Unipotent Algebraic Groups (Lecture Notes in Mathematics)

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

414

T. Kambayashi M. Miyanishi M. Takeuchi

Unipotent Algebraic Groups

Springer-Verlag Berlin. Heidelberg • NewYork 1974

Prof. Dr. Tatsuji Kambayashi Department of Mathematics Northern Illinois University DeKalb, IL 60115/USA Prof. Dr. Masayoshi Miyanishi Department of Mathematics Osaka University Toyonaka, Osaka 560/Japan Prof. Dr. Mitsuhiro Takeuchi Department of Mathematics University of Tsukuba Sakura-mura, Niihari-gun Ibaraki-ken 300-31/Japan

Library of Congress Cataloging in Publication Data

Kambayashi, Tatsuji, 1933Unipotent algebraic groups. (Lecture notes in mathematics ; 414) Bibliography: p. Includes index. i. Linear algebraic groups. 2. Group schemes (Mathematics) 3. Commutative rings. I. Miyanishi, Masayoshi, 1940joint author. II. Takeuchi~ Mitsuhiro, 1947joint author. III° Title. IV. Series: Lecture notes in mathematics (Berlin) ; 414. QA3.L28 no. 414 [QAi71] 510'.8s [512'.2] 74-20780

AMS Subject Classifications (1970): 13B10, 13D15, 13F15, 13F20 14G05,14 L15,16A24,20G 15 ISBN 3-540-06960-7 Springer-Verlag Berlin • Heidelberg • New York ISBN 0-387-06960-7 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 £3 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 1974. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.

PREFACE The geometry and group theory of unipotent algebraic groups over an arbitrary ground field were successfully pioneered by Rosenllcht in the late fifties and early sixties. In the subsequent years not very much was added to the knowledge in this area, with only a few notable contributions such as those by Russell and by Tits. Lately, however, there have been indications of growing interest in this and related subject areas (affine space, its automorphisms, purely inseparable cohomology theories, ... ). Even as the present paper was undergoing the final redaction, a graduate student in Tokyo settled our conjecture in Section 5 by constructing an elaborate counter-example; one of the coauthors established the absence of nontrivial separable forms of the affine plane, confirming an earlier announcement of Shafarevich; another found a description of the category of all commutative affine group schemes over an imperfect field by extending Schoeller's work; and still another obtained an algebraic characterization of the affine plane. The material presented here might be made into two or three separate research papers of a more polished character. Instead, in view of the rapid developments as indicated above and because of our belief in the unity behind our work, we have chosen to publish our results as one whole and as quickly as possible. We are thankful to the editors and the publishers of the Lecture Notes series for providing us with an ideal outlet for our joint work. It is our sincere hope that this publication will serve to stimulate further research in this field full of deep and fascinating problems. Finally, our grateful acknowledgements are due to the Research Institute for Mathematical Sciences, Kyoto University for the hospitality extended to one of us while the research for the present paper was conducted during the year 1972-73; to the young ladies on the Institute's staff for the carefull and efficient typing of the manuscript; and to the National Science Foundation for partially supporting the final preparation of the manuscript through a research grant. June 1974 The Coauthors

TABLE OF CONTENTS

Introduction Part I:

(i-4)

§i.

Notations,

§2.

Forms of vector groups; groups of Russell type ( 17 - 28)

§3.

Decomposition theorems for central extensions of

conventions

and some basic preliminary facts(5 - 16)

commutative group schemes; application to the twodimensional unipotent groups ( 29 - 39 ) §4.

Wound unipotent groups ( 40 - 45 )

§5.

The question of commutativity for two-dimensional wound unipotent groups

Part II: §6.

Forms of the affine line and geometry of the groups of Russell type

Appendix:

( 46 - 57 )

( 58 - 107 )

§7.

Actions of unipotent group schemes ( 108 - 130 )

§8.

The underlying scheme of a unipotent group

§9.

The hyperalgebra of a unipotent group scheme ( 141 - 145 )

Central extensions of affine group schemes

Index of Terminology Table of Notation Bibliography

( 156 - 161 )

( 162 - 163 )

( 164 - 165 )

( 131 - !40 )

( 146 - 155 )

LEITFADEN

§3a

=

3.1

to

3.4;

§3b

-- 3.5

to

3.7

On the theory of unipotent 9~er an arbitrary

algebraic

~ruups

Ground field

by Tatsuji Kambayashi

Masayoshi Miyanishi

and Mitsuhiro

Takeuchi

Introduction

This paper reports on our joint the group-theoretical potent algebraic field.

Let

a field of An

G

k. G

and the geometric

be such a group, connected

In case

k

is perfect,

is known to be k-isomorphic n = dim G

§8),

the underlying

if

~n

group is unipotent

the geometric

group is completely over an imperfect

field then the result-

-- a fact due to Lazard

structure of a unipotent

connected unipotent

algebraic

leads to

As for the group structure

algebraic

group

(cf.

ground field, while

field the study of the structure A n.

It is

is given a structure of

known over a perfect

the study of forms of

variety

to the affine space

group over quite an arbitrary

Thus,

of uni-

and defined over

(cf. §8 and Appendix),

also known that, conversely,

ing algebraic

structures

of both

groups defined over an arbitrary ground

of dimension

algebraic

investigation

of a

G, one knows that if

the ground field

k

is perfect

G

possesses

a central

of k-closed s u b g r o u ~ in which every successive quotient k-isomorphic and §8).

Ga

k, except when

dim

G < 2:

dim G = i, and at dimension

by

vector group

Beyond that, practically nothing

a perfect G a if

to the one-dimensional

Ga

all such

G

groups,

with dim G = 2 (cf. §3).

forms of

is k-isomorphic

to of

so that one knows

The state of our knowk

is imperfect:

determined all connected one-dlmensional

extending Rosenlicht's

(cf.§l

2 the central extensions

ledge in this regard gets even worse when Russe~has

is

is known even over

has been completely calculated, G

Ga

series

unipotent

earlier discovery of nontrivial

Ga; but at dimension

71

little else has been known.

We have extended the results summarized above in various directions, gation.

and have also started some new lines of investi-

The contents of the present paper will now be explained:

Very roughly,

Part I

is concerned with the group structure

of the unipotent group, while Part II does the geometry of such group. determine

in

In more detail:

Extending Russe~'s work, we

§2 all k-forms of the vector group

G a (n copies, n ~ i).

To a nontrivial k-form of

in terms of a standardized two-indeterminate given the name of a "k~group of RusselI type". the structure of

EXtcent(B,A),

G a x ... x Ga

expressed

equation we have In

§3 we study

the equivalence classes of

central extensions of commutative k-group schemes

A, B.

After proving a general decomposition we make the theorem more precise both

A

and

B

theorem of

EXtcent(B,A),

in the special case when

correspond to k[F]-modules.

It will be shown

how this theorem together with the results of classification of all two-dimensional

§2 yield a sort of

unipotent k-groups.

In §4 we introduce after Tits the important concept of "k-wound unipotent k-group" and give a new proof to his basic result on the concept.

Section 5 (§5) is concerned mainly

with the conjecture that every k-wound unipotent k-group of dimension

2

is commutative.

to the truth of the conjecture,

We present a counter-example but we also present a number

of results on the central extensions between k-groups of Russe~type,

which will show the plausibility of the conjecture.

Section 6 (§6) ultimately aims at determining all k-forms of A I.

We fall short of the goal, but we have determined all

such forms that are k-rational,

of genus

0

or of genus

I.

We have also calculated the Picard groups of the underlying schemes of certain R u s s e ~ t y p e

k-groups.

criterion in terms of Demazure-Hochschild

In §7 we give a cohomology in order

for the action of a unipotent algebraic k-group scheme on an affine k-scheme to have an affine representable Using the criterion

in part, we characterize

in

quotient. §8 the

underlying scheme of a unipotent k-group both in terms of k-forms of vector groups and in terms of the make-up of the affine algebra of its underlying k-scheme.

The last section

(§9) gives a characterization

of unipotency of an affine k-

group scheme in terms of its hyperalgebra as introduced by one of the authors. generalities

In the Appendix which goes over the

on central

extensions

and proves a six-term

exact sequence, we also discuss the splitting of extensions with kernel

Ga

and some applications.

Some of our results are valid over arbitrary ground fields.

Others lose their significance

has characteristic

0.

if the ground field

In the main, our theory is of interest

over an imperfect ground field. Important previous contributions present [8],

article

[9],

include three pioneering papers of Rosenlicht

[i0], Russell's paper

[15; esp. Chap.

on the subject of the

IV, §4].

made a heavy use of

DG

[ii] and Tits'

lecture notes

For reference material, we have and SGAD

(see References

at the end

for the abbreviations). Our notations are explained in

and conventions §i below.

are rather conformist,

and

Let us reiterate here only that

our "k-group" is synonymous with "k-smooth k-group scheme".

Pai I .

Notations~ conventions and some basic preliminery facts

1.0.

Notaions and conventions.

Throughout, k

field of arbitrary characteristic.

The letter

to represent the characteristic of

k

number. k

p

denotes a

is reserved

when this is a prime

The algebraic closure and the separable closure of

are respectively denoted by

k

and

ks .

The reference to

the ground field or the ground scheme will be usually omitted if it is respectively to

k

algebra means k-algebra and ~k

over

or to S p e c k . ~)

Thus, e.g., an

stands for the tensor product

k.

Categories are denoted by sanserif letters. among them are the following:

~k

commutative, unital k-algebras;

~

:= the category of all groups; abelian groups; ~ k ~~k~~

:= the category of all := the category of all sets; ~

:= the category of all

:= the category of all k-schemes;

:= the category of all affine k-schemes.

are categories, ~ w i l l to

Principal

~.

If

~,~

mean the category of all functors from

Thus, for instance, ~ k G ~

is the category of all

k-group functors. The letter object in

~k'

R

will be used exclusively for the general

namely a typical algebra over

schemes are affine in this paper.

k.

All group

We view an affine k-group

scheme as a representable functor from

~

to

~,

and

we call it algebraic if the representative k-algebra is

finitely generated over said to be unipotent

k.

An algebraic k-group sheme is

if it admits a k-monomorphism

k-group of all upper-triangular fixed size

(see DG-IV,

~2, No.

to the

unipotent matrices of some 2).

With the exception of

§6, all our schemes are affine over the base field and are likewise regarded as representable If

G

functors

is an affine k-group scheme,

affine algebra representing

G.

O(G)

to ~ .

O(G)

denotes the

is considered a Hopf

algebra in the usual fashion.

The underlying k-scheme of

G

= Spec O(G).

is denoted by

G;

thus,

k-group scheme smoo~h over Therefore, classical

or

curtly as a k-group.

our "algebraic k-group" may be regarded as a linear algebraic

For a k-scheme XB

k

X~

B

X

group defined over

and a k-algebra

to denote the B-scheme

Similarly for a k-group scheme We shall always denote by vector group R, and by

We refer to a

Gm

B, we write either X ~ S p e c k (Spec B).

and k-algebra

Ga

B.

the one-dimensional

R I > Ga(R):= the additive group of the ring the one-dimensional

the multiplicative k-homomorphism

G

k.

group of the units

Ga ÷ Ga

is denoted by

F.

while that of

F-I

split torus

given by

The kernel of is denoted by

in

R.

R I > Gm(R):= The Frobenius

x ~Ga(R) I > x p ~ G a ( R ) F

is denoted by (~/pZ).

~p,

Just

to be on the

tions whose meanings conventions

safe

side,

let us e x p l a i n

s h o u l d be o b v i o u s

and f r o m t h e i r context:

some n o t a -

a c c o r d i n g to the c u r r e n t

For a c o m m u t a t i v e

r i n g A,

X

A

denotes

the m u l t i p l i c a t i v e

A;

~(k)

denotes

in the f i e l d the f u n c t i o n X, @(D) we w r i t e

the set of all

k; for a k - i n t e g r a l f i e l d of

:= [f 6 k(X) ~y

for its s t r u c t u r e

A 'D = ~ x

D(x)

A

means

= 03

is a k - G - m o d u l e ,

n o t e d by or

:

We w r i t e

scheme

sheaf;

D

stands

submodule

on this

for

on such a s c h e m e scheme

when a derivation

if

G of

last,

D is

D-constants,

is a k - g r o u p G-invariants

consult

Y,

DG-II,

viz.,

functor is de~

1-2

S.

We

I. card S

have u s e d the s y m b o l

for the c a r d i n a l i t y :=

to m e a n

of the

"by d e f i n i t i o n

set

e q u a l to".

Finally, a list of special objects denoted by skeleton letters will follow:

of

with entries

X, k(X)

the s u b r i n g of

; lastly,

the

A G - for d e t a i l s

SGAD-Exp.

m× m matrices

elements

: (f) + D > 0 3 ; for an a r b i t r a r y

A' , A 'D

~ A'

of i n v e r t i b l e

X; for any d i v i s o r

g i v e n on a r i n g

and

group

~n

:= Spec k[Tl,...,Tn] , the affine n-space over

~n

:= Proj k[T0, TI,... , Tn], the projective

£

:= (...,

~+

:= (I, 2, ...}, the positive

N

:= {0, I, 2, ... }, the natural numbers

i.I.

characteristic

k[F]-modules.

given

k[F]

~, B~k[F]

(~, 4)

where

ring

as a noncommutative

~PF

for

F

all

subject

Endk_gr(Ga)

polynomial

to the relation

X ~ k.

has a right division algorithm.

To wit,

with deg ~ ~ deg B, there is a unique pair

of elements

B = ~

is assumed to have a

p.

k-algebra with one indeterminate

The ring

k

integers;

The endomorphism

is known to be identified

FX =

n-space

-I, 0, i, 2, ...}, the integers;

In the rest of this section, positive

k;

in

+ ~

k[F]

and

subject to

deg ~< deg ~ ,

deg ~, the degree of any

the largest exponent

to

F

~k[F],

of all nonzero

is by definition terms of

~.

v

The proof is easy by induction on

deg~.

Therefore:

A

I. i. 1.

LEMMA.

Every left submodule

Every left ideal of of a free left

k[F]

k[F]-module

i__~sprincipal. is free.

If

k

is perfect,

algorithm,

too.

k[F]

allows a left division

From this and from Goldie's

theory on

quotient rings follows the next lemma: 1.1.2. generated, DG-IV,

If

torsion-free

k

is perfect,

left

every finitely

k[F]-module

is free.

Let a k-algebra

A

be given.

k[F]-module by defining

Fa:= a p

Turn for all

Let us agree that in the context of k[F]-module k-algebra way.

When we say that a left k[F]-module that

into a aEA.

theory every

M

inn this

is a k-algebra

Fm=mP=(the p-th power

with respect to the ring multiplication every

A

shall be considered also a left k[F]-module

it must be tacitly understood

of

M)

holds for

m EM. 1.3.

module,

Frobenius homomorphism.

and let

A

: k ÷ A.

product of the right k-algebra k

Let

M

be a left k-

be a left k-algebra by ~ : k ÷ A

a right k-algebra by ~

over

(See

§3, 6.10 for a proof.)

1.2. left

LEMMA.

We denote the tensor A

and the left k-module

by

which we consider

to be a left k-module via

a ~ ( k ) ( ~ m:=a~) kin,

and

¢.

~(a ~ ) m ) : = ~ ( 7 ~ ) a ~ m

Thus,

M

I0

for all

aEA,

m~M

Typically,

and

X~k.

we take the case when

A=k, ~=id k

~=

and

n

fn: x I > x p

for all

1.3.1. (k,f n) ~

x~k

DEFINITION.

M

(sometimes

and for a fixed

For each k-module

denoted also by

n~N. M, M (pn) :=

k®

nM

and its

P members by

r Xi

Q

n mi)" P

1.3.2.

DEFINITION.

FM:= the mapping for all of

~k,

For each left

M (p) ÷ M

m~M.

FM

defined by

FM( a ~ p

is called the Frobenius

M,

m):= ~Fm homomorphism

M. 1.3.3.

Note that

FM

is a k-linear mapping which is

also a k-algebra homomorphism Furthermore,

if we turn

through the definition a k[F]-module 1.3.4. k':=k I/p X 'P

k[F]-module

M

is a k-algebra.

into a left k[F]-module

F( ~ % m ) : = ~ p o p

Fm, then

FM

is also

homomorphism. LEMMA.

and

Consider

'P @Fro.

M (p)

in case

Then,

Let

f : k' + k M':=k'®M

M

be a left k[F]-module. the homomorphism

given by

a left k [F] -module via

there is a canonical

Let

isomorphism

f(x'):=

F(~'~m):= o_ff k[F]-

module s

(k, f)

~k'

M ' ~ v M (p),

~®(X ' ~ m) I

> a~ ' p ®

m

Ii

which is also a k-algebra

isomorphism in case

M

is a k-

algebra. We omit the proof of the lemma as it is a routine. f Note only that (k¢-~ k' -+ k) = (the p-th power homomorphism k ÷ k)

and use the transitivity of tensor products.

