite presentation too (])..Gal, 1.3, 1.4. Thus B

implies that

is faithfully flat and of finite presentation over

C ,end of finite type over

by assumption. These data imply the finite generation of

Cover k

k

by (D­GroJ,

Exp. V, Prop. 9.1. In particular, by remark (1.6), C is of finite type over

kif G

is unipotent. For arbitrary

G and not necessarily free operation let

N

G be a normal

k­subgroup. Then

Moreover

ration over pairs

GIN) operate reductively on B (resp. NB). Hence the finite gene­

N k

of the invariant ring for the pair

(N,B) and

(G/N,N

B) .

(G,B)

follows from that for the

This argument is used three times with

N or GIN finite

or unipotent where the finite generation holds by CD­Gal, 111.2 , 6.1 ,Crespo the above argument). For there is a finite normal subgroup If

G is smooth then the 1­component

is finite. If cal of where

G then

G is smooth and connected and G/R u (G)

N such that

GIN is smooth. GIGO

GO of G is smooth and connected and (G)

denotes the unipotent radi­

is reductive. Thus one reduces the problem to the case

G is reductive. Modulo Haboush's theorem CHab) , finite generation of the

invariant ring in this case is due to Nagate

e s g , (FogJ,

rn.

5.56).11

III ­ Non­affine quotients of non­affine algebraic schemes This part is new and was not mentioned during my talk in Paris. I indicate how the theorems of the first section can be generalized to the non­affine case. The results are inspired by the corresponding theorems for linearly reductive groups, due to Mumford (MumJ, Ch.I, §4. 122

-12The following k

remain in force throughout this section. As in section I ,

denotes a field of arbitrary characteristic,

a k-group with affine algebra

A

=

A(G). Let

k

an algebraic closur9 of

neither necessarily affine nor separated. The sheaf of k-algebras on denoted by If

OX' Let

k and G

X be an algebraic k-scheme which is X is as usual

V: X x G ---+ X be an operation of G on X from the right.

X is affine and if

G operates reductively on

I shall also speak of a reductive operation of

A(X)

(compare section I) then

G Q£ X • Theorem A (1.5) of I can

then be generalized to the following result.

(3.1) Theorem: Situation as above. If the operation of

G on X is free, the fol-

lowing assertions are equivalent : p: X--+ Y with group

(i) There is a principal bundle

Y and a faithfully flat morphism pr ': X x G

p

such that

p: X x G

G, i.e. a k-scheme X and

induce an isomorphism

(v, pr ) :

X x

X x X

Y

X = \J U. of X by affine, open, G-invariant i£I l G operates reductively.

(ii) There is a covering subschemes

U. l

on which

If (i) and (ii) are satisfied, then

Y is algebraic, p

is affine and

is exact in the category of all locally ringed spaces. In particular is a universal geometric quotient of

p: X---+ Y

X by G in the sense of Mumford (Mum),

DeL 0.7 .11

Condition (ii) of the preceding theorem keeps its meaning if the operation is not necessarily free, and gives rise to the next result.

(3.2) Theorem: Situation as above. Let affine, G-invariant subschemes i E I . Let

Pi: Ui theorem B (1.7).

(i)

Vi

X=

such that

U

U be a covering of X by open, ie:I i G operates reductively on the U. ,

be the quotient of

Assume that in addition the following condition is satisfied

123

l

in the sense of

(3.3) For all i,j _ I U.nu.

=

J

1

the subset

is open in

1

1

p: X---+ Y of

Y is algebraic and

(Mum], Def. 0.7). Moreover mersive. The sets

p(U.)

Y and

are open in

1

X by G exists (cumpare

is affine and universally sub-

p

p-1 p(u.).

U. 1

1

(ii) The condition (3.3) of (i) is satisfied if for all 1

:

and

1

J

1

Then a universal categorical quotient

p.

U./G

p.(U.f'\U.).

U.

V. = U./G

1

1

1

is surjective or, equivalently, if Then the morphism

p

is I

is even a geometric quotient, i.e. if Pi

induces a bijection

the morphism U. x G (p,pr») U. xU. 1

1

u.

1

V.

1

1

from (i) is also a universal geometric quotient.O

For the next result I need the notion of G-linearized °X-module.(Mum] , Ch. I, §3 . Consider the groupoid of k-schemes

Xx Gx G

Let

y be a quasi-coherent

OX-module. A G-linearization of

is an isomorphism

0XXG-modules which satisfies the cocycle condition (X x r) * (F)

If

Y

pr*(y)

F :

of

X.

P.

X = Sp(B)

(pr

=

is affine the equivalence

1,

2)*(F) (r x G)* (F)

V..........-+ '" V between B-modules and quasi-

coherent OX-modules induces an equivalence between

(G,B)-modules and G-linearized

quasicoherent 0X-modules. This is due to the fact that a the B-module

A: V ---+ V

V is given by a comultiplication

linear w.r.t. the diagonal

Ii: B

B

(G,B)-module structure on

A , Le. by a

V G: (B G: A) ---+ V G: A = V G: • • (B B,a B,lnJ i.e. by an

Ox x F

which is semi-

B G: A -linear map

e:

A) ,

map

r

V G: (B B,A

#'

e: A)

,.,

= r* (V) ---+ V G:

B,inj

124

(B G: A)

pr* (if)

-14In particular, in the affine case a G-linearized invertible OX-module is just a (G,B)-module which is projective of rank one as B-module. V: X x G

In the non-affine case the operation A(V) : A(X) which is a G-algebra structure on of global sections of Ox-module then V

X induces a comultiplication

A(X x G)

A(X). Here

=

A(X) :

A(X) & A =

ox(X)

denotes the k-algebra

OX' Similarly, if 1. is a G-linea.rized quasi-coherent

F: \1*(1.)

pr*(1.)

induces

(G,A(X»-module structure on

[J

= 1.(X) via V

= 1.(X)

(x x G)

\1*(1.)

1.(X)

&

A=V

s

A

(see (Mum], p, 32).

= \J u.

is a covering as in theorem (3.2) then the modules V(U.) i I l l (G,A(U.»-modules and the restriction maps V(U.) are rr-linear,

If moreover are

X

l

-

i.e. A-colinear. Since

V(x)c 1T V(U.) :

-

when

If

i -

l

i

l

I

(v

=

(v I

I

is a G-submodule and

te 1!(X)

1! is an invertible Ox-module and

x(..ft) : Here

l

X is algebraic I assume I finite w.l.o.g. Then

!tx e 1 xr-x 1m 1x

t(x)

[x

EX;

let

01.

t(x)

.

(3.4) Theorem - Situation a.s above. 1et 1! be a G-linea.rized invertible Ox-module, and let

z,

[te G1!(x)

; X(..t)

rates reductively on Assume that ample and (i)

is affine (and of course G-invariant) and X(t)

G ope-

l .

X is covered by the

X(t), t

£. . (In particular then 1! is

X is quasiprojective).

The covering

X

= U [X(t) ; ..te 11 satisfies the conditions, in particular

(3.3), of theorem (3.2), hence a universal categorical quotient exists, p is affine and universally submersive and

125

p

Y is algebraic.

X---. y

-15-

=

= Sp(GA(X(t)))

Moreover

p(X(t))

and

= p-1 p(X(!)).

(ii)

X(t)

There is a unique invertible

is an open, affine subscheme of

0y-submodule

= GA(X(£)) (.tlx(,t)) G1(X(t))

=

c. 1(X(t))

=

of

p*(1)

y

with

G [A(X(.f,)) (tlx(..e))J

p* (1) (p(x(t))).

G-linearized Ox-modules.

(iii) For .(61:, is ample and

y

G1 (X) = one has is quasiprojective.H

(3.5) Main application: Let 1

-e. =-N on

V

n:;o.1

[D

yet)

p(x(t))

which is affine. Thus

be a G-linearized invertible OX-module and

GL@n(X)·,X(D) -

is affine and

G operates reductively

X(!)].

Let

be the set of "semi-stable points of X L". This is an open subscheme of Ss of X. Since X is algebraic, x (1) is covered by finitely many L , i

l.

= 1, ... ,r. Since X(J.,.) = l.

I assume w.l.o.g. that

1&N1xSS(1)

t.

l.

l.

G &N L (X) -

for all

l.

.

for the same

is a G-linearized invertible module on

N-.1 , i = 1 , ••• ,r . Then SS(1) X which satisfies the

hypothesis of the preceding theorem. Hence the universal categorical quotient

eXists, p

is affine and universally submersive and

SS(1)/G X

is quasi-projective .•

(3.6) Corollary: If the equivalent assertions of theorem (3.1) hold true, the following assertions are equivalent : (i)

Y = X/G is

(ii)

There is a G-linearized invertible OX-module 1

126

such that

X

= XS S(1).