1.3.5.

COROLLARY.

The notations

and assumptions being

the same as in 1.3.4, we have: (i)

M'

is k'[F]-free

if and only if

M (p)

is k [F]

-

free; (ii)

In case

M

is also a k-algebra, M'

if and only if

M (p)

is reduced

is reduced.

The corollary is immediate from 1.3.4 if one merely notes that

f:k' - ~ k

Frobenius and Verschiebung homomorphisms

1.4.

commutative scheme,

is an isomorphism.

Let

group schemes.

and denote by

X (p)

X=Spec A be an affine k-

the affine k-scheme Spec A (p). FA: A (p) ÷ A (see 1.3.2)

The k-algebra homomorphism a k-morphism

X--~X (P) which we denote by

the Frobenius k-morphism of the k-scheme G = a too.

k-group scheme, For more details, 1.4.1.

Then,

G

LEMMA.

Let

of

F G : G ÷ G (p) consult G

F X. X.

gives

It is called In case

X =

is a k-homomorphism,

DG-II, ~7.

be an affine k-group scheme.

i__{sk-smooth if and only if the Frobenius k-morphism

12 FG: G + G (p) Proof that

G

is faithfully flat.

(sketch).

Let

G = Spec A. Notice on one hand

is k-smooth if and only if

and on the other that FA: A (p) ÷ A

FG

G ~

k I/p

is reduced,

is faithfully flat if and only if

is an inclusion,

a special feature of homomorphism

of group schemes over a field.

Then,

the rest is taken care

of by 1.3.5-(ii). 1.4.2.

Let

X

be an affine k-scheme and

symmetric product of

p

copies of

X

of the symmetric group operation on

~3, 4.2).

scheme and denote by

Let

~: G p ÷ G

g = (gl''''' gp) ~ GP(R) I > k-morphism G

G

~ : X ÷ G

the

obtained as the quotient X p = X×...xX

There is known to be a natural k-closed EPx (see DG-IV,

zPx

(p factors).

immersion

be a commutative

X (p)~ k-group

the k-homomorphism p-id G :

gl'''gp ~ G(R).

Then, for every

there is a unique k-morphism

zPx ÷

making the diagram P A

X

> X p --

X (p) c,

> zPx

> Gp

; commutative, 53, 4.3).

where

A

I ...........

> G

is the diagonal morphism

The composite

X (p) ÷ E P x + G

and is called the V erschiebung

(see DG-IV,

is denoted by

(or the shift)

of ~.

~V

In case

13

X

too is a commutative group scheme and

it is easily seen that further special case of in place of

(idG)V

sV

~

is a homomorphism,

is a k-homomorphism.

In the

X = G, ~= idG, we shall write

VG

and call it the Verschiebung of

G.

For a commutative k-group scheme

G

From these follows 1.4.3. and

the

LEMMA.

k-homomorphism~

FG : G ÷ G (p),

VG : G ( p )

÷ G,

the

identities

VGF G = p-id G

hold.

FGV G = p-id

G(P)

(Cf. DG-IV, §3, 4.6.) 1.5.

Let

and

~

Commutative k-group schemes with null Ver~chiebung. denote the category of all commutative affine k-

group schemes with null Verschiebung, the category of all left G~,

and let

k[F]-modules.

~[F]

be

For each object

define

M_(G) : = HOmk_gr(G , Ga)

which is a left

Endk_g r (Ga)-module and hence a left k[F]-

module in a natural fashion. variant functor let

M

~

÷ ~k[F]

G I

be an arbitrary left k[F]-module,

and

gives a contra-

in an obvious manner.

as a p-Lie algebra by defining m' ~ M

> M(G)

m [p] : = Fro. Define

[m, m']

Next

and consider it : = 0

for all

m,

14 U(M)

: = the universal

enveloping k-algebra of

M

as a p-Lie algebra, which is further made into a Hopf algebra with antipode by introducing a Gomultiplication from

m~

> m®l

U(M) ÷ U(M) ® U(M)

arising

+ 1 ® m ( m ~ M C U(M), PBW Theorem :).

Let ~(M)

: = the affine k-group scheme corresponding to the Hopf algebra

which gives a contravariant 1.5.1.

THEOREM.

functor

U(M),

~k[F]

÷~"

There is a natural isomorphism

HOmk_gr(G , D(M)) ~

HOmk[F],mod(M,

functorial with respect to the variables The adjoint pair

(~, ~)

of the categories

~

~(G))

G ~,

M~k[F].

of functors gives a__n_nanti-equivalence

and ~ [ F ] "

Furthermore,

anti-equivalence ~ the objects i__nn ~

which are algebraic

correspond precisely to the objects i__n_n~ [ F ] finitely generated k[F]-modules.

under the

which are

(For the proof, consult

DG-IV, §3, 6.7.) 1.6. of exponent equals

0G

A commutative k-group scheme p

if

for all

p.id G = 0, i.e., if x~G(R).

G

is said to be

x+...+x

(p summands)

15 1.6.1. G

LEMMA.

i__ssof exponent Proof.

A k-smooth commutative p

if~ and only if

By 1.4.3, G

p.id G = 0

k-smooth,

FG

is an epimorphism by 1.4.1. Remark.

Examples

VG=0.

implies p-id G = 0.

if

1.6.2.

and

VG = 0

k-group scheme

then

like

VG = 0

~p

Conversely,

because then

: = Ker F G

show m

that the k-smoothness

assumption cannot be dropped

Even among the unipotent + Ga

by

groups,

y(T, T'):= W(T,T'),

provides us an example of V G # 0.

Let

X0

An object

said to be a k-form of of

The k-form to by a

X 0.

X

k

X0

X ~ K ~ LEMMA~

k-group of exponent

X0

[(X+Y)P-xP-Yp] for which

k-group

object defined

scheme or a k-algebra.

X0, defined over

if for an algebraic

K

y: ~p × ~p

see Appendix, A.4.)

is said to be trivial

(K/k)-form of

1.7.1.

p

we have a k'-isomorphism

If an extension

such that

-I

of exponent

X, of the same type as

k'

:= p

with

be an algebro-geometric

k, such as a k-scheme,

field

W(X, Y)

(For the notations here,

1.7. over

G

G=~p×yG a

in 1.6.1.

is

extension

X ® k ' ~ - X 0 ® k'

if already k-isomorphic

is specified,

any object

k

X

we shall mean

defined over

k

X 0 @ K. A connected commutative p

affine algebraic

is a k-form of a vector group

(Ga)m.

*Compare Prop. I and Prop. 2, p.688, in~ "Extension of vector' groups by Abelian varieties," Amer. J. Math.8__O0(1958), 685-71A, by M. Rosenlicht.

16

Proof.

Let

V G = 0, so that

G G

finitely generated One can take that

M

be a group as described.

By 1.6.1,

is k-isomorphic

to

k[F]-module

by virtue of 1.5.1.

HOmk_gr(G , Ga)

M

for

is torsion-free because

D(M)

for a s~itable

M, from which one sees G

is connected k-smooth.

By extending the ground field to the perfect closure of

k

and by applying 1.1.2 one finds that

k'[F]-free and hence group.

G ~ k'

k' ~

is k'-isomorphic

M

k' is

to a vector

2.

Forms of vector groups;

In this section characteristic

M

k

has positive

p.

such that

where as always M

(§2), the ground field

In this section we shall describe

2.I.

modules

groups of R u s s e ~ t y p e

k

g~M

denotes

is

all left

k[F]-

l?[F]-free of finite rank,

the algebraic

closure of

Let

k.

be such a k[F]-module. 2.1.1.

LEMMA.

Proof.

Since

finite

If M

k

Klk

such that

is free.

K®M

of a free k[F]-module

left PID),

it is enough to show that

Let

be a k-basis

(e i}

M

is finitely generated,

field extension

Since any submodule

is perfect,

of

K.

there is a is

K[F]-free.

is free

(k[F]

K[F]

is k[F]-free.

KIk

is separable,

Since

is a

n

{e~ } means

is also a k-basis that

{e i}

of

K

for any integer

forms a k[F]-basis

of

K[F].

n ~ O.

This

(The lemma

can be proven also by 1.I.~ 2.1.2.

COROLLARY.

extension

Klk

2.2.

Let

The k[F]-linear

such that M(pn) map

There

is a finite Purely

K~M

is K[F]-free.

= (k,f n ) ® M ,

where

M (pn) + M, X ~ x l

(To see this apply the functor

inseparable

~ )

fn: k ÷ k, X~--+ ~P

~ xFnx

is injective.

We shall denote

its

n

18

image by

M [n~

It follows

free for sufficiently

large

from 2.1.2 that n > 0.

M En~

is k[F]-

In the following

let

m

n

be the rank of 2.2.1. rank

over

k[F]/F n

LE~IA.

M/M En~

M/M cn]

is clearly a left k[F]/Fn-module

is local Artinian.

k = k[F]/F

is

extensions.

of

m

elements

M/M En~

Suppose

smallest such that k[F]-basis

of

M/M EI-~ over

Lemma, M/M En~

as a left k[F]/Fn-module. over

that MEn]

M En] .

M / M En~ , where

is not free. Let

(~l,...,Tm)

Yi ~ M.

Then

that

M/M En~

m.

is free.

Let

the

k, which is also left invariant

of rank M

is generated

By counting

one can easily conclude

is a free k[F]/Fn-module 2.3.

The dimension of

and

m, since it is left invariant by field

by field extensions,

and

is a free k[F]/Fn-module

Hence by Nakayama's

dimension of

of

~[F].

m.

Proof.

by

k®M

M

Let

n ~ 1

(Xl,...,Xm~

be the be a

be a k[F]/Fn-basis

is generated by

Yi'S and should be determined by a set of

m

x 1' s

equations

of the form

F n Yi = ~. ~ijxj J Let

A

be the matrix

(with

~.. Ij ~ k [ F ] ) ,

(~ij) with entries

in

i = l,...,m.

k[F]

and write

19

A = A 0 + AIF above

Then we can write

+ ... + ArFr , A i ~ ( k ) .

relation

the

as Fny = AX

where

Y = (yl,...,ym)t

and

X = .(Xl,...,Xm%.

t

are column

vectors.

2.3.1.

LEMMA.

is not c o n t a i n e d

Proof.

in

Since

i = ~,...I ,m, there

A0

is invertible.

means

xi's that

AI,...,A r 2.4.1

M En~ are

B

k[F]-module,

For an integer k[F]

2m

generators

by the following

set of

that

Thus in

let

m

A0

~(kP),

calculations)

and

such

Fnyi ,

that

1 = BF n + CA.

This

is invertible.

If

it follows that

from

M En-l~

is

a contradiction.

n > 0

in

Fnxi

= (BF n + CA)X.

it follows ).

by

C~..~(k[F]).~

+ CFny

(or from direct

entries

on a set of

and

are all c o n t a i n e d

a free

2.4.

free,

is generated

1 = CoA0e~(k

below

already

with

are

AI,...,A r

~m(kP).

X = BFnX

Since

One of

M(n,A)

and an

(mx m ) - m a t r i x

be the left

Xl,...,Xm, relations

A

k[F]-module

yl,...,y m

defined

20

Fn !

=A

~I

kYmJ

for which, the

as

following

AiE@m(k

(cijP)

).

9

For

shorthand

a matrix

will

be

C = (cij)

in

PROPOSITION.

(i)

M(0,A)

:

Fny

= AX.

+ ... + A F r r

A = A 0 + AIF

put

~(k)

~ ~ A (v) = A v) + A V)F + . . .

and l e t

In

with C (~)

(V)F r + Ar

=

for any

is a free m o d u l e

of

mo

(ii)

For any

(iii)

M(n,A (1)) % M(n,A) (p)

In the

following

A0~m

(k)) •

X~m(k[F])

suppose

, M(n,A) 2 M(n,AX).

that

(iv)

M (n,A)

(v)

M ( n , A (I))

(vi)

If

AI,...,Ar~(kP

for

some

M(n-I,B) ~m(k)

our

let us w r i t e

2.4.1. rank

in 2.3,

xm

A0

i__ss i n v e r t i b l e

(that

i__ss,

is t o r s i o n - f r e e . % M(n-I,A)

if )

B = B 0 + BIF

n > 0.

and

M(n,A)

n > 0, then

+ ... e ~ ( k [ F ] )

with

BOG

• (vii) M(n,A) (pn)

is a free

k [F] -module

equivalently

kp

~M(n,A)

is a free

m.

Proof.

of rank

m,

or

-n

-n

(i) and

(~) are

clear.

kp

[F]-module

of r a n k

21

(iii)

Let

generators of Fn(I~Y)

M(n,A).

= A(1)(I®X)

,i ® y m )t IP@x

Xl,...,Xm,

= l®Ix

yl,...,y m

Since in

Fny = AX, it follows that

M(n,A) (p)

l®X

= (l~x I

in

M(n,A) (p)

be the canonical

where

,i ~Xm)t. l~k

I®Y

= ( l ® y I,

(Notice that

x~M(n,A)

)

Hence we

have M(n,A) (p) ~ M(n,A (I)). (iv) of

Let

M(n,A).

xi' Yi' 1 ~ i ! m, be the canonical generators

Define a filtration

{M~}z> 0

on

M(n,A)

as

follows:

Mz

:

Xi k[F]FzYi + [i k[F]xi' if

M~ = ~i k[F]FZ-nxi'

Then we have map (For Let

FM~C__M~+I C _ M ~

-F: M~/M~+ 1 ÷ M~+l/M~+ 2

and

if

~>_n.

~M~

= 0.

The induced

is injective for all

= n-l, use the assumption that

~
A0

Z ~ 0.

is invertible.)

Y = cdFd + Cd+l F d+l + ... be a non-zero element in

Suppose that

c d ~ 0.

Then the induced map

c: M~/M~+ 1 ÷ MZ+d/M~+d+ 1 which is equal to cdFd: Mz/M~+ 1 ÷ MZ+d/M~+d+ 1

is injective for all

Z > 0.

This means that the map

k[F].

22

y: M(n,A) M(n,A)

is injective,

Let

M(n,A(1)).

elements

xi' Yi'

OM2

= 0.

1 <__ i <_ m, be the canonical

Thus we have

Zl,...,z m

in

Fny = A(1)X.

M(n,A (I))

A0Z = Fn-Iy

Hence

A;1)FZ = F n y -

Hence,

FZ = X

torsion-free,

and

this proves

m

(A 1 + ... + A r F r ' I ) x

(A~I)F + . . .

Fny = A ( 1 ) X

Fn-Iy = AZ;

Define

generators

by

Z = (Zl, ... ,Zm )t , a column vector.

where

k[F],

since

is torsion-free. (v)

of

÷ M(n,A)

as

Then we have

+ A(1)Fr)X = A;1)X.

= FAZ.

Since

(Zl,...,z m)

M (n ,A (I))

is

are free over

that

M(n,A (I)) a, M(n-I,A). (vi)

Notice

...e~(k[F])

that

AA; 1 ; B (1)

with B0e@~(k).

f o r some

Hence

N(n,A) ~_ M(n,AA; 1) 2 M ( n - I , B ) .

(v~)

follows

from

(i~) and

(v) directly.

B = B 0 + BIF +

23

2.4.2. matrices

the

n, n'

with entries

be integers

in

k[F]

with

Ai, Bj ~ ( k ) .

k[F]-modules

and

We shall give a necessary that

M(n,A) First,

and

~(k[F]) modules

THEOREM.

Let

b_~e such that M(n,A)

and

n

B (n))

M(n,B)

of

k[F] m.

in order

(v), we have and

°

that n = n' ~ 0

and

Then the

are isomorphic

A, B k[F]-

too each other if

X ~Om(k[F])

. and

Y, Z

such that

Here the matrix

Y

Y0 ~ @ m Proof.

Xm,Yl,...,y m}

(k)

+ YI F +

and

Suppose and

+ Fnz.

can be so chosen

Y = Y0

systems

2.4.1

A0, B 0 ~ G ~ m ( k ) .

B = Y(n)AX

with

,

Then

to each other.

bee a__nninteger

and only if there exist matrices ~(k[F])

c on di ti on

= M ( n + n ' , A (n')) = M(n+n'

be

m x m.

are k-forms

be isomorphic

we may and shall assume

2.4.3.

A, B

A0, B 0 E @ m ( k ) .

M(n',B)

in view of P r o p o s i t i o n

M(n',B)

and

B = B 0 + BIF + ... +

and sufficient

M(n',B)

M(n,A)

Therfore

and

Suppose

M(n,A)

~ 0

of the same size

A = A 0 + AIF + ... + A r F r

Write B sF s

Let

"'"

of the form

+ Yn-i Fn-I

Yi ~ ( k ) .

that

M(n,A)

= M = M(n,B).

{Ul,...,Um,Vl,...,Vm}

of generators

of

M(n,A)

and

Let

{Xl,... ,

denote the canonical

M(n,B)

respectively.

24 Thus we have Fny = Ax

where

x =

M In]

of

module

(Xl,...,Xm~, M, which

of rank

m

y =

bases

M In]

and

with

etc.

M (pn),

2.4.1

(Ul,...,u m}

M/M In]

m

Since

by Lem_ma 2.2.1.

Ymm), by c o u n t i n g

M/M In]

the d i m e n s i o n

{y~l,...,Ymm }

and

of

Therefore

Y E M~(k[F])

is a free

{Vl,...,~mm} there

over

The submodule

is a free

(vii).

2.4.1

is generated

by

of rank (~i'''''

one concludes

form two free exist matrices

k[F]-

(v)).

k[F]/Fn-module

k

k[F]-

It is easy to

form two free

(cf. the proof of P r o p o s i t i o n

On the other hand,

and

(yl,...,ym)t,

by P r o p o s i t i o n

{Xl,...,x m}

M/M In]

Fnv = Bu

is isomorphic

see that of

and

that

k[ F] /F n- ba se s

x ~t~m(kEF])

such that

x = Xu

and

v z Yy

mod.

M [hI

and hence we have v = Yy + Zu for some

Z ~ M~(k[F]).

Here we can take the matrix

be of the form

Y : Y0 + YI F + "'" + Yn-I Fn-I with

Y0 ~ ~ m

(k)

and

Yi ~ ( k ) .

Bu = Fnv = Y(n)Fny and hence

Now we have

+ Fnzu = (y(n)AX + Fnz)u

Y

to

25 B -- Y(n)AX

Conversely matrices

suppose

that

X E G~m(k[F])

and

...,Xm,Yl,...,y m} In order Since

denote

to show

M(n,A)

+ Fnz.

B =

Y(n)AX

+ FnZ

Y, Z e M ~ ( k [ F ] )

the

canonical

= M(n,B),

for some and

let

generators

we can assume

of

that

{x I ,

M(n,A). n>

0.

then

B0 : Y0(n)AoX0, it follows

that

If we put

u = X-Ix

M(n,A)

Y

relation If

result

called

by

with

(n,A)

integer

n ~ 0

M(n,A) (ii)

height

of then

2.5.

n e ~

such that

(i)

M(n,A). M(n,A)

if

For

of pair

implies

an integer

A = A 0 + AIF + ... + Ar Fr ~ ( k [ F ] )

our

case. will

(ii)

be

A0 ~ m

Let

M, we mean

(n',A')

~ M(n',A')

Hence

M

the

(k)

be a

smallest

= k[F] m.

k-form

admissible

Hence

or

the

M(n,A) = M(n,B).

A ~(k[F])

n = 0

of

k[F]-module with

that

in this

is an a d m i s s i b l e

Conclusion.

the

X ~(k[F]).

M (pn)

Every

(n,A)

[ ~(k[F]/Fn).

: k x.

and

By the height

for some

l~f

proves

[ll~Prop.2.3]

(i)

then

~(k[F])

whenever

COROLLARY. with

This

if either

k[F] m.

that

{ U l , . . . , U m , V l , . . . , v m}

with

AX ~ ( k P [ F ] ) of

v = Yy + Zu,

Fnv = Bu.

admissible

k-form

and hence

m = i, then

coincides

2.4.£. A pair

pair,

and

is g e n e r a t e d

defining

and

~ G~m(k)

0

k[F] m

is isomorphic

(n,A). pair,

then

is another

n

is the

admissible

n = n' n > 0 with

and a matrix

A0 ~ ( k )

and

26

M(n,A)

Ai ~(k),

the left k[F]-module

generators

{Xl,...,Xm,Yl,...,y m}

Fn

on a set of

2m

defined by

= A

[YmJ

(XmJ

-n

is a M

(k p

is a

then the matrix whenever

Once

is uniquely determined by

2.6.

M (pn)

In light of 1.5.1, the conclusion

commutative,

~[F]-free

of rank

M(n,A)

and

module

M(n,A)

AX ~ ( k P [ F ] ) the integer

n is the smallest

M;

interpretation:

integer

2.5 above admits If a k-group scheme

(Ga ) m , then

G

m.

G ~ ~(M(n,A)).

(viz., V G = 0)

By the foregoing Conversely,

as in 2.5, the k-group

k-form of a vector group.

is affine, p; hence,

k-smooth and of exponent

by I.~,I,G has null Verschiebung is

is not itself free,

is so chosen,

A

is a k-form of the vector group

algebraic,

M

(n,A)

is k[F]-free.

a more algebro-geometric G

If

can be so chosen that

A

if a k[F]-module

then there is a pair

M = M(n,A).

X ~ GLm(k[F]).

such that

Conversely,

k[F] m

(~/k)-form of

as above such that

n

k[F] m.

/k)-form of

and

~®M(G)

theory, ~(G)

for each ~(M(n,A))

W__eehav___~ethus determined

k[F]is a all

27

k-forms result

of the vector group

A pair

Consider

(n,a)

with

called admissible ai ~ kp

the case hEN

where

we saw just now, nontrivial

By a k-group

unipotent

(n,a)

k-forms

a

subgroup in

or

group

of

Ga, up to k-isomorphisms.

scheme of

and

it therefore

(Ga)2

Let

represents

whose underlying

by the equation

a i ~ kp

for some

is the factor module

by the single

2.8.

By what

r

M(n,a)

(k[F]) 2

n > 0.

= a0X + alXP + ... + a r X p

a0 ~ 0

module

and

of Russell type we

and

n

yP

where

(~) a 0 ~ 0

of R u s s e B type are all and

of this type;

~2

be

will

in the form of

is admissible

the k-groups

be a k-group

is given

a = ~ a i F i E k[F]

i > 0.

~(M(n,a)) k-closed

and

(i) n = 0

for some

~(M(n,a))

m = i, treated by Russell(ibid.).

if either

mean a o n e - d i m e n s i o n a l

scheme

Russets

[Ii].

2.7.

~

(Ga)m , g e n e r a l i z i n g

We prove

relation

at this point

1 ~ i ~ r.

The k[F]-

of the free module

Fn(0,1)

= a(l,0).

a few p r e p a r a t o r y

results

on R u s s e ~ type groups. 2.8. i. If

LEMMA.

n > Z, then

Let

(n,~),

(~,~)

HOmk[F] (M(n,~), M(Z,B))

b_e_eadmissible = 0.

pairs.

28

Proof.

Since

2 k[F], we have Indeed let one of

integer

and

n~k[F]

al,...,a r

k[F]

be an admissible such that

we have

If

2.8.3.

LEMMA.

pairs where

Proof.

Applying

made FmmFn

into

Since g = n = 0.

and

is the smallest

2 k[F].

(n,~)

for some

and

(£,B)

~,BEk[F]-

£ < n, then

be two admissible

__If n > £, then

= o. ~

the functor

a(A2k[F])

the assertion

* For k[F]-mo d u l e s

n > 0

n

Then

HOmk[F](M(l,~),

F( A 2 k [ F ] ) ( ~

torsion-free,

with

that

= 0.

a contradiction.

and

AZM(£,B))

only to show that to see

pair.

Let

n,£ ~ 0

Homk[F](M(n,~),

(n,a)

H(n, ~)(pn)

~ k[F],

ag = F~.

k p, it follows

Let

M(£,B) (pn-l)

Homk[F](M(l,a),k[F])

M(n,a) (p£) ~ k[F]

M(n-£,~)

and

be such that

is not in

COROLLARY.

Proof.

~ M(l,a)

only to show that

g

2.8.2. ~

M(n,a) (pn-l)

M I ~ M ( p n - b , we have A2k[F])

= 0.

Since

It is easy

A2k[F]

is

follows.

M, N, the k-modules

k[F]-modulesvia

for m e M, n e N.

= 0.

F(m @ n)

M ~ N and

M ~ N

:= Fm ~ Fn, F ( m A n )

are :=

3.

Decomposition

co~utative

theorems for central extensions of

group schemes;

application to the two-dimensional

unipotent groups

In this section an arbitrary field;

(§3), at first

from 3.3 until the end, k

have a positive characteristic assumed to be affine. extensions, 3.0.

p.

k

i_!s

is assumed to

All group schemes are

For the generalities

on central

see Appendix below. In this section, we first prove a direct sum

decomposition tative

(in 3.1 and 3.2)

theorem of

k-group schemes

result more elaborate

EXtcent(B,A ) A, B

in case

(see 3.4.1).

for arbitrary commu-

(see 3. 2 ). A = ~(M),

We then make the

B = ~(N)

for some

Finally, we shall show how

k[F]-modules

M, N

these results

in conjunction with our §2 yield a classifica-

tion of all two-dimensional Let

3.1.

denote by from

~(B,A)

B × B

belongs to f(y,z), B(R). f 6

to

Further,

=

0

A.

~(B,A)

f(x,y+z)

~(B,A)

f(x,x)

A, B

unipotent

be commutative

k-group schemes,

the set of all biadditive Thus, a k-morphism if and only if

= f(x,y) denote by

+ f(x,z)

and

k-morphisms B × B ÷ A

z) = f(x,z)

hold for all

8 °(B,A)

x 6 B(R).

f:

f(x+y,

which are antisymmetric, for all

k-groups.

+

x, y, z

the set of those i.e., which satisfy

(This last implies

f(x,y)

=

30

-f(y,x), while the converse monomorphism.)

By defining

(f+g)(x,y):=f(x,y)+g(x,y) ~(B,A)

and

~(B,A)

has a natural 3.2.

modules

into additive

A, B

Moreover,

be commutative

2"id A

each

k-group

has a right

schemes.

Endk_gr(A )-

function):

÷ Extcent (B ,A) ÷

(1)

~(B,A).

inverse

in

Endk_gr(A),

(i) is split exact to give the direct sum decom~0sition:

Extcent(B,A ) = EXtcom(B,A ) ~

~°(B,A)

(2)

Endk_gr(A)-modules. Proof.

(i)

Let

0 ÷ A ÷ G ÷ B ÷ 0

sion, and consider the commutator G × G ÷ G Since

defined by

B

because

A

function

(x,y) ~ G(R)

is commutative,

which factors

YG:

groups.

Endk_g r (A)-module.

(arising from the commutator

If

is a

through the rule

There is an exact sequence of left

(ii)

as

2"id A

for all x,y ~ B(R), we turn the sets

Let

0 ÷ EXtcom(B,A)

then

f + g

structure of left

THEOREM

(i)

is true provided

B x B ÷ A.

exten-

[-, -]:

x G(R)~-~ xyx-ly -I C G(R).

it gives a k-morphism

through the canonical

is central.

be a central

k-morphism

As a result,

G × G ÷ A, G x G ÷ B x B

one obtains

One can verify without difficulty

a k-morphism that

YG

31

is a n t i s y m m e t r i c YG = YG' of

B

G

if

by

EXtcent

and biadditive.

A.

and Thus, into

(B,A)

G'

are equivalent

Gl

> YG

on

structure

see that

~

gives

EXtcent (B,A) ÷ one computes discovers by

@

of

the make-up

EXtcent , one can also

a homomorphism

Finally,

the c o m m u t a t o r

that

of additive

for any

function

it is the c o m m u t a t o r

on

groups

~ ~ Endk_g r (A),

~,G

and quickly

function

on

G

followed

itself. (ii)

Let

~°(B,A).

extensions

a mapping

By re me mb er in g

group YG

defines

central

~°(B,A).

of the additive GI

Also easy is to see that

Assume now that

f ~

group

~°(B,A)

structure

2.id A

and define

by means

has a right

inverse ~/~.

on the k-scheme

B × A

a

of the formula

(b,a)(b' ,a ') = (b+b', a+a'+ / 2)f(b,b'))

for all

a,a' ~ A(R)

biadditive

k-morphism

condition,

B × ~

denote by

B ×(i/2) f

k-split

central

= f.

B × B ÷ A

thus becomes A.

extension

w h i c h we calculate B x B.

and for all

the desired

fl

satisfies

a k-group

> (the extension

splitting

of

to

every cocycle

scheme which we

0 ÷ A + B × ~/2) f

the c o r r e s p o n d i n g

Since

the usual

We have an obvious,

It turns out to be equal Thus,

b,b' ~ B(R).

geometrically A ÷ B + 0

biadditive

function

for on

(i/2) f + (i/2) f = 2 ~ / 2 ) f)

class of

B × ~/2~ f A) gives

(i) and we have e st ab li sh ed

(2). Q.E.D.

32

3.3. M ~ N

Let

L, M

as a left

y e N.

Let

and

map

~:

view

T asa m o r p h i s m

seen

to be biadditive.

3.3.1. + D(L)

LEMMA.

comes

Proof. schemes Hopf

Let

algebras.

The map

We

map,

U(H)

D(N)÷

A = @(X),

Z

algebra

A~

B

f(P(C)) C_ P(A) ~

Fy, x ~ M,

map.

Extend

this

If we

then

r

is easily

f

with

(x,y) l

morphism M~

D(M)

x D(N)

N ÷ L.

be c o m m u t a t i v e C = O(Z)

k-group be their

be a b i - a d d i t i v e

an algebra

map

~ (f(x,y),y),

affine

k-

f: C ÷ A ~ is c l e a r l y

B. a

the map

c~) b~---~ f ( c ) ( l ( ~ b)

map.

Since are

Let

B,

Similarly

f(P(C)) C__ P(A) ~ any b i - a d d i t i v e

P(B).

morphism

map

P(C)

the p r i m i t i v e

P(A) ~

B.

from a k [F] -linear

map

f: X x y ÷ Z

algebra C.

D(L),

B = O(Y),

or e q u i v a l e n t l y

in

y) = F x ~

® U(N).

bi-additive

and

identify

elements

that

F(x~

We view

Conversely,

X, Y

C(~) B ÷ A<~)B,

is a B-Hopf

+

D(M)x

Every

Let

k[F]-modules.

be a k [F] -linear

U(L)

X x y ÷ Z x y,

Y-group

Hence

N

by

from a k [ F ] - l i n e a r

and let

morphism.

be left

k [F] -module

f: L ÷ M ~

to an algebra

N

L ÷ M~

denote

elements

it follows we have Since D(M) N.

the p r i m i t i v e of the B-Hopf

from above

f(P(C))C__ P(U(M))

× D(N)

that

A ( ~ P(B).

= M, this means

÷ D(L)

comes

33

3.4. M2

Let

G I = =D(M~)I

and

G 2 = $(M2) , where

Define

are left k[F]-modules.

E2M2

as follows: if

is the anti-symmetric

elements

in

is the k-subspace

M 2~

generated by

x, y ¢ M2, and

of

~ $ixi~

M2

M1

M2Q~ M 2

and E2M2

p ~ 2; E2M2

x~y

+ y~x,

such that

xi' ~i ¢ k, x i 6 M 2

1

~iFxi M2 ®

p = 2.

= 0, if

This is clearly a k[F]-submodule

of

M2 •

3.4.1.

THEOREM.

With the notations

as above, there are

two exact sequences: 71

0 ÷ EXtcom(G2,Gl)

0 ÷ Ext

÷ EXtcent(G2,Gl)

[F](MI,M2)~+

+ Homk[F] (MI,E2M2)

Extcom(G2,Gl)

has a natural

÷ HOmk[F] (M1,M P)).

(i)

If

p ~ 2,

section.

(ii)

If

M1

i__ssfinitely generated,

o

i!s surjective.

(~i) I_ff M 2

is finitely ~enerated,

o

has a section

which i_ssnatural with respect t__oo M 1 . (iv) Hence if

p ~ 2

and

M2

is finitely generated,

then we have

EXtcent(G2,Gl)

= Ext'[F] (MI,M2)~

which is natural with respect

to

M I.

H°mk[F] (MI,M~P) ~ A2M 2)

34

Proof. 3.2-(i)

Construction

of

7, p

and

and 3.3.1, we can construct

0 ÷ Extcom(G2,Gl)

For each extension the image of

This is clear if

÷ Extcent(G2,Gl)

M 1 ÷ M 2~

p ~ 2, since

By virtue of

an exact sequence:

class represented

~(E):

o:

M2

~ HOmk[F] (MI,M 2 ~

by

E, we must show that

is contained

~(E)

M2).

in

E2M 2.

is anti-symmetric.

If

p = 2, the composite

M1

~(E) MZ® MZ_ ~ g(Mz) x ~ Yl

is clearly

zero.