-16(iii)

£ i- J

There is a G-linearized invertible OX-module

1

such that

X = U[X(t)

where

x(t)

is affine and

G operates reductively on

These equivalent conditions are satisfied if

G is a k-subgroup of a k-group

X and X/G denotes the homogeneous space.1I

Literature

[

BB]

[n-

Ga]

[D-Gro]

Bialynicki-Birula - On homogeneous affine spaces of linear algebraic groups, Amer. J. Math. 85 (1963), 577-582 M. Demazure and P. Gabriel - "Groupes algebriques" , Masson, Paris, North-Holland, Amsterdam, 1970 M. Demazure and A-Grothendieck - "Schemas en groupes I and II", Lecture Notes in Mathematics N° 151, 152, Springer Verlag 1970

[Fog

J

1. Fogarty - "Invariant theory" - Mathematics Lecture Notes, W.A. Benjamin,

[Hab

]

W.I. Haboush - Reductive groups are semi-reductive, Ann. Math. 102 (1975) 67-84

[Mum

]

D. Mumford - "Geometric invariant theory", Erg. d. Math. 34, Springer 1965

[Nag

J

M. Nagata - "Lectures on the fourtheenth problem of Hilbert", Tata Institute Bombay 1965

[Ses

]

C.S. Seshadri - Mumford's conjecture for Gt(2) and applications, in "Algebraic Geometry", Oxford University Press 1969

[vo

]

D. Voigt - Endliche algebraische Gruppen, Habilitationsschrift, Bonn 1975.

Inc. 1969

Manuscrit regu le 10 Janvier 1977 M. Ulrich OBERST Institut fUr Mathematik Universitat Innsbruck Innrain 52 A - 6020 Insbruck

127

LES BASES DE HODGE DANS LA THEORIE DES INVARIANTS

by

Claudio PROCESI

§I - The Grassman variety The theory of combinatorial bases in invariant theory has its origins buried in the invariant theory of last century. The main starting point is the study of the Grassman variety and more precisely the quadratic equations satisfied by its coordinates, in the canonical projective embedding. Let therefore

V be a vector space of dimension

n

over a field

in fact, as it will be clear, we can work always over the integers) and k-th exterior power. We select a basis f\

e.

/I

••.

««,

e

, ••• , e n k of 1\ V •

(i " i " ••• js

jk

We then apply the quadratic equation relative to this pair of Plucker coordinates and to the

k+l

indices

is .•. i k

to replace the product

JI

Jk

I

jl .•• js . The net result of this equation is

by a sum of similar products which are necessa-

rily lower in the lexicographical ordering. In fact: jl.:.jZ V&l- 2 by contraction on the given indices but we have also a map : V&l- 2 ----:> v lilm given tensoring (in the two given indices) by the invariant element 1lii Vlil2 corresponding to the form. The result is a map 1:'.. : V&l ----:> V&l If G is the group of the form I we have &l Theorem 5.2 - EndG(V )

is generated but the elements

1:::..

and the symmetric

group.

One can give a

(somewhat obscure) description of this ring (the Brauer-Weyl

algebra) by standard bases.

References [11

c.

De Concini, C. Procesi - A characteristic free approach to invariant theoryAdvances in r1ath. 21, 330-354

143

(1976)

-17P. Doubilet, G.C. Rota, J. Stein - On the foundation of Combinatorial theory Vol IX , pp. 185-216. Studies in Applied Mathematics Vol 53

(1974)

W.V.D. Hodge - Some enumerative results in the theory of forms, Proc Cambridge Philos. Soc. 39

(1943) - 22-30

J. Igusa - On the arithmetic normality of the Grassmann variety - Proc. Nat.

Acad. Sc. U.S.A.

40

(1954), 309-313

H. Weyl - The classical groups. Princeton Univ. Press, Princeton, N.J.,

Manuscrit

144

1946

Ie 14 Fevrier 1977

INTEGRAL REPRESENTATIONS OF FINITE GROUPS

Irving REINER

Introduction Let

G be a finite group, and

ZG

its integral group ring. By a

ZG-lattice we mean a left ZG-module which is finitely generated and projective as Z-module. A basic problem in given a group

theory of integral representations is as follows

G, classify (up to isomorphism) all ZG-lattices. It is easily seen

that every lattice is expressible as a finite direct sum of indecomposable lattices, though usually not in a unique way, since the Krull-Schmidt Theorem need not hold true for ZG-lattices. The basic problem may be split into three parts : I) For which groups

G is the number

n(ZG)

of isomophism

classes of

indecomposable ZG-lattices finite ? II) When

n(ZG)

is finite, determine a full set of indecomposable

ZG-lattices. III) When are two direct sums of indecomposable lattices isomorphic ? The solution to (I) has been known for many years (see the discussion in

[2, Chapter XI]), and is as follows Theorem - There are finitely many isomorphism classes of indecomposable ZG-lattices if and only if for each rational prime of

G are cyclic of order

p

dividing IGI, the Sylow p-subgroups

p or p2

Jacobinski t6J has generalized this result to the case of RG-lattices, where

R is the ring of algebraic integers in a number field. 145

-2Problem (II) is much harder, and its solution usually requires knowledge of ideal class groups in algebraic number fields, as well as congruence properties of units in such fields. The problem has been solved only for the following few cases i) G cyclic of prime order

iii) G metacyclic of order p

2

,

(see(2

2p , where

ii) G dihedral of order

iv) G cyclic of order

p

is prime t9J

p

pq , where , where

P

Chapter xI], or (3J,[12J)

p,q

are prime [ll]

is prime (see [14]-[16J).

To complete the list, we mention the work of Nazarova{lOJ, who solved problem (II) for the case where though

n(ZG)

G is an elementary abelian (2,2) group, even

is infinite for this case. She also treated the case where

the alternating group

G is

A

4

We turn finally to the most difficult problem (III), which is almost untouched. For cyclic groups of prime order, the solution has been known for many years (see (3.2) below) ; the problem has also been solved for case ii) above. In this article, we shall describe the solution of (II) and (III) for cyclic groups of order

p2; detailed calculations may be found in [16].

Let us recall the definition of genus : two ZG-lattices same genus (notation : M V N) if their p-adic completions

M, N are in the

M and N p

p

are

ZpG-isomorphic for each prime p dividing IGI. In trying to classify ZG-Iattices up to isomorphism, one usually begins by giving a full set of genus invariants. One must then find additional invariants which distinguish the isomorphism classes within a fixed genus. Often, these additional invariants are ideal classes of some kind. In the cases considered below, we shall find an invariant lying in some factor group of the group of units in some finite ring. Furthermore, a Legendre symbol will also appear as a possible invariant of a ZG-Iattice.

§l - Extensions of lattices Throughout, let

R denote a Dedekind ring whose quotient field

K is an

algebraic number field ; let A be an R-order in a finite dimensional semisimple K-algebra A • For each maximal ideal P of R , let denote the P-adic completion of R , and A the completion of A , etc ••. We may choose a finite nonP empty set SeA) of P's , such that A is a maximal Rp -order in for each p 146

-3-

p ¢ S(/\). (For example, when II. is an integral group ring choose for

S( /\)

a II. -lattice, let

RG , it suffices to

any set which includes all prime ideal divisors of IGl). For End/\(M)

denote its ring of

group of /\-automorphisms of

/I. -endo1llOrphisms, and

M, acting from the left on

denote the external direct sum of

n

copies of

M. We use

M

Aut,,(M) the M(n) to

M

Let us begin with a simple lemma (see (IJ or [5J) (1.1) Lemma - For

S.

].

i I , Z , let

E Ext/\I (N.,;) determine a ]. ].

Then

XI

Xz

if and only if :

M and N be 1\ -modules, and let i i 1\ -module Xi . Assume that Hom" (M I ,NZ)

for some 1\-iso1llOrphisms

(1.2) Corollary - Let let

Ii

M,N

A -modules

be

determine a

l

for some

(1.3)

Let us call

and

51

fied, and write

1\ -module

such that Xi

Then

'6:

M I

MZ '

$:

NI -;: NZ

Hom/\(M,N) - 0 . For XI

i

1,2,

Xz if and only if :

6 Aut" (M) ,

strongly equivalent when condition (1.3) is satis-

'z . The isomorphism classes of extensions of

thus in bijection with the strong equivalence classes in the orbits of

o.

under the actions of

For each maximal ideal

I

N by Mare

Extl\(N,M) , that is, with

Aut/\(M) and Aut,,(N) .

P of R , we have

The right hand expression is zero for each

P t

s ( A), since for such P we know

that

Ap is a maximal order, and thus the Ap-Iattice Np is Ap-projective. This I shows that ExtA(N,M) is a torsion R-module, whose torsion occurs only at the

primes

p in S( 1\). It follows at once that if I '" Ext,,(N,M) =

M' V M and

N' V N , then

Ext...I. cNI,M')

Indeed, we may give such an isomorphism explicitly, as follows

1\ -exact sequences

Lemma (see [13, (Z7.1))), we can find

o -")

M

M'

T

by Roiter's

0

o

,¥' ----7' N I --'-+ N __ U __ 0

147

-4in which both

T

P

and U

induces an isomorphism (J .4)

are zero for each

p

t :

,

(41, '1')

P 6 S (1\). The pair

(N,M)

then

(N' ,M')

which we shall call a standard isomorphism We wish to show that under certain mild hypotheses, the strong equivalence classes in

I

ExtA(N,M)

depend only on the genera

called an Eichler lattice if (see (13, (38.1)J). When R

EndA(K

M)

of

M and N . A

A -lattice M is

satisfies the Eichler condition over

is the ring of all algebraic integers in

Eichler lattice if and only if no simple component of definite quaternion algebra. Certainly

EndA(K

M)

R

K, M is an is a totally

M is an Eichler lattice whenever

End,,(M)

is a matrix ring over a commutative ring. The following result is established in t161:

(J .5) Theorem - Let

and let

, 12

N be Eichler

M

M'V M , N' " N Let (N,M) Then

t

such that

1\

Hom,,(M,N)

,

be a standard isomorphism as in (1.4) and let

Thus there is a bijection between the strong equivalence classes in and those in

0

I

Ext" (N,M)

(N' ,M').