The Poxncare-Bxrkhoff-Witt

that the kernel of is exactly

)xy

E2M2 .

M2~

M 2 + U(M2)

given by

(i.e.,

E

is commutative).

C

C

killed by the Ver~hiebung, E

x ~y

I

) xy

Thus we get the first exact sequence.

Next suppose given a commutative G2 ÷ 0

theorem implies

extension

0 ÷ Gl + E +

Since the

Gi

the VerChiebung

are

homomorphism

of

factors as

VE: E (p) _.+ G 2(p)

and we may view

o(E)

~E)

G1 ÷ E

as a k [F] -linear map

M 1 + M~ p).

Tile

35

map

EXtcom(G2,Gl)

÷ HOmk[F] (MI,M2(P))

given by

EI

>~(E)

also is a homomorphism whose kernel is exactly the image of the map

P: ExtI[F](MI,M 2) ÷ EXtcom(G2,Gl)

defined by (0 ÷ M 2 + M + M 1 + 0 ) ~ Since

~

(0 ÷ G 1 + ~(M) ÷ G 2 ÷ 0).

is clearly injective, we have the second exact

sequence. (i)

Now, in more detail: In case

E2M 2 = A2M 2

and

p ~ 2, we have the identifications Homk[F](MI,E2M2 ) =

this case, 2-idGl = Endk[F](Ml)

~° (G2,GI).

Also, in

is clearly an isomorphism in

(notice

2F = F2).

Endk_gr(Gl)

Therefore, by 3.2-(ii),

we have a splitting of our exact sequence. (ii)

First consider the case

Ext~[F](k[F], M2) = 0, we have HOmk[F] (k[F], M~p))" = M~p).respect to

M1

M 1 = k[F].

~: EXtcom(Gz,Ga) C--~

The naturality of

~

implies that this is k[F]-linear.

order to show that this is surjective, 1 ~ u ~ Im(~)

for all

(E 0 )

÷G a

0÷Ga

Since

u E M 2.

with Hence, in

it suffices to see

Now let

x -W G a + G a ÷ 0

be the central extension determined by the 2-cocycle (-I/~[(X + Y)P - X P

YP].

It is well-known that the

-W =

36 & Vershiebung that

of

G

a(E0) = i ~

is (x,a) J > (0,x). This means x G a -W a (p) Let u: G 2 ÷ G a be the group 1 c k[F]

homomorphism associated with have clearly

k[F] + 2.12, i]

o(E0u ) = i ~ u.

o:EXtcom(G2,

Ga)

÷

~ u.

Then we

Hence we have

M P).

In general every left k[F]-module has a free resolution of length < 2, since

k[F]

[i, page 81, Remark].

is a left principal ideal domain If

M1

is finitely generated,

is a finitely generated free resolution

there

0 ÷ P ÷ Q ÷ M 1 ÷ 0.

Observe the following commutative diagram: 0 + 0 ÷ ExtI[F] (MI,M2) +~ EXtcom(G2,Gl) ~ HOmk[F] (MI,M~P))

EXtcom(G2,D(Q))

-~ HOmk[F] (Q,M~ p))

+

+

EXtcom(G2,~(P))

All of its rows and columns are exact ÷ ~(Q) ÷ ~(P) ÷ 0 that the map (iii)

a Since

is exact).

-~ HOmk[F] (P~M~P)) •

~otice

that

0 ÷ G1

Therefore one sees immediately

is surjective. M~ p)

is finitely generated,

from (ii) that there is a commutative extension

it follows E: 0 ÷ G~p)-

37

÷ E ÷ G2 ÷ 0 the map

such that

g i v e n by

H°mk[F] (M I,M~ p)) ÷ EXtcom(G 2,G I)

is a section of that

x~

3.5.

then

M1

relation

for

This is clear since

y - y ® x

f~D(f)E

and natural with respect to

~(f) C HOmk_gr(G~P),G I) (iv)

if

Then

~(E) = I ~ Homk[F](M~P),M~P) ).

M I.

(Notice

f ~ HOmk[F] (MI,M~P)).)

A2M2 ~= E2M2

via

....

x A Yl

p ~ 2.

We saw in §2 that if

G1

is a k-form of

is generated by two elements Fnu = ~v, ~ E k[F].

u

and

For such an

v

Ga with one

MI, it is easy

to check

1 EXtK[F](M1,M 2) = M~(FnM2 + ~M2).

3.6. let

n

M

For a left

denote the kernel

HOmk[F](k[F]/Fn,M)

3.6.1.

A2M 2

~ n M.

Examples.

-~Mz/FnMz ~ n (M~p) ) M~ p) ~

k[F]-module

if

n(M2(p)) @ n(A2M2 )

of

M

Fn: M ÷ M.

Notice that

EXtcom(G2,Ga)

if

M2

p ¢ 2.

and an integer

is

f.g..

n > 0,

Then

a n = ~ (k[F]/Fn)" P = M~ p)

EXtcom(G 2,~ n) P

EXtcent(G2,Ga)

=

EXtcent(G2,~ n) = M2/FnM2 P if further M 2 is f . g . (cf. I~ ~II,§3,

38

4.6,

III,

§6,5.3,

3.7. known

7.7

Two-dimensional

(cf.

unipotent some

7.6,

§] g DG-IV,§2) k-groups

admissible

).

unipotent that

connected

are k-forms

pair

(n,a)

groups.

with

of

It is well

1-dimensional Hence

Ga

and

n > 0

they come

from

~ ¢ k[F] : G

~(M (n, ~) ). Let

G

be a c o n n e c t e d

Then c l e a r l y

G

2-dimensional

unipotent

is of one of the following

i)

g

is not commutative.

ii)

G

is c o m m u t a t i v e

three

C

k-group.

types:

•

and

V G (= the VersAhlebung

and

V G = 0.

of

G)

of

G.

0.

iii) In case

G

is c o m m u t a t i v e

i) let

In case

ii)

bung map.

let

Then

1-dimensional Hence and

G 1 = [G,G]

there

G 1 = VG(G(P) ) be the G1

and

unipotent

G 2 = G/G~ schem~s k-groupAand

are two a d m i s s i b l e

B i ~ k[F]

and we have

be the c o m m u t a t o r

such that

a central

pairs

G i = ~(Mi)

image

subgroup

of the Vers~hie-

are c o n n e c t e d

k-smooth

hence

of

k-forms

(ni,~i) where

with

that

p ¢ 2

extension

for simplicity.

ni ~ 0

M i = M(ni,B i)

0 ÷ G 1 ÷ G + G 2 ÷ 0.

Suppose

G a.

2hen we have

39 nI

EXtcent(G2,Gl)=M2/( F

by 3.4 and 3.5. Hence 0.

nI ~ n2 Since

nI < n2 ty

M2+BIM2)~HOmk[F]

In case i) we have by 2.8.3.

nI = 0

from

Homk[F](MI,A2M2)

In case ii) we have if

M~P) = M(n 2 - 1,~2)

or

(MI,M~P)~ A2M2 )

2.8.1.

~ O.

Homk[F](MI,M~P))

n 2 > 0, it follows

that

In either case the possibili-

cannot happen.

nI > n2

v

In case iii) there is a pair ~(k[F]) that

G (n,A)

where

is by 1.7.1a k-form of with

A 0 c~(k)

G -~ D(M(n,A)).

n >__ 0 and

and

(Ga)2

A ~ A 0 + AIF + "'"

AI,-'-~(k)

In rather a formal sense,

is a complete

description

type in terms of

k [F]-modules.

Hence

such the foregoing

of the groups of the envisaged

4.

Wound

In this positive

unipotent

section

groups

(§4),

characteristic PROPOSITION.

• ..,

Xn])

be a n o n c o n s t a n t

f:

I

to the image immersion.

of

has

a

f

~

k-morphism,

where

f

: N1

and

X = Spec(k[Xl,

and d e c o m p o s e

N1

~

X0 c

> X

X0

it as

is the e p i m o r p h i s m

is the closed

Then

(i)

tl

X0

the n o r m a l i z a t i o n (ii)

X0

not

k ( X O)

Let

> X)

X0

k

field

p.

4.i.

x0c

the ground

is X0

of

a finite in

on

and

A1

ins

k(N1);

is a rational

finite

morphism

curve

X 0 comes

from

and the only place

of

the

of

infinite

place

k (~i) ; and

(iii) is

X0

with

A1

if and only

if

X0

k-normal. Proof.

Let v

i_ss k - i s o m o r p h i c

k(X0) over

Only

the

= k(v),

k.

rational

the only place

under

that place, it is clear

4.2.

with

Ga,

set that

COROLLARY.

of

function

v = g(t)/h(t)

Then,

gives

Then

'if' part

not finite

~ on

u = I/(v-a); k[X0] Every

and the kernel

(iii)

requires

field

in one v a r i a b l e

k(Al),

if

and

If

X 0.

a proof.

v

v

t

~ >aEk

> ~ , set

u = v.

= k[u]. quotient

of

of the canonical

Ga

is k - i s o m o r p h i c

homomorphism

i__ss

41

given a_~s the zeroes of a p-polynomial Proof.

The first assertion

over

follows

the second is verified by the well-known

k.

from 4.1-(iii),

and

calculation.

The corollary above is, of course, very well kno.~. 4.3.

Let

G

As noted in §I, group.

be a commutative G

is then a

It is well known

PROP., p.102]) p-polynomials

k-group of exponent

k-closed

(cf.

equations

in several variables.

III, 3.3.1].

G

consist of

(In case

k

is infinite,

k-group.

tq

one in the ambient vector group--see

THEOREM.

Then,

Let

G

G

to be

Tits

[15;

though.)

be a unipotent

algebraic

the following are equivalent:

(i)

Every

(ii)

G

k-homomorphism

contains no

Ga

k-closed

> G

is constant;

subgroup

k-isomorphic

Ga; (iii) Every (iv)

to

one may take

We do not use this result,

4.3.1.

[9;

of

which is really the case that counts, of codimension

subgroup of a vector

[15; Ill, 3.3, p.120 ff],

that the defining

p.

G

k-morphism

contains no

A1

> G

k-closed

is constant;

subscheme

k-isomorphic

A1. The theorem i s e s s e n t i a l l y

p.151 ff].

due to T i t s

We offer a simpler proof which,

yield the decomposition original proof

(ibid.,

Observe first that

theorem

[15;

IV,

however,

4, does not

in its course as did Tits'

IV, 4.2). (iii). >(iv)--~(ii)

is trivially true

42

and ( i i ) = = ~ ( i )

is immediate from 4.2.

shown is the implication

Therefore, what remains to be

(i)~(iii).

We shall conduct i t s proof by

proving two lemmas: 4.4 LEMMA. Let F,

and l e t

k'/k

be a Galois extension with Galois group

G be a commutative

H = HOmk,_gr((Ga) k, , Gk,)

k-group scheme. Then, the set

has a natural r i g h t

and also admits a natural action of

(f~)a = f a j

Furthermore, the HOmk_gr(Ga, G) The f i r s t

for a l l

F on i t s e l f

in such a manner that

~ ~ k ' , a ~ F,

F - i n v a r i a n t elements

which is a

k'-module s t r u c t u r e

k-form of the

HF of

and

f ~ H.

H coincides with

k'-module

H.

h a l f of the lemma hardly requires proof, while the

second h a l f is the standard Galois descent (see, e . g . , [15; I , 5.3.2, p. 42, and Errata, p . l ] ) .

43

LB~A.

4.5.

G

be a c o m m u t a t i v e

p, and let there

exponint

k-morphism. ~ : Ga

Then,

there

exist

exists

~

: ~i

k-group

of

> G, a n o n c o n s t a n t

a nonconstant

k-homomorphism

>G.

Proof. as a

Let

By what we r e m a r k e d

k-closed

subgroup

by the e q u a t i o n s

¢~ E k [ X l ,

~

...,

of

= 0

G a × ... × G a (n (yEN

is a

Xn]

in 4.3, we m a y c o n s i d e r factors)

= an index

p-polynomial;

set),

G

defined

where

each

thus,

for each

a~

.

v,

one can write

~v

~i

+ "'" + ~ n

By a s s u m p t i o n , least one

there

exist

is n o n c o n s t a n t

~

(f(r))

=

for all

yEN.

Assume,

and has

a term

and for each

~

...,

fn E k[T]

of w h i c h

at

such that

+

...

as we may,

with

1 < i < n

b ~ 0.

+

~ n(fn[T))

that

=

fl(T)

Write

s = up

0

(i)

is n o n c o n s t a n t m

, (u, p)

= i,

define

gi(T) := the sum of all h

~vi = "

fl'

l(fl(T))

bT s

up

'

for any

terms

of

fi(T)

of d e g r e e

h > 0 h

and set

gifT1~.:= 0

if no terms

is p r e s e n t

in

¢~l(gl(T))

+ ..... + ¢ n(gn(T))

manner

the

fi(T).

gi(T)

of the form

It follows

are built.

= 0

from

cT up

(I) above,

for all

~ ~ N

(c E k) then,

that

by the

One can now c o n s t r u c t

44

0: Ga ÷ G by defining all

@(R)(y) = ( g l ( y l / U ) ,

....

gn(yl/U))

~ G(R)

for

y ~ Ga(R). Proof of 4.3.1.

given a nonconstant

We now prove ( i ) ~ ( i i i ) k-morhpism

to the separable closure the image of g ~ GK(K)

Us

K = ks

~

x P + Us(X)g Us(X)-Ig -I

to obtain

Us = U 0

the image and replace

whose image is contained in the center of is not contained in the kernel of the x (p factors) on

morphism and repeat the process u n t i l p-th power endomorphism.

po

may apply 4.4 : k-form

In a l l ,

GK.

Since the

nonconstant

ks-module

take a point

by the morphism ks-morphism is

@s: ~

-~ GK @s

p-th power endomorphism G,

compose @s with that endo-

the image of

~

is k i l l e d by the ks-morphism

k-closed subgroup

Z~ G

n s ¢ HOmK_gr((Ga)K, ZK)

and

HOmK_gr((Ga)K, Z K) ~ { 0 } ,

its

cannot be reduced to

k-homomorphism

If

Next, i f the image of

we have a nonconstant

By 4.5, we may assume

HOmk_gr(Ga , Z)

Us

ks-morphism

1 ms : ~K -+ ZK ~ GK with a suitable central of exponent

GK ,

GK"

By repeating t h i s process as long

as necessary, one has at hand a nonconstant

p-id : x~-~ x . . . . .

K : ~÷

The image of the new nonconstant

[G K , GK] = [G , G] O K .

Suppose

and extend the ground f i e l d

is not contained in the center of

not c e n t r a l i z i n g

contained in

U : ~I÷

of the theorem.

n : Ga ÷ Z ~ G.

{0}

and there is hence a

45

4.6.

DEFINITION.

said to be

k-wound

A unipotent algebraic

k-group

is

if any one of the equivalent conditions

of 4.3.1. holds for it. 4.7.

unipotent

PROPOSITION. k-group.

Let

Then,

G

be a connected one-dimensional

the following are equivalent

(i)

G

is

(ii)

G

is a nontrivial

k-form of

Ga

is a nontrivial

k-form of

~i

[iii)

k-wound

:

;

This is an easy corollary to 4.2, 4.3.1, and to the fact that

~I

admits a unique

of the origin

[2~

§3].

k-group structure up to the choice

5.

The question of commutativity

for two-dimensional

unipotent groups

In this section

(~5), the ground field k is assumed to

b__ee imperfect of characteristic 5.0.

p.

In spite of what we saw in §3

(see, esp.,

the state of our knowledge on two-dimensional groups leaves much to be desired.

unipotent k-

In particular,

of those which are k-wound presents considerable for obvious conjecture

reasons.

Among other things,

in some quarters

connected two-dimensional In the present

the study difficulty

there has been a

to the effect that every k-wound

unipotent k-group

is c o m m ~ tative.

section, we show that the conjecture

false but under mild,

additional hypotheses

turn out to be commutative

itself is

all such groups

(see 5.8 to 5.8.4).

with some explicit calculations groups of Russell type.

3.7),

We start out

of homomorphisms

between k-

The calculations pave the way for

later results in ~5, but they are also of some independent interest. 5.1. §2.

Let

G

be a k-form of a vector group discussed

By the height o f

least group.

n ~ 0 By

such that

G, denoted ht(G), we shall mean the G (pn)

is k-isomorphic

2.5, this is equivalent n ~ 0

to the vector

to saying that -n

the least

in

ht(G)

is

-n

such that G~k p is k p -isomorphic to -n a vector group over kp In the present paper, we shall

47 actually use this concept only when i.e., when

G

G

is a k-form of

is k-isomorphic to either

Ga

G a'

or a k-group

of Russe~ type. 5.2. where

Let

G=D(M(m,~))

~ ~k[F], m > 0

of 2.7.

be a k-goup of Russell type

and

Thus, ht(G)=m

(m,~)

by 2.5.

is admissible in the sense Write

and (by a slight abuse of notation) ~x=Fmy.