This result shows that, under suitable hypotheses, there are as many isomorphism classes of extensions of

N by M as are of

N' by M' . We conjecture that

this same result holds even when HomA(M,N) f 0 , and whether or not Mana N are Eichler lattices. As an easy consequence of the above theorem, we obtain

(1.6) Corollary - Let

M and N be Eichler lattices for which

let

Mi V M, N V N, i = 1,2 , ... , r . For each i determine an extension X. of

i , let

HomA(M,N) 1.

0, and

E Ext"I(N.,M.) 1.

1.

1.

be a standard isomorphism as in (1.4), for invariants of the

'" -lattice

I 4-i4Sr • Then a full set of isomorphism

Xl \& ••• \& X

r

are as follows

148

-5i) The isomorphism classes of

Ell M.

ii) The strong equivalence class of the matrix diag (t 1 ( in

, ••• , t ( lr) )

r

under the actions of

GL(r,A) and

GL(r,r), where

A = EndJ\(N)

End"(M)

§2 - Exchange formulas the notation of §I ; by an R-Iattice we mean a finitely generated

Keep

projective R-module. Steinitz's Theorem (see (2 , Chapter IIIJ) gives the structure of R-Iattices :

Theorem - Each R-Iattice of fractionalR-ideals its

in

K . A full set of isomorphism invariants of

n , and the ideal class of the product

is called the Steinitz class of

••• GL

M).

n

+

+

M is isomorphic to an external direct sum

Mare

. (This ideal class

A special case of this theorem gives

This formula is easily generalized to the case of

II -la t t i ce s , where"

is an

R-order, and we obtain (see t13) or t17J) (2. I) Proposition - Let

L,M,N

A-lattices in the same genus. Then M Ell N

for some

L'

Now let

in the genus of

L .

M and N be arbitrary

If

the left on tension class

L Gl L'

A-lattices; the ring

X is an extension of

E Ext(N,M), and if

E End,,(M),

A-lattice which corresponds to the element

¥

p

Eo Aut

1\

p

(Mp)

for all

PES ( " )

149

acts from

then we shall denote by

)fJ co Ext(N,M).

condition (2.2)

Endll(M)

N by M corresponding to the exIf

¥

lIX

satisfies the

the

-6-

'6 X is in the same genus as X. The method of proof

then it is easily seen that

of (2. 1) then yields (see (16J) :

(2.3) Exchange Formula - Let

X and Y

N by M , and let l

satisfy condition (2.2). Then

End,,(M)

be

f\ -lattices which are extensions of

X ED llY

Similarly, we obtain (2.4) Absorption Formula - Under the above hypotheses, we have

The preceding results show at once that the Krull-Schmidt Theorem need not hold true for

1\ -lattices, and that usually "cancellation" is not possible. The

proofs of (2.1)-(2.4) are elementary, and depend only on the "Strong Approximation Theorem" in algebraic number fields. There is a much deeper version of (2.1), due originally to Roiter (181, and proved in a different manner by Jacobinski (7] (see also tI7J). Roiter's result is as follows:

(2.5) Theorem - Let

Land M be

f\ -lattices in the same genus, and let

F

be any

faithful'" A-lattice. Then L ED F

for some

§3 -

F'

in the genus of

p

F

be prime, and let

h.

z txJI (xp

=

J

e". J

M ED F'

Cyclic p-groups Let

where

';t

j -

1)

,

R. J

pj

is the cyclotomic polynomial of order is a primitive

integers in the field

pj-th root of

K. J

=

J

lover Q , so Thus

This means that no non zero element of

R.

J

R. J

Then

R.

J

';t Z

(w. '], where J

is the ring of all algebraic

is a Dedekind ring, and Steinitz's

1\ can annihilate

150

F

•

-7Theorem gives the structure of R.-lattices. J

G is a cyclic group of order

If

• For

j

=

O,I,Z,

viewed also as a

R j

Now let

[m £

o --. N

MIL. Here, L

=

is a

to classify all 2G-lattices RZ-lattices (Note that

(x

p

with the ring

' and so each Rj-module may be

M be any M

ZG

,we may identify

-

AZ-lattice, and set

I) m

hz-exact sequence

(3. I)

where

z

is a factor ring of

L =

Then there is a

p

L

----+

M ---"t N

hI-lattice, and

----+

0

N an RI-lattice. Thus, in order

M, we must classify all

AI-lattices

L, and all

N, and then determine all strong equivalence classes in HomZG(L,N)

=0

in the present case, so Corollary (j.2) applies here).

By Steinitz's Theorem, the RZ-lattice RZ-ideals. The isomorphism invariants of

N is a direct sum of fractional

N are its RZ-rank and its Steinitz

class. On the other hand, the structure of the

AI-lattice

L

is known from the

results of Diederichsen and Reiner (see (Z, Chapter XI]), and can be described in the following manner : both viewed as

Z and R j are factor rings of AI' so they may be Aj-lattices. For each fractional Rj-ideal , viewed as Aj-lattice,

we have (Z,.&) rt Z I

where

Z=

zjpz

(Z,-G: ; I)

Let

ie z ,

to the extension class

E (0(; )

brevity. It turns out that isomorphism class of

(3. Z) Theorem - Every

)

denote the extension of

and let us denote

(Z,o(;. ; I)

Z by

corresponding E (-e. )

is always in the same genus as

depends only on the ideal class of

AI-lattice

..e.

by

L

e&-

Al

,

for and that the

Then one has :

is isomorphic to an external direct sum

(a) • 'pI • +"j+'"

where the are the integers

are fractional RI-ideals. A full set of isomorphism invariants of L a,b,c

(which determine the genus of

the product

,

)

-I r) denotes an Z) Z Z' r extension of R with class (l, '7I ) Z Ell iiI , using the isomorphism Z by Z Ell "I

Here, (z

,

Similar definitions apply to the other cases in (4.1).

156

P

13We may then determine all indecomposable ZG-Iattices by calculating all lattices in the genus of each of the lattices listed in (4.1). This calculation depends on determining strong equivalence classes of matrices (set [16J for details), and we shall need some additional notation in order to state the results. Let

Uk be the group defined in (3.7) ; if

morphic image of of

u(R ) , where

R

=

RI/p R

I I . We may therefore choose a subset

R

I of representatives of the factor group

and

0 $ k "p-I, then u(R )

I Uk of u(R

I

Uk is a homo-

denotes the group of units

such that Uk is a full set I) AI It is easily seen that each u E Uk

Uk

may be chosen to satisfy the condition that

=I

u

(mod '?\), where

x

ZC.'lI1/('l\p-I), 'h I -t.:l . Likewise, U is a factor group of u( I)' and I l -p . I (mod we may pick a full set of representatives Up 1n u( such that u

R

for each

u G

=

ITp

Finally, let

(4.Z) Theorem - Let

n

0

be some fixed quadratic nonresidue mod p .

range over a full set of representatives of the

hI

ideal classes of R I, over the h ideal classes of R . The following is Z Z a full list of indecomposable ZG-Iattices (up to isomorphism) : i)

Z,..c, Ec-e.) ,;m + a n_ 1

est caracterisee par a

(mi"'"

avec

0, chaque representation

exactement une fois (voir [I] page 161).

170

"

intervenant

- 9

IT' I sR-(n-1 ,a:)

PI sHn-] ,a:) Done

P

restreinte a

sR-(n-I,a:) se decompose en somme de representations

irreductibles de poids dominant :

La mUltiplicite de la representation irreductible de dominant

m; 2]+ ••• +

dis tincts

B. Etude de

dans

n_2

pls2(n-I,a:)

est egale au nombre d'entiers

tels que

k

c'est-a-dire

2

s2(n-l,a:) de poids

k

a+j avec

a

P (a:) n- I P - (a:) est, en tant que variete differentieIIe,

L'espace projectif complexe isomorphe au quotient de

n I

SU(n,a:) par

S(U

1

x

l,a:) sous-groupe de

SU(n,«) des

matrices de Ia forme

avec

L'algebre de Lie de Xa:

= s(u l

S(U

1

N

U(n-I,a:).

x Un_l,a:), s(u j x un-I ,a:),a pour complexifiee

x un_l,a:)

n-r l

= sR-(n-I,a:)

a:

a:(

Notons que Ie centre de I'algebre de Lie

I

e

i=l

- (n-l) e nn).

ii

n-l

est

a:(

I

e.. r

i=1

i.