Write

~=a0+alF+

k(a p

@'=k'[F].

(Gk' ' (Ga)k')

...

isomorphic Xp ® Fm

to

~(G)

M(G)

calculations

is @'-

m ~-~ 1 @ m

as an E-submodule of the @'-module of rank one.

~x=Fmy

A free base

M'(G)=O'z as above.

x=Fmz and (see 2.4

Let

~'

be the left total quotient

and imbed the torsion-free left @'-module

~i'@ r@,

imbedded

~'(G)

(-m) . )

Proof (Sketch).

into

~'(G) :=HOmk,_gr

Through the mapping

~(G)=~@y,

for the notation

~'

E~k[F]

@'-module via F(X ~ m):=

may be chosen so that

y=a(-m) z, where

ring of

Let as above

m~M(G) .

which i s @ ' - f r e e

M'(G)

in

be a

"

made into

PROPOSITION.

(G)=k' ® of

M(G)

X~k',

one can identify

{z)

)

r

Consider the @'-module

k'@

for

5.2.1.

ap '

It is easy to verify that

"

k'

-m '

and let

M~(G)=HOmk_gr(G , G a ) ~ + +a F r, and let

-m

field containing

O(G)=k[x,y], @~k[F]

(G)

in the standard fashion

inside

~'(G)

~5)@,~'(G)

~2['(G)

Having already

as indicated above, we perform to find

48

i Q x = F m ( F - m Q x) 1 ~y

whence

= F-marx

(I)

= a(-m)(F-m~

x) ~ @ '

~__'(G) C ~ ' ( F - m ~

(~ M_~'(G). But, by virtue

of the right divison algorithm in {¢'e@'

:

J' = ~ ' y ' J'

•

¢'(F-m~

since

Fm

J'~'.

see

DG-IV, 5.3.

Ga

y'

F -m~

x

M'(G)

serves as our

z.

~(-m)&

is separable and

x) = ~ ' - f r e e

of

For more details,

§3, n°6.

Let

C 1 = D(M(m,~)),

m = ht(Gl)

C 2 = D(M(n,B))

are

With these notations, we have:

If

m > n

(ii)

If

1 ~ m <_n, then to

(n, fl)

n = ht(G2).

(i)

group

(m,~),

be either

~ = a0+alF+...+ar Fr' fl = b0+bl F+''''+bsFs"

and

THEOREM.

5.3.1.

additive

and

~(-m)

= ~(F-m~

or k-groups of Russell type, where

Thus,

F m ~ J'

so that

must be a nonzero constant 6 k'

admissible pairs and

as

is principal,

By (1) we have

Therefore,

rank one, and

the left ideal J'=

is purely inseparable and

as k'-morphisms, hence

@',

x) E M='(G)}~ 0'

y'6~'.

with

x)

then

the

HOmk_gr(G2,

group

GI) = (0} ;

HOmk_gr(G2,Gl) of

all

elements

and

i_ss isomorPhic ¢

=

k'[F]

-n

with

k' = k p

such that

(23

49

where

~ = k[F]. Proof.

(i)

(See 2.4 for the notations This is clear since

Hom(9((G1), by 2 . 8 . 1 . functor

By means of 5.2.1, one way write:

= ~u~t,

with

mapping

to an

M(GI)

~'-module

into

~'

homomorphism

Since an ~'-module

is completely

determined

is in one-to-one

such that

5.4.

Let

by

n.

x=Fnz, y=~(-n)z

and

proposition

5.4.1.

Fm~w~(G2)

As before, ~G2)=~h+~t

~'(G I) +

homomorphism

~'

such

correspondence = ~Fn+~

(-n) )w

and

~(-m)

be k-groups

let us write , u=Fnw,

with the set

of

M(Gl)=~x+~y

t=~(-n)w.

comes handy in pair with 5.3.

PROPOSITION.

t =

homomorphisms

Gl=~(M(n,a)) , G2=~(M(n,B))

the same height

following

u = Fnw,

M(GI) ÷ M ( G 2 ) , it

z ~......> ~w, the set of all ~ m o d u l e

~(GI) ÷ ~(G2)

and

~(G2) , and vice versa by the

of homomorphism.

M'(G I) ÷ M'(G2)

with

homomorphism

l) (-m) z;

x = Fmz, y =

Bu =Fnt, ~'(G 2) = ~ ' w

uniquely

restriction

of

= ~'z

Given an Q-module

corresponds

that

the Frobenius

times to the situation.)

~x = Fmy, ~'(gl)

M'(G2)

G I) =

by 1.5.1 snd this last is reduced to (0)

n

B(-n)w.

HOmk_gr(G2,

(This can also be proven by applying

(ii)

~(G2)

~(m), ~(-m).)

With the notations

The :

as above, we

have: (i)

If

n ~ I, then

HOmk_gr(G2,

GI)

is isomorphic

50

as additive group to the group of all elements for which there exist

~E~

~¢ = ~(n)Fn

Such a triple M(G2)

(ii)

If

(~, ~, D)

x I

q~O

such that

(3)

gives the homomorphism ~

~u, y [

>

M(G I) ÷

~u + nt.

n > I, then the mapping which sends the triple

subject to (3) to the triple

realizes the Frobenius h0momorphism f(P) E

Homk_g r

,

(~, ~(1)F, ~

).

calculations.

We omit the details.

a monomorphism if and only if

f(P)

is such.

and

Note only

that the homomorphism in (ii) above is injective and

f # 0

(1) )

f E HOmk_gr(G 2, GI) I >

These are immediate consequences of 5.3.1-(ii) elementary

k[F]

+ rl(n) B

(~, ~, n)

defined by

and

@ ~=

f

is

Note also that

is always an epimorphism. We list a simple fact ~Iready used in 2.8.1)as a lemma

here for the sake of easy reference: 5.5.

LEMMA.

least one of ~

= F~

Let

~= i a i F i ~ k [ F ]

al, a2,.., be a non-p-th

for ~ E~, ~ ~

be such that at power in

implies ~ = ~ = 0.

k.

Then,

(Proof is omitted.)

Note that this lemma shows among other things that the n

appearing in (3) of 5.4.1 is never zero for any nonzero

k-homomorphism

G 2 ÷ G I.

51

5.6.

If

an overfield

G = ~(M(n,~)) k' ~

k

is a k-group of R u s s e ~ t y p e ,

satisfies

-n

only if

k'~

k(a~

'''''

aPr

-n

The f i e l d

if and

)

see

[11; Lemma 1 . 3 ,

p.529].

,...,

aPr

)

is called

t h e minimum s p l i t t i n g .

G.

5.6.1.

PROPOSITION.

R u s s e ~ type, and let containing

K

the minimum

exists a power of

Let

G

be a k-wound k-group of

be a field containing splitting

p, d = d(K)

field of

= pV

AUtK_gr(GK)

= {x~K

EndK_gr(GK)

= {yE K : yd = Y }.

: x d-I = I}

k ~ K1 ~ K2

ing the minimum

field of

splitting

integral power of the last becomes

d(Kl)

k

G.

but not

Then,

there

with v ~ I, such that

M0reoyer , for any fields

over

k,

-n

k(a~

field of

G Gk'~(Ga)

-n

and

,

none of which contain-

G, d(K 2)

and a f o r t i o r i

an equality whenever

is a positve

d(K I) ~ d(K2); K2

is separable

K1 . The proposition

p.536].

Therefore,

our method Proof

duplicates

Russell's result

we shall merely sketch its proof,

is somewhat different (Sketch).

the case

and look at

G K.

though

from Russel~s.

By 5.4.1-(ii)

it is enough to consider G = ~(M(I,~))

[II; Th. 3.1,

and remarks

following

it,

ht(G)=l.

Let, therefore,

The defining

equation for

52 r

GK

is

YP = a0x+alxP+'''+ar xp , but one can quickly

that one may assume with no loss of generality a r ~ K p.

That being assumed,

let us look for

that

ascertain a0=l,

~ C K[F]

satisfying

(4) for some

~ 6 K[F],

by 5.4.1-(ii). because

neK[F].

Then

q

= y~K

4, too, must be an element of

a r ~ Kp, so write

~=c.

and a0=l, whence

c=y p.

=~(1)F

x I .... ,x r ~ K

for some

We may assume

Then, ~

Now we have and

K

-q(1)~=~c-yP~=~(1)F

~yP-yP~=x~F+...xPFrr y ~ K.

Thus, our

y

must satisfy the condition i ai(yP)P

for all

i - aiyP = ai(yP

1 ~ i ~ r, or equivalently

for which

a i ~ K p.

gives a solution of

Conversely,

(s)

- y)P ~ K p

each

(4) if one sets

i yP -y=0 y e K ~=yP.

for every satisfying

i (5)

This shows what

we asserted above. 5.6.2. AUtG

REMARK.

: ~k

where as before is a splitting Aut G ~ a

) ~ G

Consider the automorphism given by

k', whence

AutG(R):= AUtR_gr(GR),

is a k-group

field of Aut G

functor

G, Then

of Russell type.

If

k'

Aut G ~ k' = AUt(Ga)k ,=

is a k-form of Aut G

It is easy a

53

to show that

Aut G

is non-representable,

and therefore

a

Aut G

is not,

either.

Notice,

however,

that

Aut G (K)= a

Kx =Gm(K )

for

any field

5.6.3. EXAMPLE. G2=~(M(I, GI)

K~k.

Let

aek,

l+aF2)), both of height I;.

by means of 5.4.1-(i).

we look for all and

c&k

~ E k[F]

such that

8 = l+aF 2.

and

with c

We compute

for which there are

~-cPB

HOmk_gr(G2,

= ~(1)F, where

Further

~

~ 6 k[F] ~=l-aF,

is of the form

show that 2 is subject to the condition that c+ac p 6 k p.

One then concludes

x6k.

GI=~(M(I,I-aF)),

As in the proof of 5.6.1,

Immediately we learn that

= c p + xF x=-c

a ~ k p, and let

that

calculations

HOmk_gr(G2,

GI)

as additive group to the k-rational points

is isomorphic G2(k)

of

G 2.

The fact stands valid if k is replaced by any other field -I not containing ap , in particular by k s . Thus, card

[Hom k _gr((G2) k , (GI) k )] = ~ S

S

in this example.

S

(Cf. 5.10 below.) 5.7. i.e.,

We now turn to the main question of this section,

the question of commutativity

unipotent groups.

Let

G

be a k-wound connected unipotent

algebraic k-group of dimension

2.

If

there arises a central exact sequence where G2

are

G 1 = [G, G] k-forms of

and Ga

for two-dimensional

G 2 = G/G I. and

G1

G

is non-commutative,

0 + G1 ~ G ÷ G2 ÷ 0 Clearly, both

is k-wound;

G1

and

hence so is

54

G2

by virtue of 3.7.

Thus,

to study the groups like

above, one must look at central of R u s s e ~ t y p e

by another

such group.

prove a number of sufficient noncommutative 5.8.

central

THEOREM.

scheme and let commutative).

G2

extensions

Let

G1

of a k-group

By doing so, we shall

conditions

extensions

G

for the absence

of the envisaged

be a commutative

be a k-group of dimension

of

type.

k-group one

(necessarily

Suppose that

(6)

card (H°mks_g r ((G2)ks , (Gl)ks )) < ~.

Then,

EXtcent(G2,

extensions

o_~f G 2

Proof.

by

G1

By 3.2-(i),

antisymmetric So, suppose

GI) = EXtcom(G2,

biadditive

GI),

i.e.,

all central

are commutative. it suffices function

to show that every

G2 x G 2 ÷ G1

given a nonzero biadditive

function ks

is zero. B : G2 x G2

÷ GI; by extending

the base field to

we still have a

nonzero biadditive

function on (G2) k , and therefore we shall S

assume without nonzero, that

loss that

there exists a point

B(a, b) ~ 0.

a nonzero

onto

S

already.

the function

G 2 ÷ GI, hence

x ~ G 2 ( k ).

Since

B

(a,b) ~ (G 2 x G2)(k)

Consequently,

k-homomorphism

only finitely many x ~G2(k )

k=k

is such

B(a,

B(a, x) = 0

-)

is for

On the other hand, map each

B(-, x ) E HOmk_gr(G2,

G I)

to obtain a

55

homomorphism (finite); many

G2(k ) (infinite)

its kernel is infinite;

x 6 G 2 ( k ) , we must have

hence a f o r t i o r i

5.8.1

o__~r (ii)

therefore,

i_ss k-wound.

Proof.

as function and

Let

G1

and

G2

xeG2(k).

b_~e one-dimensional

and suppose that either

card(H°mk Then,

for infinitely

for infinitely many

-gr ((Gl)k

ht(Gl)

>

= ~

and

s

Extcent(G2,

Under the condition

obviously from 5.3.1-(i).

(i)

' (GZ)ks))

s G2

GI)

Q.E.D.

COROLLARY.

u nipotent k-groups,

HOmk_gr(G2,

B(-, x) = 0

B(a, x) = 0

This is a contradiction.

ht(G2)

~

GI) = EXtcom(G2,

GI) holds.

(i), the assertion follows

Now assume

(ii) o

The conclusion

comes then from 5.8 and from the next lemma. 5.8.2.

LEMMA.

unipotent k-groups, and

G2

i__ssk-wound.

Proof.

Let

G1

and

G2

and suppose that

h_ee one-dimensional card(HOmk_gr(Gl,

Then, HOmk_gr(G2,

G2))=

GI) = {0}.

Assume that there were a nonzero k-homomorphism

: G 2 ÷ GI, which would then be an epimorphism clearly. Then, for all Endk_gr(G 2) first

Hom

~EHOmk_gr(Gl,

G2)

the k-homomorphisms

would have to be mutually distinct.

~

6

Since the

set is infinite whereas the second is finite by

5.6.1, we have gotten a contradiction.

56

5.8.3. unipotent

COROLLARY.

k-group.

Then,

This is obvious 5.8.4.

G

be a k-wound one-dimensional

EXtcent(G, G)=EXtcom(G , G).

in view of 5.6.1 and 5.8.

COROLLARY.

two-dimensional

Let

Let

noncommuative

E

be a k-wound connected

unipotent

k-group.

Then,

card (Endk s -gr (Ek s ) ) = ~"

Proof. that

k=k s.

central G2

There is no loss in making the working hypothesis By the noncommutativity,

exact sequence

1 ÷ G 1 ÷ E ÷ G 2 ÷ i, where

are one-dimensional

by virture

of 5.8.1-(i).

is infinite.

Then, as

an inclusion map consequently

HOmk_gr(E,

E)

k-wound k-groups Moreover, E ÷ G2

HOmk_gr(G2,

HOmk_gr(E , GI)

a monomorphism,

we can construct

by 5.8,

GI)

>

GI=[E , El,

ht(G I) ~ ht(G 2) HOmk_gr(G2,

is an epimorphism

inclusion

But

G I)

there arises

HOmk_gr(E , GI)

is infinite.

so that another obtains,

and

a

and

G1 ÷ E

Homk_gr(E,

is

G I)

which shows that the last Hom set is

infinite. 5.9. unipotent

Let

be a k-wound connected

noncommutative

may be written of

E

k-group.

By 5.8, we know that W__e_econjecture

We have seen that

in the form of a central

Ga, 1 ÷ G 1 ÷ E ÷ G 2 ÷ I, with

two-dimensional E

extension of k-forms

1 __< ht(Gl)__< ht(G2).

HOmks_g r ((G2)ks , (GI) k s ) is infinite.

that the converse

is valid.

To wit:

If

57

EXtcent(G2,

GI)

= EXtcom(G2,

GI) , t h e n

c a r d [H°mk s - gr ((G2)ks

(*) (el) k )] <

~.

s

5.10.

Example.

We p r e s e n t

example

of a n o n c o m m u t a t i v e

k-group

of d i m e n s i o n

(see below) Let to

k-wound

It was

by S h i z u o

Endo.

a ~ k, a ~ k p,

and

0 £ m £ n.

We p r o d u c e

2.

Let

a nonzero

G2 × G2 + G1 m x ax p and

let

y]

assignments

x t

,>

connected

discovered

m, n

be

~ m up

~)

k[u,

t],

k[u,

t] Q

u

first

for

integers

m=n=l

subject

G 2 = ~(M(n,l+aF2m)).

biadditive

Writing

the

unipotent

first

antisymmetric

U ~ = Spec

k[x,

is p r o b a b l y

G 1 = ~=D(M(m, l - a F m ) ) ,

as f o l l o w s :

a function

what

function

m G 1 = Spec k [ x , y ] , yP = n 2m tp = u + au p , we d e f i n e k[u, m

u ® up

t]

through

the n-m

and

Y I

that

this

>

u @ tp

n-m tp

@

u.

It

antisymmetric using

f

is

mechanical

biadditive

calculated cf.

(*)

5.6.3

and

GI)

a central

k-group.

in case

gives

: G 2 × G 2 ÷ G I.

The m i d d l e

unipotent

HOmk_gr(G2,

f

construct

1 ÷ G 1 ÷ G 2 x f G 1 ÷ G 2 ÷ I. k-wound

verify

function

as a 2 - c o c y c l e ,

noncommutative

to

term

an

Then,

extension is the

Recall

m = n = i

desired

that we already

--

5.9 above.

In early 1974 a counter-example to this conjecture was constructed by Mr. Tsutomu Oshima, a graduate student at Tokyo Metropolitan University.

Part 11 6.

Forms of the affine lin e and geometry of the groups

of R u s s e ~ type

In this section characteristic 6.0. of

A I.

(§6), k

is an imperfect field of

p.

We investigate

in the present

section the k-forms

After some preparations on purely inseparable descent

and derivations,

we firstly present

in 6.5 a necessary and

sufficient condition in order for a k-form of trivial.

Next,

rational k-forms of

A I.

In particular,

6.7.9 of all kwe determine all

k-groups of R u s s e ~ type in 6.9.2.

classify completely all k-forms of genera are either points

to be

in 6.7, we consider the function field of

such a form and give a characterization

k-rational

~i

0

or

1

whose arithmetic

and which carry k-rational

(see 6.8.1 and 6.8.3).

explicit calculations

A1

Thirdly, we

Lastly, we make out some

and determine the Picard groups of the

underlying k-schemes of certain R u s s e ~ type k-groups and the paragraphs and 6.12.2). EXtcent(G,Gm)

that follow it -- especially 6.10.1,

6.11.1

We have also appended a remark in 6.13 concerning for k-groups

G

of R u s s e ~ t y p e .

Notations used only in this section following:

(see 6.10

For a k-scheme

classes of invertible

X, Pic(X):

sheaves on

(§6) include the

= the group of isomorphism

X; for a k-algebra

A,

59

C(A):

= the divisor

= Spec

k[r,T "I]

class

= %

= (A 1

Pic and Aut are used,

Our useful

thanks

facts

owe 6.7.3.

forms

of

A1

k'-algebra. k'

~

D

Dp = 0

on

A'

Let

of

Then:

A = A'

that

of 6.7.7

k-derivation so that

on

D( ~l i ~ D(X)

element

a'

respectively.

out

some

In particular,

of purely

a k-suba!gebra

= i.

Dp = 0

A k' ai)

and

such

A

we

inseparable

D(%)

Let

A'

be a

such that D

on

A'

such

for each k - d e r i v a t i o n

= I, the k - s u b a l g e b r a satisfies

as above.

that

inseparable

A-derivation

Conversely,

be given

=

be a purely

%P 6 k, % ~ k.

is a unique

D(X)

i

Dp = 0 ,

$i:

to him,

A = A'D( = the ring o f D-constants)

Let

and Also,

for pointing

and examples

with

Given

and

proof.

point}).

see 6.7 and 6.9,

k' = k(%)

k

there

with

[12];

A 2.

LE~.~4A.~

extension

as in

section.

by some remarks and

A

to Mike A r t i n

to this

and a proof

6.1.

simple

are due

of

{k-rational

for which

relevant

We begin

group

d(k)

Xd(%i) ® a i.

Let

= i.

Let

A'

= k'

®

d

be the unique D = d ~

It is obvious

A.

I,

that

1

= 1

of

and

A'

A ~ A 'D

in the form

To p r o v e

A 'D

Pi I i a' = . 0ai ~

A, write with

any

a i e A.

1=

Then

* This

D(

Pi I " Pilia. li-I aikl ) = i= 0 i= 0 z

lemmaVfollows also

easily

Thus

ai = 0

for

i=l, • • • ,

"

from

Cor.7.10

in the next

section.

60

p

- i.

Hence

a' = a 0 £ A.

Conversely see

that

A = A'

any

element

of

(0 < n < p) i > n.

let D

be g i v e n

as b e f o r e .

is a k - s u b a l g e b r a

A'

such

Then

D

Since that

A'.

D p = 0, t h e r e

D n(a')

D n(a') ¢

of

# 0

A, and

and

D n(a'

It is e a s y Let

a'

to be

is an i n t e g e r D i(a')

= 0

- n ~1 D n ( a , ) ~ n )

n

for all = 0.

By

c a n be n, we k n o w t h a t any e l e m e n t a' of A' p -i xi In w r i t t e n in the f o r m ~ ai . The e x p r e s s i o n is u n i q u e . p-i i=O i n be the i n t e g e r s u c h that fact, let _~ a i k = 0. Let p-i i 0 i > n. Then Dn( [ ai ki) = n ! a an ~ 0 and a i = 0 for all i=0 n = 0. Thus a = 0. This is a c o n t r a d i c t i o n . Hence a0 = induction

on

n

....

a p l_=

0.

6.2.

This

inseparable the

first

forms

group

6.2.1.

D(k)

A1

Let

A' t.

k[x,y]/(yP

A2

of R u s s e ~ t y p e f o r m of

Let

= k'[t] Let

= I, D p = 0

k[tP,t+~tP].

or

Write

= x + ~ x p).

in c o n s t r u c t i n g

We s h a l l

is a r e c o n s t r u c t i o n

Example.

one v a r i a b l e by

of

inseparable

~ k.

is u s e f u l

o n e of w h i c h

unipotent purely

lemma

and

two e x a m p l e s

of a o n e - d i m e n s i o n a l

and the o t h e r

is a n o n - t r i v i a l

A2.

k' = k(~)

with

= the p o l y n o m i a l D

give

purely

ring

be a k - d e r i v a t i o n D(t)

y = t + ~t p

= t p. and

k p = ~ ~ k,

Then

over

on

A'

k'

in

defined

A = A 'D =

x = t p.

Then

A =

61

6.2.2.

Examp!e.

~2 = ~ E k,

X ~ k.

over

k'

D

A' by D(X)=I,

on

that

Let

Let

on

x

D(x)=xy A'.

zation domain. their

decomposition

that

t, s, u, v

tu =sv.

Hence

6.3. condition

for a purely

be trivial. which

Our main

we shall

recall

of which we shall Let

next

A'

make

K'

is given

a derivation

Let

K = K'

Denote and form

by

D

and

C(A')

let and

below

where

P

D ~ 0

in

A.

When

However

domain.

and sufficient

A1

or

S1

Samuel's

to

Theorem,

convenience

of c h a r a c t e r i s t i c

field on

of

A'.

K' K.

such A

and

An element

of

p # 0

Assume that

that

there

D(A') ~ A'

is then a Krull

the divisor

over

factori-

generalization.

A = A' ~

runs

k-

it is clear

factorization

form of

is

indeterminates.

elements

for the reader's

domain

is not

v=y+Xy 2.

for a n e c e s s a r y

a slight

C(A)

and

is the following

be the quotient

A, respectively. YnpP, ~

tool

A

is considered,

inseparable

be a Krull

and let

look

to see

is not a unique

irreducible

ring

A = A 'D

of two

u=x+Xxy

is not a unique

We shall

It is easy

k

A

k'[x,y]

with

a k-derivation

show that

ring over

t=y 2, s=xy,

are all

Define

D-constants

We shall

in

A

y.

to show that

Let

k' = k(~)

be the p o l y n o m i a l

and D(y)=y 2

to a p o l y n o m i a l it suffices

and

The ring of

k[x2,y2,xy,x+Xxy,y+~y2]. isomorphic

and let

A' = k'[x,y]

in two variables

D2 = 0

For this

p = 2

class

C(A)

all prime

groups

is w r i t t e n ideals

domain.

of

A'

in the

of height

62

1

of

A

and

almost all

np

P.

is an integer such that

Since

A 'p ~ A, above each

and only one prime ideal cation index to

ep,

np = 0

P' ~ A'

is either

of height

there is one i, whose ramifi-

or

p.

Assigning

~npP, we have the canonical map

j:

C(A) + C(A').

To describe group

L

z 6 K'

X

1

P

for

Ker(j), we shall introduce the abelian

consisting of logarithmic derivatives

and

consisting

D(z)/z

of

~npep,P'

¢ A'

D(u')/u'

Let with

L0 u'

be t h e

a A'

×

D(z)/z

subgroup of

With t h e s e

with

L

notations,

Samuel's Theorem states: 6.3.1.

LEMMA.

With the notations that 1

D(A')

in

i

i

C(A)

sends where

corresponding

of height

Vp, t_OO P ' .

6.4.

1

J

D(z)/z

if the ramification P'

as above, assume moreover

in any prime

ideal of height

Then we have an exact sequence.

0 ÷ L/L 0

(p' ~ A)

[12; Theorem 3.2, p.62]).

and assumptions

is not contained

A'

The map

(P. Samuel

of

C(A').

(modulo

LO)

is the discrete The map index

ep,

j is

t__O_O ~p,

valuation

is surjective 1

on

K'

if and only

for all prime

ideals

A'

To generalize Samuel's Theorem,

following situation:

(z)/ep,).

consider the

63

Let

k be a f i e l d

connected O(G)

k-group

is f i n i t e

the h e i g h t N-times Let

A'

field the

of c h a r a c t e r i s t i c

scheme.

dimensional.

iterated

Frobenius

be a K r u l l

domain

of A'.

Assume

k-scheme

X'

the

) A'

can c o n s u l t If we put A'

the

E k for any

every

uniquely

extended

too,

Since

K = K 'G, we h a v e

of A'

ep,

of P'IP

index

is d e f i n e d X:

K'

X

it f o l l o w s

that

the

is t r i v i a l .

G acts

quotient

freely

on

via

coaction.

action

and

from DG-III,

(The r e a d e r coactionO

§2, n°6

A-module

that N fP

of r a n k rk(O). Since N that a 'p ~ A for any a' E

of K'

is of the

Therefore

K ~

~ (K'

Let A'

the

form b/a

coaction

A'.

with

p can be

divides same

¢ O(G)

K be the

= A. H e n c e

of h e i g h t

through

in the

> K'

K'

clearly.

ideals

those

N be

to a c o a c t i o n

is free

with

scheme

of free

A'

which

the p r i m e

let K' be the

algebra

~ X'

it f o l l o w s

p:

and

and

concepts

element

(0) and b ~

Let

N such

on the r i g h t

G

the

) G (pN)

k-group

x

that

G

associated

projective

f ~ 0(G),

In p a r t i c u l a r

X'

integer

be the

A = A 'G, t h e n

is a f i n i t e

a ~ A -

¢ O(G)

§7 for

FN:

= Spec(A')

u:

p: A'

map

G a finite

= [O(G):k].

smallest

k-algebra

that

we m e a n

Put rk(G)

of G,defined as the

affine

Let

By G f i n i t e

p > 0 and

clearly.

fashion

@ O(G)) x ,

A <

zl

~ P'

.....

;,

x(z)

of A.

domain

bijectively

The r a m i f i c a t i o n

The map

as above.

field

A is a K r u l l

i of A c o r r e s p o n d

P = P' ~ pN

quotient

j: C(A)

Put

= (z ~ l ) - 1 p ( z ) .

> C(A')

Then

X is a h o m o m o r p h i s m

clearly. p

N

Hence

x ( K 'x)

of

abelian

= K ' X / K x is

groups an

abelian

and

Ker(x)

group

of

= Kx exponent

Therefore

L:= x(K 'x) /~ (A' (90(G)) is

a subgroup

of

x ( K 'x)

and

contains

the

subgroup

X

L 0 := x(A'

6.4.1.

The multiplicative groups

prise as special cases

Samuel's groups

introduced in 6.3 above. g = ~

× ... x ~ P

L0

L, L 0

we defined just now comof logarithmic derivatives

To see that fact, consider the case when

(r factors).

O(G) on

Let

•

A' •

=

D. I

for each

0: A' ÷ A'

Then, for each

I 0
v(i) = v i

where and

L,

Write O(G) = k[r I .... , rr]/(T ~ ..... T~) =

P

k[tl,., ., tr] of

)•

--

O O(G) = A'

[t I ,

.... t r]

be the coaction

a' e A' , one may write

(DI(1)''" DVr(r)(a')/vl!'''vr!)tl (I)''" tv(r)r l O
for each i < i < r

i ~ i ! r

ranges over the integers

stands for the derivation on

0,i,..., p-l,

A' caused by

65

the action of the

i-th

factor

~

of

G = ~

P

§2,

6.4;

cf.

7.1.2

D.D. = DoD° i ] ] l

family

for all

{D.:

a eoaction

and hence obtains

It

is

rather

i ~ i, j ~ r.

1 < i < r}

1

can d e f i n e

below).

p

O(G)

an action of

that

Conversely,

on

with

that the action of

U l'~ . . . ~

U r'

1 < i,

j < r.

of

A'

G

such that

This follows,

D~ = 0 1

× ... x e

commuting

for every

1

A' by means o f t h e

G = ~

and

given a mutually

D~ = 0

P observe

(see DG-III, P

obvious

of derivations of

× ... x ~ P

i,

one

formula above,

on Spec

A'.

Let us

P

is free if and only if there are elements Di(uj)

= 613 ..

(Kronecker's

for instance,

from 7.9.1

delta)

for all

in the next section. T×

Now the above formula for X(Z) e A' Q

O(G)

also shows that This

(coordinate-wise

shows that for any

if and only if X(z) = 1

fact permits

P

additive)

Di(z)/z e A' for all

if and only if

us to identify

z e K

L = X(K

Di(z)/z = 0 '×)

n (A' Q

group of logarithmic

we have 1 ! i ! r;

it

for all 1 ! i ! r. O(G))

with the

derivatives

,×

{(DI(Z)/Z . . . . .

and that as in

L 0 = x(A

TM)

z e A TM . 6.3.

Dr(Z)/Z):

z e K

with the subgroup of all

If in particular

, Di(z)/z e A'

for all

(Dl(Z)/Z ..... Dr(Z)/Z )

r = i, we recover Samuel's

L

and

i}

such L0

66

6.4.2.

(Generalized Samuel's Lemma (6.3.1)). With the notations

LEMMA

and assumptions as in 6.4, there is an exact sequence

0 ÷ L/L 0 where the map P = P' N A

i

Proof. Samuel's [npP.

X(Z)

Our proof

L0)

t__o_o

is e s s e n t i a l l y

the

6.3.1.

j([npP)

with

(modulo

~(Vp, (z)/ep,)P

and

6.3.1.

Theorem

Then

divisor

sends

as in

C(A'),

C(A)

Write

an element

= 0 4--9 [npep,P'

a ' E K' x

same

of C(A)

= (a'),

Thus ep, t V p , ( a ' )

as that

for

of

in the form

a principal all

P'.

Therefore,

×

for

any P',

with

P'~

there

A = P, i.e.,

element

of A'p,.

p(A'p,)

C

Since that

exists

A'p,

Thus

a' = au', x(a')

@ 0(G),

P' is arbitrary x(a') E

an a a K

u' being

= x(a)x(u')

it follows

and A'

A' ® 0(G),

such that

$.e~,

that

is a Krull

Vp,(a')

= Vp(a)

an invertible

= X(U'). X(U')

Since

~ A'p,

domain,

@ 0(G)

it follows

x(a') E L. ×

Moreover, (

~ x(a'/w')

i.e., map

[npP

x(a') ~

L0 <

= i, i.e.,

is a p r i n c i p a l

¢: Ker(j)

> L/L 0.

F x(a')

= X(W')

a = a'/w' ~ divisor.

KX - ~

Thus

with w' ~ (a')A,

A'

= (a)A,

we have an injective

67

We shall next element

that the i n d u c e d

coaction

the a s s e r t i o n w i l l

6.4.3.

SUBLEMMA.

as in 6.4.2,

assume

X(Z) E A' ~ O(G),

Proof. Hence

ep,

First

= i or p. Let

X

X

be such that

We can a s s u m e

valuation

such that

Vp,(Z).

G is of h e i g h t

z ~ K'

that p~n.

i. T h e n ep, Ip. X(Z) E A' ~ 0(G) Write

z = ut n

w i t h u E A 'x. T h e n x(t) n = X ( z ) x ( u -I) ~ A' ~ O(G). is an a b e l i a n g r o u p Since

P' = tA',

induced Since

coaction

A/P ~

of e x p o n e n t

it f o l l o w s

that

p: A'/P'

p, we have

x(t) E

Since L A' ~ 0(G).

p(P') C__ P' ~ 0(G).