(n--L) e

nn

)•

On a

n-I

sMn,a:) ou

!Pa:

est Ie

=

Xa: $fa:

=

1Ja: $

{.IJ=I

a:

a:-espace vectoriel engendre par

171

n-l

e.

+

In {e.

J,n

I a:

j=1 }

U

{e

e .L nJ

.},

n,J

s

j

n-I.

-

On a

=

10 -

en-I,n-l-(n-I) e nn). La representation

adjointe de

sur

est definie par: g

Adg

--.;>

Adg(h)

[g,h]

gh - hg

d'oil ici e

[ek,Q,

,

J ek,n

j n]

e .] nJ

n-Z

[ 1:

n-Z [

1:

n-I

1:

i=1

1:

i=1

.

e ..

-

(n-i l ) e

e ..

-

(n-i l ) e

n-] [

n,)I, n-Z

( 1:

=)1,. 1, i+l)' e j n] J

A. (e .. - e.

i=1

[

-

Ai (e i i

i=]

e

J

On voit que

-

) , e .] nJ

nn

nn

n e. In

,

- n e nj '

e .] nJ

agit sur

e. In

)1,. J

, e. ] In

Ai (e i i - e i+ 1 ,i+l»

i=1

en une representation somme directe de

deux representations irreductibles e.

de poids dominant

e

de poids dominant

J,n

Vz

sur -)I,.

poids sont

e

Vz

a Dz'

n

n,j

)1,1+" .+ )l,i_1 + \+1+" .+ )I,n-I

On voit que tion par

e

,

e n-I,n-I -(n-l) e nn )

V

et sur

z

par la multiplication par

I

(les autres poids sont

)1,1+"'+ )l,n_Z(les autres i

n-Z).

agit sur

Vj

-no De plus

par la multiplica-

-* 11

1

est egale

On a done

PROPOSITION 11.5

Si

e.

In

et

vZ =

e . ; la representation nJ -

se decompose en somme direete de deux representations qui sont les suivantes :

172

0

PI et DZ

- II -

agit sur minant

£1

et sur

V 2

VI

comme la representation irreductible de poids do-

comme la representation irreductible de poids dominant

.(e j j+ •.. + en_l,n_I-(n-l) e nn) agit sur comme

VI

comme

n Id

et sur

V2

-n Id.

C. Spectre de En utilisant les resultats du theoreme 1.8 ; nous sommes amenes les representations irreductibles contient l'une des

W ..

p

de

telles que

Or l'algebre de Lie

a

p[s(u(l)

rechercher x

u(n-I))

est la complexifiee de

Dans ce cas, il y a bijection entre les representations de un espace vectoriel complexe et les representations de

dans

dans ce meme es-

pace vectoriel considere comme espace vectoriel reel (voir [9], VIII 9). II nous suffit done de rechercher les representations l'une des

de

p

iii'

PROPOSITION 11.6. Une representation irreductible

a

lorsqu'on la restreint

!

qui contiennent

p

et contient

e l 1+"'+ en_l,n_j-(n-I) e nn

de

n

n 1d

3, contient

lorsqu'on la restreint

si et seulement si son poids dominant est de la

forme k n £j

k E fN*

*

kErn. H

Dans Ie premier cas la multiplicite de

p

dimension de l'espace de la representation de xieme cas la multiplicite de

pest

2

Preuve : La representation irreductible U 1

lorsqu'on la restreint

a

dans P

est

)

de

kn

«(I +k)n-J)) ; dans Ie deun-l

et sa dimension est p

est

(l+kn) «k+l)n-2). n-2

contient la representation

; elle a un poids dominant de la forme 173

- 12 -

a

)(,1

(d'apres la proposition 11.4) ; e 11+ " , + en_1,n_l-(n-l) e n,n

est un element de

1 "a l.gebre ab e l i enne maximale de

,+

o

{j

definie au II.A. ,.

e

1I

+"

e

n-r l , n-l

-(n-l)e

nn

agit done dans ehaque sous-espaee de poids par une eonstante. Les poids de la representation irreduetible de a)(,J

st(n,[) de po ids dominant

sont de la forme : INn - 1

= a - k _

k

n 1

si

a - k

Si

a

=

n- 1

bn

de oultipliete

n_ 1

(n-I)

n = n, alors

avee

b E

a - k _ n . n 1

a = (k

n_ 1+l)

on sait que

n

d'ou

a

O[n].

a)(,J - k()(,l - )(,2)

est un poids de

pour (voir [4]

Done

bn Q,j

(b-l) (Q,j - )(,2) est de multiplicite I. II en est de meme de )(,

n

est l'element de

page 119),

)

ou

O(2,n)

w qui permute Q,2 et Q,n'

On a alors e

= bn

- (b-l)

(b-l) (n-l)

n-l,n-l

bn - (b l ) n r

De meme les poids de la representation de a t

l

+ Q,2

sont de la forme

174

n

-

done

a

O[n] ,

sQ,(n,[) de poids dominant

p

-

13 -

n-I I) E IN • ( k •••• k I • rr-

=a

+

I-k

n

n-I

a :=

d'ou

=

I [n ], Si

a>, I, ail + i cite

n

pour

2

- k(i

0.(.

1

I [ri ] , c'est-ii-dire s i

a-

- i )

=

bn-I

avec

bE IN* •

est un poids de cette representation de mu1tip1i-

2

= a-I done l est poids de mu1tip1icite I. 11 en est de meme de l

i , i

+

2

a - (n-l) - b' - b'(n-I)

a - (n-I) - bn

bn - I-n+l - b'n = (b-b'-I) n pour

b'

= b+2

on a 1e resultat.

La mu1tiplicite de

H

est egale ii la multip1icite de G c'est-ii-dire, d'apres 1a proposition 11.4, a :

dans

kn

p

dans

pour

p

de poids dominant

kn i]

pour

p

de poids dominant

kn

- i] + i 2 •

Ca1culons la dimension de la representation de kn

en uti1isant une formule de Weyl (voir [4] page ]39) : 1a dimension de la

representation de poids dominant

A est

II (A + a >0 IT

a > 0 ou a

est une racine et ou Or pour

s

0

0 vaut

0 =2I

o,a)

(0, a)

est 1a demi-somme des racines positives.

les racines positives sont {i.

et

de poids dominant

n

I

i=l

-

(n-2i+l) £.

J

,

I .$ i

< j ;( n}

Done :

175

- 14 !I

a > 0

(kn £] + 6,a)

(kn £1 + 6, £

!I

!I (6,a) a > 0

- £.)

1

]

I M,E), la famille i A est filtrante decroissante et i?r N fOJ. D'apres la proposition 1* i de (1) ,3iEI tel que N. = foj . Comme E est un cogenerateur on en d8duit

modules de type fini de M

que

M

M

l

= M. • l

Rappelons les definitions suivantes Definition 1.2 - Soient

A un anne au ,

M

A-module

a gauche.

On dit que

M est

lineairement compact (9), (resp. semi-compact) si toute famille filtrante decroissante

(x.+M.). I,oupourtout i,xl.E.M et l l l E (resp. l'annulateur dans M d'un ideal a gauche de

estunsous-moduledeM A), a une intersection non

vide. Definition 1.3 - On dit

A-module

fp-injectif) si pour tout A-module

=0

a

gauche

M est absolument pur de presentation finie on a

(6)).

(voir

Un A-module

a

a gauche

gauche

M est injectif si et seulement si

M est fp-injectif

et semi-compact. Voir (6) et (4).

2 - Sous-modules purs Definition 2.1 - Soient A-modules

a gauche.

A un anne au , 0--,> N..-l4 M "'""""'l'P ---..,0

On dit que cette suite est

199

une suite exacte de exacte (ou bien que

-3u

est universellement iniectif, ou bien que

pour tout A-module

a droi te

Q la suite

N est un sous-module pur de

0

Q Iil: N A

est exacte.

Proposition 2.1 - Soient

Q Iil: M

A un anneau cOillillutatif,

Q

A

@

A

A-algebre (non necessairement commutative),

(S) : 0

0

M

P

0

E un cogenerateur pour la

categorie des A-modules, B N

M) si

P

une sui te exacte de B-modules

a gauche.

Alors

les conditions suivantes sont equivalentes : 1) la suite

(S)

est universellement exacte

2) la suite

Hom A(P,E)--7 HomA(M,E)

a droite

suite scindee de B-modules 3) Pour tout B-module

o --'t

a gauche

HomB(F,M)

F

par

est une

de presentation finie, la suite

HomB(F,P) --'t

°

est exacte.

Demonstratjon - Voir la proposition 9.1 de (5) en rempla9ant Q/-Z

°

HomA(N,E)

Z par

A et

E.

Definition 2.2 - Soient

A un anneau, N un A-module

a gauche.

pur iniectif, si pour toute suite universellement exacte

HomA(u,N)

On dit que

0---7

Nest

M_ M"---70 ,

est surjectif. Alors tout produit de modules pur-injectifs et tout fac-

teur direct d'un module pur-injectif est un module pur-injectif.

Proposition 2.2 - Soient

A un anneau cOillillutatif, E un cogenerateur iniectif pour

la categorie des A-modules, B

Ynf

A-algebre, N yn B-module

a)

N est un sous-module pur de

b)

N est pur-injegtif si et seulement si

tel que

N soit facteur direct de

Theoreme 2.3 -

b) si

Alors :

HomA(HomA(N,E) ,E) il

existe un B-module

a

droite

M. Alors :

est de type fini,

P

HomA(p,E).(Voir (5) proposition

A un anneau cOillillutatif semi-local, M Yll A-module,

sous-module pur de a) si 14

a gauche.

N l ' est aussi

M est de presentation finie, N est facteur direct de

200

M.

N Yll

-4Demonstration - Soient

les ideaux maximaux de

D'apres la proposition 2.1, HomA(N,E)

est facteur direct de

quant la proposition 1.1, on en deduit que si Si

MIN

sition 2.1 3°), Nest facteur direct de

=

n

HomA(M,E). En appli-

M est de

M est de presentation finie, alors

Theoreme 2.4 - Soit

A, E

fini, N l'est aussi.

l'est aussi. D'apres la propo-

M.

A un anneau commutatif. Alors tout A-module est un sous-

module pur d'un produit de modules de type cofini et pur-injectifs.

Pour montrer ce theoreme nous avons besoin du lemme suivant. Lemme - Soient A-modules

A un anneau, (M., 4'.. ).

a -gauche,

l

(M,4'.).

I

llE

Jl l;:;'

@

I

--

ker =(t:> lnt(A)/lnt(A)

f"\

lnt(A)

n Intt:(A) )/«'2::? () Int(A)/< C7{)

(:-1 , 't:'(t. ) = pour tout et t:'(r) = r In ,n • On a ?:cr = 0- e ,t:(K) t K , car x e-(K). Sur K on definit 1 la t:'-derivation interieure S(y) = , pour tout yE.K. Soit E Le o

plus petit sous-corps de et les [YiR,'xo'J pour tout 'l:(E)s.E

et

J'

K, contenant

k , les

et tout

,s

n'est pas interieure, car

xi ' i

.

1 , les

Alors E

LXi 0 ,

E

et que par ccne equen tEl: t

quasi-simple.

244

1

(voir (3)).

Les lemmes suivants permettent de montrer que Ie couple hypotheses du theoreme 3 sur Le corps

'

S(E)s.E,

verifie les

:'l:,SJ

est

-11Lemme 9 - Soit

,

entier 10

aEi:K

. On suppose

qu'il existe un entier

, ou

at 1Y=t' 1 Ya n n-J,

-lll. aEoE verifie t:(a)

Remarque - Le centre de

E

est

j = a

a

S (a)

et

a

d'ou l'egalite

,

tel que pour tout aEk

et

alors

.

.

a6 k

d ne sont interieurs et par

k, puisque ni

le tMoreme 1, Z(E)e.Fix(e/KerC»

,

N>O

k = Z(E).

§4 - Extensions d'Ore d'ordres dans des anneaux artiniens

a gauche

On sait qu'un anneau de polyn8mes sur un ordre dans un anneau artinien est un ordre dans un anneau artinien (16). Nous allons exposer quelques resultats dfis A.V.

et (13»

s'agit des extensions

qui montrent que ce

A tt

a

reste vrai lorsqu'il

A est un anneau semi-simple ou plus genera-

ou

a

lement un ordre dans un anneau artinien

gauche et

un endomorphisme injectif

A. A la fin du paragraphe nous donnons un exemple d'extension d'Ore d'anneau

de

semi-simple, qui verifie la condition 5° du theoreme de classification. On considere un anneau semi-simple

A, qu'on decompose en somme directe

d'ideaux bilateres minimaux. (Pour les questions concernant ces anneaux, voir (2». Soi t

!J?> l' eriaemb Le fini, de cardinal

On designe par bijection

l'intervalle

q:

designe par

(1

m, des Ld eaux bilateres maru.maux de

d3. Posons Cf(i) l'idempotent de

Proposition 4 -.§.ill

rn • Une

,m1 dans = B.

alors

1

santes simples E .. ,E 1,· k

bilatere et

t..e k

R =

,telle que

B.

1

=

m

i=1

0

Z (A) ,

A

son centre et

une suite exacte de A-modules a gauche.

I 4& C-Ass Al1}

I EC-AsSAM' }

un sous-module

est vide

M est

M est

H

f I II

Alors l' ensemble

A-module a gauche

est un anneau noetherien a _gauche,

diviseurs de zero dans

soit

A

es t contenu dans la reunion des ens em-

£InZ(A),

IEC-ASSAH"}

un ideal a gauche central-premier associe aM. 11 existe donc N de H

isomorphe a

M"; done

xC; Nil f(M'),

, un element tel que

X" 0

st

A/I .

un so us-rmod ul.e de

IEC-AsSAH"

annulateurs d ' elements non nuls de

N i\ f (11') = (0), Nest isomorphe a Supposons

Nof(H')" 0

et soit

AnnA x soit maximal dans l'ensemble des

N 1'\ f (M'). Alors

gauche central-premier et central-associe a

M'

JAnnA (x)

est un ideal a

et on verifie que

I "Z (A) = J" Z (A) •

Lemme 1.9 - Soit

A

un anneau noetherien a gauche, M un A-module de type fininon

=

M: (0) = M S M S •.• M = M , o l m est un ideal a gauche central-premier

nul. Alors i l existe une suite de sous-A-modules de telle que de

Mi/M

i- l A. De plus, si

tlnZ{A),

A/J

i Z{A)

IEC-AsSAM]

'

, oil

i designe Ie centre de

est contenu dans

culier l'ensemble !InZ(A),

J

I EC-ASSAM]

A, lfensemble

fJ "Z{A) i est f i.n i ,

254

i = 1 , .•• , m] • En parti-

-4J resulte du lemme 6 de (15J. D'apres 1.6, i IE c-AssAI1 es t contenu dans la reunion des ensembles

Preuve - L'existence des I' ensemble [111 Z(A)

J

tI"Z(A),

IEC-Ass A A/Jd pour i est l'annulateur de la classe de

J.

1.

a

ideal x

tel que i tout element a

J

AnnA(x)

du centre de

done s i et seulement s i, i

1"\

r

dans ri

o

x modulo J

si et seulement si

i On a done fI" Z(A),

• Pour i axE J ' i A/JJ

Z(A)J, d'ou Ie r es u.l t.a.t ,

1,r 2

, ••• ,r

I.

dans A/I

m

I

un ideal

On a pour

est un A-homomorphisme. si

A

o

ri bE;I o

Proposition 1. 11 - Soit

on a

a

A un anneau noetherien

A

non contenue

a

gauche par

Done la multiplication

a, bE A : (ario -ria)bEI

precedent est injectif.

gauche.

gauche central premier d'un anneau

i Min [i , r ¢ IJ . Alors la multiplication o i est un A-homomorphisme injectif.

par

rio

a

une famille centralisante d'elements de

Preuve -

b

:

I ; done I' homomorphisme

gauche et

M un

M est injectif si et seulement si il verifie la condition sui-

vante : quel que soit l'ideal

a

gauche central-premier

1

II est bien evidemment que l'ideal

a

a

I I reste

forment un systeme centralisant. Soit

I

6.= r·

u - u r. I

(r· u

et soit

I et n

1

...

-

i,

Yl"Y-Z"" 'Yl?

k, que

n

• Lorsque k=l , Ie resultat

Gv k-l

.

verifier que, pour l'ordre lexicographique,

-r.

d'entiers compris entre

. A tout

et on pose

et Le r e s u l.t a t d emout r e pour Gv k est engendre par les

les

r .

Gv

k

k

gauche

y .... = r·

A= r·

.• , . )

. On va montrer, par recurrence sur

est un systeme de generateurs centralisant de

y""

Lei, (E A (M) )

de generateurs de

(. I" . O

est une suite exac t e de A-modules

gauche de longueur finie, la suite de groupes abeliens : 0= Extn-I(M',A)_ Extn(M",A) _Extn(M,A) --;>Extn(M',A) _;:.0

est exac t e ; d'ou I.e r e s ul t a t ll c

Dans la suite, on notera

AMod

la categorie des A-modules

a

gauche,

f (resp. AMOdlf) celle des A-modules a gauche de type fini (resp. de lonMod A gueur finie) et Ab la categorie des groupes abeliens.