~ A'/P'

(A'/P') G, w h e r e

valuation

ideal w i t h t a

is an e l e m e n t

divides

that

is

and a s s u m p t i o n s

A' is a d i s c r e t e

If z E K'

assume

Vp,(Z).

~ 0(G)

is a d i s c r e t e

be its u n i q u e m a x i m a l

t h e n ep,

and put n ~ Vp,(Z).

~ A'p,

W i t h t h e same n o t a t i o n s

parameter.

divides

It

f r o m the f o l l o w i n g :

f u r t h e r that

r i n g and let P' = tA' uniformizing

p: A'p,

X(Z) be an

i of A'.

i n d e x ep,

Since A'p,

follow

Let

ideal of h e i g h t

to see that the r a m i f i c a t i o n

free and that Ap = (A'p,) G ring,

¢ is s u r j e c t i v e .

of L. Let P' be a p r i m e

suffices Notice

show that

~ 0(G)

P = P'~

rk(G) = [K':K] ~ [ A ' / P ' : A / P ] e p ,

The

is of c o u r s e

free.

A, we have the i n e q u a l i t y

~ [A'/P':(A'/p')GIep, = rk(G)ep,.

It f o l l o w s

that

ep,

= i divides

In g e n e r a l we p r o v e the h e i g h t F:

G

Vp,(Z)

= n.

the a s s e r t i o n by the i n d u c t i o n on

of G. Let H be the k e r n e l

) G (p). T h e n H is of h e i g h t

of the F r o b e n i u s

I and the h e i g h t

map

of G/H

68

is smaller coaction

than N. Put A" = A 'H and P" = P' ~

p: A"

) A" @ O(G/H)

is easily

Let ep, jp, ep, jp,, and ep, jp denote of P'JP,

P'IP"

A". The

seen

to be free.

the r a m i f i c a t i o n

and P"JP respectively.

Then

induced

i n ~ ces

ep, jp = ep, ip,,ep,,jp"

×

Suppose

that

z E K'

is of height z = ux with field

i, we have u ~ A'

of A".

X(x)

X

Since

= X(Z/U)

it follows

is

and

such

that

×(z)

ep, ip,,IVp,(Z). x C K"

X

where

~ (A' ® O(G)) ~

Hence

Since

H

we can write

K" d e n o t e

the

(K" ® O(G/H))

hypothesis

Therefore

ep, ip = ep, ip,,ep,,l P divides

= Vp,(Z),

q.e.d.

inseparable

~ O(G).

quotient

then

from the i n d u c t i o n

A necessary

~ A'

and

sufficient

that

= A" @ O(G/H),

ep,,jp, IVp,,(x).

ep, jp,,Vp,,(x) = Vp,(X)

condition

for a purely

form of ~I or S I to be trivial

is given

in the

following: 6.5. extension

THEOREM.

Let k' be

an

of k. Let A be a finitely

arbitrary generated

k-algebra

and

let A' = k' @ A. Then we have: (i) Assume

that

there

is a finite

scheme

G which

acts

freely

Assume

further

that

A' i__ssk ' - i s o m o r p h i c

k'[t]

over

on k' in such a way that

k' in one variable

p olynomia! ' ring A is a unique

k[u]

and c o n n e c t e d

k = k 'G

to a p o l y n o m i a l

ring

t. Then A is k - i s o m o r p h i c

to a

over k in one variable

factorization

k-group

domain

u if and only

and Spec(A)

if

has a k - r a t i o n a l

point. (ii) Assum~

that

A'

i_~sk ' - i s o m o r p h i c

to k'[t,t -I] with

69

an i n d e t e r m i n a t e

t. T h e n

a_nn i n d e t e r m i n a t e

u if a n d o n l y

k'X/k ×

A is k - i s o m o r p h i c if the

Remark. ~o of k , ~ t h a t k'

holds. p(v(i))

there

= kin I] @

are

> A'

elements

that

(i) E n o u g h

@ O(G)

coaction

~(t)

= 0. T h e n K e r ( ~ )

X(C).

Here

assume

~

= k' ~ kit],

commutes

@ O(G). there

note

that

condition

such

of

(i)

where

naturally

"if"

part.

to a free

By v i r t u e

C(A)

and

freely

By a s s u m p t i o n

such

that

coaction

of L e m m a

k = ~:

A'

6 . 4 . Z we

extended

with

it

follows

k'-homomorphism

~:

We m a y

assume

is G - i n v a r i a n t ,

that

x(t)

= (t @ l ) - I p ( t )

~ A' @ O(G).

A 'x = k 'x

= i,

is a

= tA'

Hence

x(t)

there

the G - c o a c t i o n s .

is an e l e m e n t

that

.... > C ( A ' ) .

L = L 0. M o r e o v e r

~: A ---9 k. The

that

L = L0,

i =< i =< r,

> k' @ O(G)

extended

~} L / L 0

clearly

Since

the

A = A 'G

) k'

tA'

the

G acts

p: k'

A'

C_

k',

extension

x ... × ~ p ( v ( r ) ) "

Then

= 0 by h y p o t h e s i s ,

k-homomorphism

~(tA')

and~

sequence

0

C(A)

a I. E

Then

to p r o v e

s u c h that

inseparable

k = k 'G

can be u n i q u e l y

an e x a c t

Since

homomorphism

is a f i n i t e ~ m o d u i a r

= [k[ai]:k].

is a free

k 'G. T h i s

k'

G = ~p(v(1))

in s u c h a w a y

there

that

... @ k [ a r ] .

put

= pV(i)

Proof.

have

Suppose

In fact

on k'

canonical

> A ' × / A x is an i s o m 0 r p h i s m . kpurely

that

to k [ u , u -I] w i t h

i.e., that

c E

k 'x such

Replacing

t ~ A. S i n c e A = kit].

that

t by t/c A

~

kit]

is~

x(t) we

=

can

a n d k' @ A

70 (ii) then

obvious The

is

The

X' ~

"only that

"if"

if" part:

We

shall

It is

that

,

t ~ A 'x and k ' X / k x = A' /A

X

, there

u = X'-it ~ A x. T h e n A _~ k [ u , u -I] This

means

immediately

statement

of T h e o r e m

and that

A = k[u,u-l].

6.6. becomes

u for t.

X

Since

k' ~ A = k' ® k [ u , u - l ] . we h a v e

take

k ' X / k x = A ' X / A x.

part:

k'×such

We m a y

Remarks.

false give

(i) Let ~

(a) T h e

if any one two

P

first

of the

conditions

is d r o p p e d .

examples.

be the p r i m e

let k be a p u r e l y

two

6.5

field

transcendental

of

characteristic

extension

~

(t,u) P

p and

71

of

•

with variables

t

and

u.

Let

A = k[X,Y]/(YP

=

P t + X + uxP). Spec(A)

Then,

A

has no k - r a t i o n a l

a polynomial obvious

ring over

that

variable.

O k.

A

If

domains,

k' ®

but

(see 6.11.2

out another

A

k"@

and

k ( t l / P , u I/p)

A

ring

over

curious

k

fact

u I/2) O

are both unique

is not,

@

A

in one variable.

k' = k(t I/2, A

domain,

because

is

It is

in one about

this

k" = k(t I/2)

factorization

Pic(k"®

A) ~ 2Z/2Z~

below).

(ii)

Let

G = Spec(A)

A = k [ X , Y ] / ( Y p = X + ~X p)

is a o n e - d i m e n s i o n a l

type w h i c h unique

and

is not a p o l y n o m i a l

p = 2, let

Then,

point

factorization

k ( t l / P , u I/p)

Let us point

example.

is not

isomorphic

factorization

point

is a unique

to

domain,

with

unipotent G a.

although

~ 6 k, @ k p.

group

of Russell

Hence

A

is not a

it has

a k-rational

(0,0). (b)

Let

G = Spec(B)

be a o n e - d i m e n s i o n a l

unipotent

n

group

of R u s s e ~ type,

where

B = k[X,Y]/(Y p

= a0X + •.. +

r

ar

Xp )

point

with

a!s E k i

(x = O, y = O)

and some

aj ~ k p

is a k - r a t i o n a l

G-(x = 0, y = 0) = Spec(A)

with

for

point

of

A = B[i/x].

j > i. G

The

and

Then

Spec(A)

-n

isApurely and let

inseparable A'

= k' ® A.

of the canonical

form of

$i

It is easy

homomorphism

Let

k'

to show that

k'×/k × ÷ A ' × / A ×

k a(

-n

•

ap ' "'' r

the cokernel is

Z /pnz

)

72

Hence,

6.5 - (ii) shows that Spec(A)

is a non-trivial

form of

$i.

6.7. ~i.

We shall now take a closer look at the k-forms of

Let

X

be a k-form of

completion of over C

k

hI

containing

Let

P

6.7.1.

X

as a dense open set. X

Proof.

k'

of

= C-X. LEMMA.

Let

k, Ck,-Xk,

to

With the notations

C'

k'-rational.

A~,.

C'-Xk,,

extension of

P~

extension of

P

P

is

k. k.

Then

Xk,

(Use the additive and multiplicative Let

C'

is k'-isomorphic Since

as above,

be a perfect closure of

Theorem 90 of Hilbert.) Then

is a one place point,

Then we have

k'

is k'-isomorphic

point

Such a curve

up to k-isomorphisms.

rational over a purely inseparable

Ck,.

be a k-normal

is a one place point which might be singular a n ~ for any

field extension too.

C

X, i.e--, a complete k-normal curve defined

exists and is determined by

C-X

and let

be a k'-normalization of to

pl.

Hence

C'-Xk,

is

is dominated by the k'-rational

is rational over a purely

inseparable

k, q.e.d.

By virtue of the well-known

existence theorem of the

Picard scheme for a proper k-scheme consider the Picard scheme

Pic C/k

is locally of finite type over

k.

(see FGA, 236-02), we may of the carve

C, w~lich

The neutral component

73

P1Cc/k. o

of

Picc/k

is of finite type over

k.

Some of the

properties of the Picard scheme which we shall make use of later is summarized in 6.7.2. such that P.

LEMMA. HO(X, O x )

Let

X

be a proper scheme over

= k

and that

X

k

has a k-rational point

Then:

(i) Pic(S)

For any

S e~,

we have

(direct product).

on

PxS

% PicX/k(k'). (ii) Let PicXk •

,/k'

-~

Proof.

k'

Pic~/k (i)

consists of the

PicX/k(S )

isomorphism classes of invertible restrictions

Pic(Xs) ~ PicX/k(S ) ×

are trivial.

sheaves

k'

Let

morphism of a k-scheme

and

PiC~k,/k,

f: X ÷ Spec(k) X.

k.

PiC(Xk, )

~

Then the Picard scheme

Picx/k

Rlf. (Gm, X) , Moreover,

the spectral sequence

= HP(s, Rqf,(Gm,Xs)) ~ H P + q ( X s , G m , X s )

gives rise to an exact sequence~

k'.

be the structure

in the sense of (f.p.q.c)-cohomology.

EP2,q

whose

Then,

% Pic°x/k

is identified with the first derived cohomology considered

XS

In particular,

be a field extension of ®

on

74

o -+ HI(S,Gm,s)

~ H 1 (Xs,Gm, X)-+ H2

H 2 (S,Gm,s)

On the other hand, f

(Xs,Gm,Xs).

has a section

~(the only point of Spec(k)) = P. section

uS •

Therefore

HO(s,Rlf, (Gm,Xs ))

H 2 (S Gin,S) ,

~, f~ = I, such that Hence f~

fs: XS--~S }

H2

(Xs,Gm,Xs)

has a is

injective and f~

0 ÷ HI(S,Gm,s) ~ S ~ H 1 (Xs,Gm,Xs) ÷ H 0 (S,RIf,(Gm,Xs)) ÷ 0

is split-exact. Pic (S)

Pic(Xs) ~_ PicX/k(S ) ×

(cf. FGA-232).

(~i) ~ .

Thence follows that

Let

S'~,

and consider

S'

as an object of

By the isomorphism of (i), Pic.x/k(S') ~= Pic(X×S')/Pic(S')

PicXk,/k,(S'). isomorphism

Hence

Pic~/k

Picx k,/k' ~ PicX/k ® k'

® k' ~ PiC~k,/k,

The

follows from the fact

that the neutral component of a group scheme defined over a field is preserved by a base field extension. 6.7.3.

(M. Artin) LEMMA~ Let

scheme such that either

X X

generically separable over scheme.

be a k-regular proper integral has a k-rational point or k, and let

Then any rational map

everywhere on

V.

V

X

be a smooth k-

f: V--~Picx/k

i__ssdefined

is

75

Proof. f

Let

ks

be the separable closure of

is defined everywhere on

V ~ k s + Picx/k ® ks

V

point

f ® ks:

is defined everywhere on

V ® ks.

we may assume that

Then in both cases, X

U

be a dense open set of

defined and let

g = flu

(i), g ~ PiCx/k(U ) on that

XxU XxU

is representable

such that is regular,

D

~

has a k-rational

~: V ÷ Picx/k

~

D

on D

X×U in

= L I X x U.

such that

is defined everywhere on

by an invertible P×U.

such that

X×V, which

Then

~[U = g"

is

By virtue of 6.7.2

~

on ~

sheaf

Note here Therefore ~

=

~(D).

X×V

such that

defines a k-morphism

Hence

f = ~.

Thus

V, q.e.d.

k-normal completion of a k-form Let us suppose that

X

of

X = C - {P }

A1

C

and let

be a P~ =

has a k-rational

P0" 6.7.4.

LEMNA.

PI'c~/~

-

is regular too.

Going back to the notations of 6.7, we let

point

f

hence locally factorial.

Then, there is an invertible sheaf and

on which

is trivial on

be the closure of

= ~(D-)

V

: U ÷ Picx/k.

there is a Well divisor

C-X.

k

P. Let

Let

Then

if and only if

Hence, without any loss of generality, is separably closed.

k.

is a k-smooth affine k-group

f

76

scheme and there is a k-morphis m

i: C-{P

that for any field extension

of

k'

k

} + Pie°c/k and for

such

Q6C(k')-

{P }, i(Q) = Q-Po" o PICc/k

Proof. = 0

(see FGA, 236-15), which

because

C

is a curve.

other hand, since k. .

for k-scheme

C

Pic~/k

O

is smooth if

is certainly

Hence

Pic~/k

H 2(C, ~ C )

the case here

is k-smooth.

On the

is affine by virtue of Lemma 6.7.2

is rational

over a purely .

(The abelian rank of

PICc/k

C

O

PIcc/k

inseparable

is zero.)

is a k-smooth affine connected

- (~),

extension of

Therefore,

algebraic

k-group

scheme. We shall prove the second statement. U

is a smooth affine k-scheme

is regular,

the divisor

by an invertible

sheaf

on

A

PoXU, where

a k-morphism point of

extension since

i': U k'

of

U

k

UxU

UxU.

such that

~

U×U

is trivial

Then

i'(P0)

Picc/k.

{P~}.

is representable

is connected,

and for

Since

such that of

= C

i'

~

defines

= the neutral

is factored

For any field

Q GC(k') -

{P~},

i(Q) = Q-Po

= (Q-Po)XQ.

Before going to the next step, we shall recall

the Riemann-Roch smooth)

CxU

i ~ PlCc/k. o ~

{A - Po×U} n ( C x Q ) 6.7.5.

on

is the diagonal

Since

Picc/k.

as follows:

on

i': U + Picc/k

U

of dimension i.

A - PoXU ~

Let

Theorem on a k-normal

complete curve

C

(not necessarily

as given in 6.7.

k-

The notations

77

being the same as at the beginning divisor on

C

degree of algebraic

whos e support does not contain

D, deg(D),

is defined

closure of

k

canonical projection. (P).

of 6.7, let

= degc

(EGA, chap. m

(¢

C~ -i

P .

The

be the

Then, the base

, (1.4.15))

closed fields

be the

Let

is smooth outside of

(D)).

Roch Theorem on a (not necessarily over algebraically

be a

¢: C~ = C®K ÷ C

and let The curve

Then deg(D):

change theorem

as fellows:

D

and the Riemann-

smooth)

(cf.

complete curve

[14; chap.

IV]) together

give: 6.7.6.

LEMMA.

With the notations

and assumptions.* a__ss

above, we have

dim H 0 ( C , ~ ( D ) ) = dim HI(c,(~C) dim Itl(c,

where

~ (a)

*

If

= the

~ ~ ~

sheaf on

i_s_sth___9_edualizing

described w

arithmetic

(see

genus of

+ 1 - ~, C, a n d

C, which satisfies:

sheaf on

C[, which can be

[14; p.78]).

i__ssa__nninvertible

n = 0, the support of

(cf. Theorem 6.7.9).