Lemme 3.3 -

a

tr

gauche annule par

s;: de

A un anneau, m6M

Alk--.+ M en posant

aeA

xET(M)

T

=

T(A/jj)

=

,

on definit

On verifie facilement que fonctorialite de

:to

(x)] (m) = T( lfl (x) pour tout A-module a gauche M, m) on definit un morphisme fonctoriel T =*HomA(-,T(A»o

($ (M) definit, grace

A, on a

(P",,) suivante :

(y)]

Preuve - Pour tout A-module

suivant, oil

. On a

un foncteur additif contravariant de

gauche

(y) = a ((T(lfl

designe Ie morphisme

et

J

modulo

un ideal bilatere d'un anneau

(P oo ) Pour tout A-module

mEM

E

une resolution injective minimale de

('I)

o --'> HomA(A/ Ax,

tion injective du

H

"!'t'

A/xA-module

M/xH Preuve

o

M et

ni dans

do(E o »

M une A-module

d

o non diviseur de zero dans

conserve les monomor-

AMod

A un anne au local,

gauche. Soit

A; alors

x

£

A E1)

HomA(A/Ax, do(E o»

un element du centre de

d1

>

...

est une resolu-

lequel est isomorphe

a

La demonstration est la meme que dans Ie cas commutatif [4J.

Ext i (A/Ax, M) A HomA(A/Ax, E j) pour Puisque

pour

0

i

, la suite (x) est exacte en

D'autre part la suite

etant exacte, il en est de meme de : 0

_HomA(A/Ax, E E

A,

E 1)

D'ou l'exactitude de la suite (x). Puisque, pour

2). est l'enveloppe injective de

d

(E

) , i_ 1

,

il resulte de 4.5 que

i i- 1 HomA(A/Ax, E est enveloppe injective du A/Ax-module i) Verifions que Ie morphisme nature I de A/Ax-modules :

HomA(A/Ax, 1m d

i- 1).

Cf: di - 1 (HomA(AJAx, HomA(A/Ax, 1 (E est un isomorphisme i- 1» If est evidemment injectif ; montrons qu'il est surjectif. Soit gE Hom (A/Ax, d (E ; alors g de f i.n i.t un element g,,-Hom (AJAx, Ei) et A i- 1 i- 1» A = ker d . Par suite on a : di(g) = d a g = 0 car 1m g S d i- 1(E i- J) i i gEker d = Lm d et i l existe fEHomA(A/Ax, E J) tel que: g = di_J(f) i_ i i 1

267

-17-

d'ou

\.f(di_l(f)

=

g. Par consequent (;0) est une resolution injective minimal

de son premier terme. Enfin, puisque

x

est non diviseur de zero dans

est non diviseur de zero dans l'enveloppe injective

M, il

de M . De plus, Eo

un A-module divisible et

x

etant non diviseur de zero dans

et la multiplication par

x

est un automorphisme de

E

etant

A, on a : Eo

xE

o

o

Considerons Ie diagramme du serpent dans la categorie des A-modules 0

Hom (A/Ax , do(E o» A

0

1

d_

0 --'l> M

1

l

d

E

il d (E ) o 0

0

0

xl

xl

w

d_

0 --'l> M

-b

1

d

'> E0

M/x11

0

xl 0

l>

1

pl

--':>

do(E o)

--;:>

'> 0

1 0

do (Eo» es t l'isomorphisme d ' identification : HomA II en resulte un isomorphisme de A (et done de A/Ax)-module oil

i

'I' : HomA(A/Ax, d (E » o

Corollaire 4.7 - Soit

0

';f

M/xM .11

A un anneau local dont Ie radical

une famille centralisante, M un A-module a gauche, x de

A non diviseur de zero dans

A ni dans

A/xA Pi (M/xM)

=

Ann (Eo) (x) , do

est engendre par un element du centre

M. On a alors :

A Pi+l (M).

Preuve - Soit

o ----'.l>

Eo

M

une resolution injective minimal de

E1 M; alors : 6l N. l

N est somme directe de module du type EA(A/I) ou I est un ideal a i gauche de A, central-premier, inter-irreductible et distinct de . Soit

ou

un ideal zero dans x eI

a

gauche central premier; si A/I

ni dans

EA(A/I)

x

I , alors

x

et par suite: HomA(A/Ax, EA(A/I»

alors par 4.5, HomA(A/Ax, EA(A/I»

I

n'est pas diviseur de 0; si

E (A/I). Enfin, toujours par 4.5, A/Ax

268

-18-

HomA(A!Ax, E A(A!'lJ1,»

E (A!'ll\,). On a done pour A/Ax

HomA(A/Ax, Ei) = AnnE. (x) =

A

A

Pi(M) HomA(Ax' EA(A!'lIt»)J

[(jj

Mi

(jj

1.

ou

M est somme directe de A/Ax-modules de la forme HomA(A/Ax, EA(A/I» i est un ideal a gauche de A, central-premier i.n t.er-r Lr r educ t.Lb Le , tel que

et

I f

que

; pour conclure que

I/Ax

a

est un ideal

=

""1.

gauche de

(M/xM)

'1.-1

A/Ax

il suffit done de verifier

central-premier, inter-irreductible et

distinct de

et d'appliquer 4.6. II est evident que

irreductible si

I

classes modulo

[r 1

sons que

I/Ax f

l'est et que

est central-premier. Soit

rI,

r I,

Ax, soit , ••• ,

Puisque

x E: I , on a

lisante

x, r

,r

rpJ $:

r.1. E

,r p

I/Ax

I f

si

Verifions que

a EA , et

done

aEI

a

A

I/Ax

dont les

, forment une famille centralisante. Suppo-

I = I/Ax e t; posons 1. 0 = ti, i=l, ... ,p , r rt 13 . i i si et seulement si r.1. E I . Done la famille centra-

, ... , r

l

Soit

est inter-

une famille d'elements de

p

ou I x s, I

n'est pas contenue dans I et p la classe de a modulo Ax. Si r i -a o

-aE>I.1I

etparsuite

.s

= Min [i , r Ii o i I , on a r i a e I ;

i E

0

Remarque - Dans Ie cas ou tout ideal bilatere de l'anneau

A admet un systeme

de generateur centralisant, on peut dans la demonstration precedente simplifier la preuve du fait que

I/Ax

donnee en 1.3. En effet

A

est central-premier. En utilisant la caracterisation A/Ax

possede la meme propriete que done

Proposition 4.8 - Soit radical

et

a

A

Si

aoubEI n

et de

gauche. Alors :

i : V1(M) =

, M), dimension du

i

A/m -espace vectoriel a gauche 2) Si

et

A un anne au local regulier de dimension

M un A-module

1) Quelque soit

aoubSI

Ext (A/'lI1., M) ; A

p1

M est de type fini,

est fini pour tout

(M)

i

Preuve - La demonstration est une adaptation du cas commutatif (4) . Considerons une resolution injective minimale de

M d

ou type

E

i

=

[(jj

A

Pi(M)

EA(A/I), I

(jj

N i

etant un ideal

tible et distinct de

m ;

o

et ou

a

N i gauche de

pour un tel ideal

est somme directe de modules de A I

269

I

central-premier inter-irreducon a: txEA/I

I 'lI\x =

oJ

= (0)

-19en effet, soit

r

une famille centralisante de generateurs de et, p puisque I " Ill\, , on peut poser i o = Min i , i = I, ... , P , r.IT J, I] ; alors s i a E A et 'In a.: I on a r i aEI; d'ou, pu i sque I est central-premier, l

, ... , r

t

o

a E I ; (remarquons que, dans Le cas oil tout ideal b i l at.e r e de l' anne au

A

est

engendre par un systeme centralisant, on peut pour demontrer la precedente assertion utiliser la caracterisation 1.3). II resulte de ce qui precede, que, si l'ideal

a

\li !xEEA(A/trrj,)

directe de

a

A/trn,

I

I\ll.x = OJ = (0). On a done

tx/;EA(A!I) = ffi

gauche central-premier IlII.x

\li copies de Done

0].

=

Ill\, ,

alors :

= [XEE i tm. x = 03 HomA(A/trn., E est somme i) qui est, d'apres 4.5, isomorphe

RomA(A/m,

Par consequent

RomA(A//ln,

J-li = dimA!'l1I.

que, pour tout

est distinct de

HomA(A!IlI\, E

Posons

i).

E_ 1 = H et demont rons

di : HomA(A!'ln, Ei)

l'application

HomA(A/'Il\,

I)

a l'aide de d i , est nulle. Soit xEiROmA(A/'l1\, Ei). Puisque est l'enveloppe injective de d I) il resulte de 4.5, que l'injection i_ 1 A canonique RomA(A/Iln, d (E » Rom (;m' E est un isomorphisme. 11 en i_ 1 i_ I i) A resulte aisement que d. (x) = 0 Considerons la suite de A/on - espaces de t i n i e

l

vectoriels

o Pour (ker Pour

i

Hom (A/'M" A

H) r.

1 , les A/ "m. -espaces vee toriels

d) / (Lm

Ext (A/rm., H) A sont isomorphes.

I) = Hom (A!1Jn, ,

A

i =0 , Le fait que

d

o

= 0

entraine que

et

est un isomorphisme.

2) D'apres la proposition 2 de [153, les A-modules a gauche A. Tor n-l (A/I\ll., M) sont isomorphes. Soit :

et

une resolution projective de type fini de A-modules l

A/I\ll. Cil Pi A et des

M)

M. Alors pour tout

i

i

Ext (A!1f'II, ,M) A

les

sont de type fini et il en est done de meme des (A!",.. , 11).11

Definition 4.9 - Un A-module

a

gauche

M est cofini si son enveloppe injective

est somme directe fini d'enveloppes injectives de modules simples. La proposition suivante est la proposition 3.19 de (21J • Proposition 4.10 - Soit

H un

a

gauche. Les conditions suivantes sont

eq uivalentes :

270

-20i) le module

M est cofini ;

ii) tout systeme inverse de sous-module non nuls de

M admet une inter-

section non nulle. La proposition suivante est Ie theoreme 3.21 de [21) Proposition 4.11

a

M un A-module

.

gauche. Les proprietes suivantes sont

equivalentes : i) Le module

M est artinien

ii) Tout quotient de

Theoreme 4.12 - Soit

M est cofini.

A un anneau local dont Ie radical

est engendre par

un systeme centralisant. Alors l'enveloppe injective du A-module a gauche est artinienne. Preuve - On adapte la demonstration du theoreme 4.3 de [2IJ. Notons

a

loppe injective du A-module

. On supposera que

gauche

artinien et on en deduira une contradiction. Soit bilateres Gv de A pas vide puisque

AnnE(0v)

tels que AnnE(O)

E

E

est suppose non artinien. L'anneau

I) Montrons que

ef.

A etant

10.

un element maximal, soit

f

et

AI rrrv

Ilfl,

, on a

•

est isomorphe au A/'P -module

EAI,p (Aj'YYlt)

l'enve-

l'ensemble des ideaux

est extension essentielle du module simple done 'Itt (;

E

n'est pas

n'est pas artinien ; cet ensemble n'est

noetherien a gauche, l'ensemble Comme

E

AnnE l'

L'extension de A/1P -modules

est evidemment essentielle. D'autre part injectif ; en effet soit

J

un ideal

a

AnnE

f

gauche de

es t un AIl' -module

Alf

et

f: J _AnnE

un A/f -homomorphisme. Alors l'homomorphisme de A-modules

f: J

prolonge en un A-homomorphisme : g : A/f

x cAlf

done

g(x)"AnnEf>

qui prolonge

bilatere non nul de

f'1

E . Mais si

l'

se

on a

est un A/1P -homomorphisme

. Par suite

g: A/f --4AnnEf'

A __ A/f

la surjection canonique et soit ;fr un ideal

f

2) Soit

E

A/f

; alors

Q, . D'apres la maxi.maLi t e de

artinien. Posons

E'

AnnE -module

E'/N

o

$

P EA/f

(Aim)

est Soc [EA!f (A!tm,)J =

Par consequent

,

. Soc(E'/N

centralisante et que que

r

SOC(E'/N

f ""

o)

,

= 0 . Puisque

A/f

; d'ou :

272

p .

A/lIfl..

est engendre par une famille

on peut trouver un element

appartient au centre de l'anneau o)

,

$

,

r E mt

Alors un tel element

r "" 0 tel r annule

-22Soc(E'/N

D'autre part, puis que A/

1>

-rnodul.e

)

£

AnnE'/N (r )

o appartient au centre de

r

bilatere non nul de

O

A/f (r.A/

A/f

est un ideal

, r.A/.p

D'apres Ie point 2) de la demonstration, Ie

f ) N

est artinien. Posons :

=fXE ...

M. Alors : A

Pi (M)

EA(AI'!r.)] ED Ni

N est somme directe de modules du type EA(A/I) ou I est un ideal a i gauche de A, inter-irreductible, central-premier et distinct de . Puisque,

ou

pour un tel

I

on a

=

(0)

et que

=

d'apres 2.6, on en deduit que:

A

D'apres 4.8, Pi (M) Done

est fini et d'apres 4.12, Ie module

est artinien pour tout

i

et il en est de meme de

273

est artinien. i

H'!Y\,(M) •

-23-

v - Annulation des foncteurs Defini tion 5.1 - Soi t

A un anneau local r e gu l i.e r de radical

A-module a gauche. On appelle profondeur de plus petit entier

Lemme 5.2 - Soit et

M est infinie.

A un anneau local regulier de radical

M un A-module

a

i

o ,

M)

o

ExtA(N,M)

pour tout A-module

a

M est de type fini, la profondeur de

M est infinie si et s eu l.e-:

M; (0)

3) Si

Preuve

n

N de longueur finie 2) Si

ment si

et de dimension

gauche

1) Si

gauche

M) # 0; sinon on

s'il existe, tel que

dira que la profondeur de

'WI- , M un

M (en. notation: profA M) Ie

M est de type fini non nul on a

n .

1) II suffit de proceder par recurrence sur la longueur de

2) D'apres la proposition 2 de (15], Ie A-module est isomorphe a

a

gauche

N • n

Ext (A!%,M) A ,M) ; 0 , il resulte du lemme de Nakayama que

. Si

M; (0). 3) Posons

p; profA (M). D'apres 2), on a . D'apres la proposip-] A Ext M) ; (0) et Tor n-p+ A sont isomorphes. Considerons une suite exacte de A-modules : tion 2 de [15J les A-modules a gauche

o --l> ou les

Pi

S

--;>

P n- p-]

... __ Po --i> M __,>0

sont projectifs de type fini. Alors

A

TorI (A!rm. ,S) ; (0)

e t d ' ap r es

Ie corollaire 2 de la proposition 5, nO 2, §3 chapitre II de [5J, Ie A-module est libre ; d'ou

dh

MoE-n-p . Pour prouver l'egalite

A par l'absurde et on suppose que jective de type fini de

o Puisque

M:

-----';>

A

a

o

d'ou (loc.cit) On a done

n".

A un anneau local regulier de radical my

gauche. On a profA(M)

Ie second membre etant

co

on raisonne

P n-p-I - ; . ...

A Tor (A!1lI1 ,P n-rp-r 1) 0 on a Tor n-p (A!fIr< ,M) M) 0 ce qui contredit la definition de p

Theoreme 5.3 -

M ; n-p

i

S

A . II existe done une resolution pro-

A]

profA M + dh A M

A-module

dh

dh

Inf ; 0

p

,

(M) # 0

pour tout

274

i

et

M un

-24- II suffi t de prouver, pour un entier tions suivantes : i)

o

(M)

pour tout

ii) profA

a

m

ii) on procede par recurrence sur

demontrer. Supposons

l'hypothese :

(M) = 0

= 0

*

.

Pour prouver i) rien

i

m 0 , I' equivalence des condi-

si

m>]

pour

i< m-I

m. Si

et Ie resultat prouve pour

i < m , on de dui.t que

e t si

m=O, il n'y a m-I . Alors de

prof A (M)" m-r l . Par suite

N est de longueur finie. Soit

k-s k '

;

m-' l (A/ IlJlk ' , M) de i e d e I a sur j. ec-: I , app 1"i c a t i.on : Extm-' l (A/ lilt k , ) M e du u i.t A . . / k' / k . . . . t i.on c anorn.que : A rm __ A lln- , est t nj e c t i ve . Pu i sq ue m-I m-I k nr-I

(M) = Itm.ExtA

H'lllo

(A!trn-, M)

Verifions que i

on a: Ext A

m et tout A-module de longueur finie

pour

i

(A/trfl"

On a done

m et pour tout entier

M) = 0 ;.d'ou

d'ou

pour

=

N . En particulier

k z- O . Done

(M)

=

i ou

1;(k)

o

lim T(K )

Alors

k, il existe

on a

k'

"Ifkk ' :

un A-homomorphisme. Il en

--l>

resulte un homomorphisme de groupes abeliens

D'ou un systeme inductif de groupes abeliens,; on notera

'¥k

Ie morphisme

canonique : --'l>

lim!;> T(M r ) r

Soit que

x

=

x

d'ou

e

lim::> T(M r ) ; il existe un entier r Cornme on a "'I'k 'lfk?:(k) = 0

x

=

"ilk

("\

0

et un element il en resulte que

'I' t:(k) k)

Ext A/ zA(-' M).

k'

Si

, on note

(j.,

Y'kk'

L'ideal

:

a

Az

k'

a.

A/zA-morphisme canonique :

tel

=0

0

b) On applique Ie resultat precedent

j

k 6 1\ If -e(k).k

X

l'anneau --;>

A

zA

et au foncteur

Q,k : Az

Ie

Q,k

verifie la propriete d'Artin-Rees. Done, si

m est un

k] m ] . r '>0 , t.' I .. z, en t i.e exa's d t e es ent i' e rs k l ' k 2"'" tels que '. r- - z "(A, Co_rom w ....m-2 k2 m-] C "" z fI &, S G" z, •..• Notons 'l:; (m) = Sup 'l.m , k] ,k 2 , ..•

!.

On a, alors

A z ()

x Az !:;(i,m • z d ' oii

x

e G"m

e t , puisque D' oii

Theoreme 5.]0 - Soit radical, x],x

Go G.Jm • z . Done, s i x " (j,?::(m) : Az , on a

.•• ,x 2, n

"Jm't;(m)

z

est non diviseur de zero dans 0

A, xA

.11

A un anneau local r e gu l i e r , de dimension

£a..m

n , on,

une suite centralisante A-reguliere engendrant qn et

277

,

-27xI ,_ .. , xm , M un A-module Ii gauche de K-dimension finie

l'ideal b i.Lat.ere de Soit entier

A engendre par

on a

i > s

"

0

Preuve - On raisonne par recurrence sur pour

(0)

demontre pour

m-j

et

m

0 , on a G,

s e-O

(0)

d' oil

et Le resultat

et raisonnons par recurrence sur

suffit d'appliquer 5.6. Supposons s

m. Si

; d'oille resultat. Supposons

modules de K-dimension au plus

s . Alors pour tout

s . pour

s

0 , il

et l e r e s ul t a t demontre pour les

s'< s . Chaque sous-module de

M etant deK-dimension

M etant limite directe de ses sous-modules de type fini on se

ramene, Ii l'aide de 2.11, au cas oil M est de type fini. Alors il existe une

M:

suite de sous-modules de (0)

telle que

M

o

A/l

M/M _

I central-premier de j

j

c. M c. ••• C M = M r

I

, j = 1 , .•• , r , oil

I

est un ideal Ii gauche

j

A. On peut done se ramener au cas au

un ideal Ii gauche central-premier de

A, avec

K-dim A/I

sant l'hypothese de recurrence sur la K-dimension. Soit dans

Le

centre

Z(A) de A de I' ideal

I I)

Z (A),

M

A/I

=s S

I

etant

, ceci en utili-

Ie complementaire

qui est un ideal premier de

Z (A).

Considerons la suite exacte de A-modules

et ou

K est Ie conoyau de la fleche

A/I

H i

de recurrence sur la K-dimension, on a : HGJ (N) sous-module de type fini

plication par si

i

xl

et

On a donc xl E I

si

xl

¢ I

et oil

i'l): s , pour tout (K)

0

pour

XjE I . Si

xl

i I

s la

H est, d'apres 5.8, un isomorphisme. Donc la multi-

xI dans

(H)

est, pour tout

yEHExt i AlxIA ( xIA +Gi., k

I" k

i (A -:K Q.

lim

278

i Ext A/

A , XIA xjA+ G-k

-28-

A=

L' anneau

A/x IA es t r egu Li e r et u,

e s t erigend r e les

=

m-] premiers

elements d'une suite centralisante A-reguliefe engendrant le radical de On a

W

G,

G,

pour

A.

k

+ Ax l = --...,--Axl

-k

donc par hypothese de recurrence sur

(AlI)

=

i ( Ext A/ x jA

k

A

k

xlA + Gc.,

A ' I)

m on a

=0

i:> s . D' ou le r es ul, tat. •

Corollaire 5. II - Soit

A un anneau local regulier, M un A-module

a

gauche

non nul de type fini. Alors on a :

Preuve

On a :

(M) ". 0

p
0

;.0

I

;.0

K

2 K >0

2/b]

--;.. >0

a

Structure of

2

algebraic surface of general type

2, I

elliptic surface of general type

X

0

0

2

4

0

2, 1,0

J

0

0

2

3

0

J

I

0

0

0

0

2

2, 1,0

0

fo

0

0

0

I

2

Enriques surface

fo

0

I

2

0

2

hyperelliptic surface

or 0 0

0

I

I

0

1

0

0

I

2

elliptic surface (Type VIl ) o (I?I X pI oder (I? 2

1

2

0

2

elliptic surface

2q

l-q 0 m

0

for all

for some

V by m

xCV) 00

if

Pm

=

m>O

Then Iitaka's theorem states (cf. (9] bf (23], theorem 6.11).

Theorem] - Let

V ba an algebraic variety of Kodaira dimension

smooth projective varieties f :

and W*

. There exist

and a surjective proper morphism

which satisfy the following conditions:

(4) For an algebraic definition and the universal properties of

283

O ; dimq;Ho(V,;tIlhn) ?

:

map

p '>

V which is unique in the birational

sense, with algebraic varieties of Kodaira dimension

on a smooth projective variety

m with

4>mi (V)

, m

1J

and for

!\lm f, (P)

m

6INCt,V),

(f o (P), ... , fN(P»,

£, -dimension of

IN(t , V) , if

V is

lNet ,V) f. f/J

IN(t ,V) ; f/J •

t. -dimension

of the sheaf

associated to

D is

X(D,V).

A theorem analogous to Theorem I holds for the

;f, -dimension (cf. (23J, §5 for

details) •

(5)

A fibre space is a morphism g: X of reduced projective varieties which is (proper and) surjective and has connected fibres.

284

-5Next, Iitaka's theorem suggests that we divide the algebraic varieties of a fixed dimension into 4 classes as follows. I) Varieties with

dim V , called varieties of general type or hyperbolic

xCV)

type ; 2) Algebraic varieties with

dim V > xCV)

3) Algebraic varieties with

xCV)

o ,

I ;

called varieties of parabolic type ;

4) Algebraic varieties with

, called varieties of elliptic type.

The birational investigation of the varieties of class 2) reduces by the theorem of Iitaka to the study of fibre spaces of algebraic varieties with a variety of Kodaira dimension

0

as general fibre.

The Albanese map is essential for the study of the classes 1), 3) and 4). The following facts concerning the structure of the Albanese map are of interest. Proposition I - For a (smooth and projective) variety irreducible components of the general fibre of

V of general type, the

V

(V)

are also of

general type. Concerning the Albanese map of varieties

V of Kodaira dimension

0 , i.e. of

class 2), Iitaka and Ueno have suggested (cf. £23), p. 130). Conjecture K ; If V is of parabolic type, the Albanese map c(; V Alb (V) n surjective and has connected fibres. Horeover, the fibre space (; V is birationally equivalent in the etale topology to a fibre bundle over whose fibre and structure group are an algebraic manifold and automorphism group

Aut (F)

of

S

of parabolic type

K

n

is known to hold for

is a relatively minimal projective surface of Kodaira dimension 0,

then if the irregularity an isomorphism. I f

q(S)

of

q (S) = I ,S

S equals 2 ,S

is an abelian variety and

is

0 W be a surjective morphism of projective smooth C n,m algebraic varieties over with connected fibres, i.e. or: V W is a fibre Conjecture space. Let

where

V

w

n

=

dim V , m

dim W . Then

is the general fibre of 11 .

Let

Conjecture

C and Proposition 2 immediately imply the following statement n,m V be a projective variety of elliptic type with irregularity q(V» 0 .

Let

V

(V----;.W

W be the fibre space associated to the Albanese map is the Stein factorization of the morphism

0(:

V __ Alb (V) .

V --t.{(V)). The general

fibre of

V --4 W is of elliptic type. Therefore, if C n,m algebraic varieties of elliptic type is reduced to

holds, the study of

I)

the study of algebraic varieties with irregularity

2)

the study of fibre spaces whose general fibre is of elliptic type.

It is interesting to note that Conjecture More precisely, if

C n,m type, the Albanese map

holds and

n,ffi

is related to Conjecture

lS

n

surjective and an irreducible compo-

is of parabolic type.

We indicate a proof of this fact. Consider the fibre space to the Albanese map. Then since

=

V ---+ W associated

0 , it is not difficult to show by the

adjunc tion formula (cf. (23J, §6) that the general fibre of variety of Kodaira dimension

K

V is a projective variety of parabolic

V

nent of the general fibre of

C

0;

Also

)l,(W)

0,)s"-)

I Ho (V,.I'l.V(log(D».

The quotient

Alb(V)

=

considered as a Lie group is

the Albanese of

V. The Albanese map

point

P e V along a path in

06 V

to

.J:

V

is related to the Albanese of

Deligne's theory (2J implies that where

r

=

q(V) - q(V)

=

V

: V _ _!>

P Alb(V)

is obtained by integration from a fixed

1----+

ill (V)

£

O

w, we H (\7,

(D» .

V by the exact sequence of groups r

K

bl(V) - bl(V). Thus

is a torus of dimension

ill (V)

r,

carries the structure of a

quasi abelian variety, i.e. is a group variety which is an extension of an abelian variety by a torus. (cf. (12J for details). The universal properties of

0 ,

+ 1 • Alors

i) i l existe

AGZ

1(E),

BE.Z

n_1

(E)

verifiant

ii) il existe des elements a , ••• ,a 6 m 1 n ""((_1)j+1 D + a

T (K) ir 1 2T1 2'()

j(B p' I x(l) (B» est un anne au local noet.he r i en de dimension 1. Suivant [6J '

Ia

(Lemme 2-1). en posant

{IO

P

n(B

p'

E: x(l)

(A)

I ;i '"'

A E x(l) (A)}

IJO'E: x(1) (B» = n (i.

361

pip

E:

p)

- 5 -

B}l- est done un anneau generalise de fractions de anne au de Krull done

B

it

A, qui est un

est aus s i un anne au de Krull. Suivant l e lennne 2, B*

est un anneau no e t he r i en avec dim B·I/-

2. Evidennnent, B Ie =