= deg(D)

= dim H0(C,m®(~(-D)),

is the dualizing

expliuitly (b)

(~(D))

- dim HI(c, (~(D))

sheaf if and only if the local

D

is allowed to contain

P

78

ring of

C~

6.7.7. such that

a_~t C~ - X~

is a Gorenstein

LEMMA.

C

C' = C ~

purely inseparable that

C

k'

2

-

be a smooth complete k-curve

i__~sk'-isomorphic

algebraic

has a k-rational

e~ual to '

Let

Then

C

extension

point

is

.

.

~M. Artin L P r 0 ~ o f ~ W e may assume that

that C'

G

is k'-isomorphic

Then the cycle

to

pP'

divisor on

-2.

n

consider

Then

be a positive

the divisor

dim H0(C, ~(pP'+nK))

= 2

Q

Indeed,

since

has a k'-rational

point

is a k-rational

is a k-rational

on

C.

Let = p

C.

K

be the

and

deg(K)

p = 2n+l

P'

=

and

Then deg(pP'+nK)

and dim HI(c, ~(pP'+nK))

= i.

= 0.

genus of

c

is zero.

positive divisor

Q

on

is linearly equivalent

I,Q

p ~ 2.

D = pP'+nK, we have

Note here that the arithmetic there

i_~s

k "

deg(pP')

on

By virtue of 6.7.6 applied to

p

~I

integer such that

pP'+nK

Assume

is a simple extension

if

p1k'' C'

C.

k.

First of all, we shall show

is k-rational

canonical Let

point

to

.

k'

[k':k] = p.

has a k-rational

of

for a

if the characteristic

.

i, i.e.,

t__o_o P~,,

k'

k-isomurphic

•

of exponent

ring.

to

Since

pP'+nK.

point of

C

C

Thus

such that

deg(pP'+nK)

which does not ramify

= in

C' In both cases Q.

(p=2

or

Then the k-rational

map

linear system

IQI

~2), C

has a k-rational

f: C ÷ P~

is a k-isomorphism,

point

defined by a complete since

1 fk': C' + ~k'

79

is such. 6.7.8.

Remark.

If

p = 2~ Lemma 6.7.6

the existence of a k-rational

point on

shows the following

Let

transcendental variables

t

and

u.

Let

C

of

Spec(A)

is a hypersurface

C

fact,

C

Spec(A)

The foregoing

such that C

Spec(A)

points, whereas

has no k-rational

~2

with

y2 = tZ2+XZ+uX 2.

However

were k-rational,

many k-rational

be a purely

Then

as imbedded canonically

defined by

k' = k(t I/2- , u I/2)- . if

k = $2(t,u)

as

A = k[X,Y]/(Y 2 = t+X+uX2).

is a smooth complete k-curve

for

is assumed,

extension of the prime field

the completion ~

example:

C

is false unless

Ck,

in

Hence

is k'-rational

is not k-rational.

In

should have sufficiently

it is easy to see that

point.

lemmas 6.7.4 through 6.7.7 combined

give

the following: 6.7.9.

THEOREM.

Let

curve carrying ~ point purely inseparable {P }

point

equivalent (i) immersion,

P0"

A I.

be a k-normal

such that

algebraic

is a k-form of

rational

P

C

P

complete k-

is rational

extension o__ff k

Assume

that

over

and that

C - (P }

C

has a k-

Then, the following conditions

are

to each other: i: C - (P } ÷ Pic°c/k and

•

O

P1Cc/k

given in 6.7.4

i_~s generated a_~sa k-group

is a closed scheme by

8O

the image of

i;

(ii)

dim Pic ° C/k > O;

(iii)

C

is not k-isomorphic to

Pi' i.e., C

is not

k-rational ;

(iv)

C-{P }

cannot b_e_eembedded into a smooth complete

k-curve. Proof. ~(iii):

(i)~> (ii)

~f.

are obvious.

o dim P ICC/k = dim H I (C,~c)

Since

~ > 0.

(iii)~(ii):

Hence

C

(ii)

(cf. FGA,195-16),

•

the arithmetic genus to

and (iv) ~ ( i i i )

is not k-isomorphic

It is well known that a k-normal

complete curve with a k-rational point and with zero arithmetic genus is k-isomorphic to

P~.

Thence follows our assertion.

(iii)===>(iv):

If C-(P } can be imbedded into a smooth complete

k-curve

should be k-isomorphic to

~, ~

then implies that

C

is k-isomorphic to

Let

k

be the algebraic closure of

let

C

be the normalization of

over

P .

Let

C'

unique singular point of ~: C' + U.

local rings of points respectively.

With

C.

P~

(ii)~

~ = C P

C'

~, we have only to show that

k

and

defined as a hyperP'

be the

We shall show that there is a

In fact, let

and

~

(i):

be the point of

Y2Z = X 3, and let

P~, P'~ 2'

Let

Lemma 6.7.6

P~.

k, let

be a curve in

surface with the equation

[-morphism

C.

and

~, ~' P~

on

and ~

~ C'

be the and

~,

~__ identified with subrings of ~'

dominates

~__. Iden'tifying

81

with tke localization of a one-parameter ~[t]

at the ideal

(t), we then have

does not dominate Then,

if

~

=g denote the completions

we s h o u l d

have

~ = ~.

is an ~-module of finite type. dim Pic___°C/k = 0 ~.

Now, t h e

phism

morphism

~:

C'

Then

F~I

Q 6 [(~)

{p }.

isomorphic

to

All,

÷ C

follows that

i

Let

rise

to

~ = i ~

~

and

C'

is a closed

a generalized Jacobian variety

J

with a module

m = (~+I)P , where

FGA, 195-16).

Therefore,

Pic ° -

scheme by the image of

-

i-.

C-It"

Hence

a k-group scheme by the image of Pic__°~/~

=~'

--~,

because

__@' dominates a k-homomor-

~: U - {P } +

pot (Q) = q - P

is a closed immersion,

The fact that

~ = ~

it is easy to show that

In particular,

~_ and

Therefore

gives

c - {p }

isomorphism.

If

This implies that

is given by

since

ring

t~ - t2~._

of

Hence

(note (ii)~=~(iv)).

~: Pic___°~/g + Pic°c,/g.

Pic°u/[.

~' = ~ + t2~.=

~, ~ contains an element of

and

respectively,

polynomial

P'

are

lot

is

a [-

immersion.

too.

Pic ° C-/[

constructed

m

for

0

from

It is C -- PI t

= dim H I(C,~C ), (cf. is generated as a k-group Pic__°C/k i.

is generated as

Q.E.D.

is isomorphic

to a generalized

3acobian variety and the results of [14; chap. V, Nos. 16, 17] imply 6.7.10.

THEOREM.

The notations

the same as in 6.7.9, Pic°c/k

and assumptions

being

is a commutative unipotent

82

algebraic k-group.

More precisely,

Pic°c/k

is a k-form of

a product of Witt vector groups of finite lengths. 6.8.

In the following

complete classification genera

two paragraphs,

of all k-forms of

(defined as the arithmetic

completions)

are equal to

of a k-rational

a

or

genus

A1

whose arithmetic

genera of their k-normal i, assuming only the existence

point on each form.

the case of arithmetic Let

0

we shall give a

First of all, consider

zero.

be an element of

k

kp

and let

n

be a

positive integer. Let ~: Pl--,P pn be the embedding of ~i ~pn tpnl n into given by t I > (l,t,.-., ,tp -a), where t is a parameter

of

pl.

Let

P~

be a point of

pl

defined by

n

tp

= a.

Denote by

Xa, n

the image

~(pl_{p

}).

Then we

have: 6.8.1.

THEOREM.

is k-isomorphic

(i)

to either

Every k-rational A1

oh -

and

*

for suitable

a e k-k p

X

is a k-rational

a~n

k-form of

A1

not k-isomorphic

~i. (iii)

m

a,n

n ~ ~+.*

(ii) to

X

-

k-form of

=

n

If

Xa, n

i_!s k-isomorphic

and there exist

~,B,y,6

p >_ 3, all k-forms of

are k-rational

(cf. 6.7.7).

A1

to in

Xb, m k pn

if and only if such that

with arithmetic

~6

genus

~y ~ 0

zero

83

and

(~a + ~)/(ya + 6) = b. Proof.

and let

(i)

C

Let

X

be a k-normal

be any k-rational

k-form of

completion

Theorem

and Lemma 6.7.1 then show that and

P

= C - X

extension of that t n tp = a

is rational

k.

P .

a 6 k, where

X.

over a purely t

to

~

inseparable

of

Suppose that n

6.7.9

is k-isomorphic

Choose a parameter

is finite at with

C

of

A1

algebraic

C (= ~ )

P

such

is given by

is the smallest non-negative n

integer such that P~

P~

is a k-rational

Assume that

is given by

point.

n > 0.

t p =a

Then

X

The divisor

P

is

with

a6k.

k-isomorphic

of

C

If

n=0,

to

has degree

~i. p

n

Hence,

dim H0(C,~(P~)) n

= pn + i.

Since

n n

(l,I/(t p -a),t/(t p -a),

n

"'" , t p - 1 / ( t P

-a))

i s a k - b a s i s of ~ e c o m p l e t e

linear

system

n

I P ~ l , the embedding by

(c

tl

~: C + PP

~ (1,t,''',tpn-l,tpn-a). P~) (ii)

which is Xa~n. n Let ¢: ~I + ~p

d e f i n e d by Then

X

i Pt

is given

i s k - i s o m o r p h i c to

be the embedding which defines n

X

a,n

and let

Then the point

k(al/pn).

be a point of ~(P )

Therefore

not isomorphic (iii)

P

If

~: Xa, n + Xb, m

to

pl

given by

is not k-rational,

X a,n

tp

= a.

but rational over

is a k-rational

k-form of

A I,

A I.

Xa, n

is k-isomorphic

to

extends to a k-isomorphism

k-normal completions

Xa,n

and

Xb,m, ~

a k-isomorphism between their

Xb,m' which sends the point t pn

84

pm

_

= a

of

Xa, n

and

Xb,m

(c~'t

+ B')/(y't

are

to the point identified + 6')

t'

with

with

= b ~.,

of

~

- ~'y'

have

(~a + ~)l(ya + 6)= b.

"if" part

¢ 0.

a',~',y',6'

~ k

genus

We shall is equal

complete

curve with

rational

point

the h y p e r s u r f a c e

Y2Z

with if

+ ~XYZ

X,~,~,y p ~ 2,3

condition point,

setting

tI

such

that

n , ...,~ = 6 'p , we

~ = ~'P

(Clearly,

~a,n

we h a v e

m = n.)

The

is obvious.

6.8.2. metic

Then,

If

is given by n

a'6'

X%, m.

next to

i.

the case where

It is known

the arithmetic

is k - i s o m o r p h i c

that

genus

1

to a plane

the arith-

a k-normal and with

cubic

a k-

defined

by

equation

+ ~YZ 2 = X 3 + ~X2Z

~ k, where and that

that

consider

we may assume

X = ~ =

the above

+ ~XZ 2 + yZ 3

plane

and we are led by direct

0 cubic

if

that

k = ~ = ~ = 0

p ~ 2.

have

Impose

a one-place

calculations

the singular

to the following

two cases:

or

(I)

p = 3

and

Y2Z

= X 3 + yZ 3

with

(2)

p = 2

and

Y2Z = X 3 + BXZ 2 + yZ 3

y ~ k 3, with

~ ~ k2

y ~ k 2. Let

conditions

C

be the plane and let

P

cubic

be the

satisfying singular

one of the above

point

of

C.

Then

85

P

= (-yl/3,0,1)

in the second C - {P }

y@

case.

case

On the other

has a structure

k~isomorphic k~group

in the first

hand,

P

= (Bl/2,yl/2,1)

by virtue

of a u n i p o t e n t

to a k-group

of Russell type

and

of 6.7.9,

k-group,

of Russell type.

hence

is

The c o r r e s p o n d i n g

is:

(i)' p = 3

and

y3 = x

yx 3

with

y @ k 3,

(2)' p = 2

and

y4 = x + ~x 2 + y2x4

with

~ ~ k2

or

k2• It is easy

it is not

to see that

straightforward

Let us therefore homogeneous (2)" setting curve and

(I) c o r r e s p o n d s

to see that

explain:

Write

y = T/V

and

x = U/V.

is given by Then,

(T,U,V)

(2)"

is

The

singular

(2)'

in a

is the equation

t = Y/Z

Summarizing

This and

and with

p = 3

Every

the a r i t h m e t i c

o f the following

k-groups and

t = T/U

u = (t 2 + y)/v.

Hence

Then we have

obtained

from

(2),

u = X/Z.

the argument

THEOREM.

Let

of the

t 4 + v 3 + 6v 2 + y2 = 0.

t 2 = u 3 + ~u + y.

(i)

(2)'

point

= (yl/2,1,0).

Let

point

of

to

= 0

((t 2 + y)/v) 2 = v + ~.

6.8.3.

However,

form

v = V/U.

setting

(i)'

(2) corresponds

the equation

T 4 + UV 3 + ~U2V 2 + y2U4

(2)"

to

and c o m p u t a t i o n s k-form ~enus

of one

AI

above,

with

we have

a k-rational

is k - i s o m o r p h i c

to one

of Russell type:

y3 = x

yx 3

with

y ~ k 3,

*After the manuscript for the present paper was completed, the authors became a~¢are of Clifford S. Queen's paper, "Non-conservative function fields of genus one, i", Arch. Math. (Basel), 22 (1971), 612-623. One can show that his Theorem I is equivalent to 6.8.3 here.

86

(2)

p = 2

y4 = x + ~x 2 + y2x4

and

with

$ ~ k2

2.

or

Remark.

6.8.4.

every k-form of homogeneous

A1

to R u s s e ~ [ii; Prop. 4.1],

with arithmetic

genus one is a principal

space for a k-group of Russe~ type.

are specifically misses

According

the case

given in our Theorem 6.8.3. (2) (p = 2) above

These k-groups

Note that R u s s e ~

in his paper

(cf.

ibid.,

p.539). 6.9. k-normal

Let

completion

the automorphism automorphisms as

AutC/k,

component, type

of

X.

functor

Since

and let

C

is projective

be a over

by a k-scheme

denoted

k, whose neutral

k-group

scheme of finite

221-10).

has a k-rational

(Rosenlicht

point

P0'

P

[I0], Russell [ii]).

= C

X.

The following

Assume that conditions

Let X

are then

to each other: X

has a k-group

structure with

P0

as the neutral

point. (ii)

k,

>the group of all S-

CS, is representable

be as above and let

equivalent

C

S ~ ~ I

PROPOSITION

C

(i)

of

AUt°c/k , is a connected

(cf. FGA,

and

A1

be a k-form of

locally of finite type over

6.9.1 X

X

X

i__ssisomorphic to the underlying

k-group of Russell type.

scheme of a

87

(iii)

AUtc/k(ks)

the separable closure of If the arithmetic tions are equivalent (iv) k-groups group

H

genus of

X

is non'zero,

these condi-

to

Pic°c/k

such that

is

S

k.

There exists a surjective from

k

is an infinite group, where

homomor~hism

to a one-dimensional

p o i

p

unipotent

is an isomorphism with

(P0) = the neutral point o__ff H.

o_~f k-

(p o i)

(For the notations,

see

6.7.4). Proof.

(i)--~(ii):

commutative,

since

Any group structure on

dim X = I.

Hence

(ii)~(iii):

isomorphic

to

give rise to AutC/k(ks)

X.

Let

G

of

G a.

(ii) ~ ~(i):

be a k-group of R u s s e ~ type

Since translations

R-automorphisms

is

it is a k-form of

Then we are done by R u s s e ~ [Ii; Theorem 2.1]. Obvious.

X

by elements

of

G(R)

CR, G c~Autc/k.

Hence

(iii)~(ii):

Russe~

is an infinite group.

[Ii; Theorem 4.2.]. Now assume that the arithmetic and let

H

be a group structure

the neutral point. (poi)(k'): and Then

Q~, Q p o i

X(k') P0

For any field

÷ H(k')

by

given on k'

X

over

X

is non-zero

with

P0

k, define

(sum with respect to the group law of

is a k-isomorphism.

homomorphism

as

P01---, the neutral point of

as a group scheme by the image of surjective

genus of

Since

Pic°c/k

i, p o i

p : Pic~/k_ ~ ÷ H.

H

H).

is generated

extends to a

This implies

88

(i) ~

(iv).

(iv) ~

6.9.2. G

THEOREM.

be a k-group

k(G)

(i): Obvious.

over

k

(Rosenlicht

of Russelltype. is rational

d e f i n e d by an equation Proof.

rational

point

tion by

P

of

at

Choose

~I _ G.

a, or

ca

2

completion

G

Hence

Let

t(~ 1

+ (d - a)a

2

Then the d i m e n s i o n

~i

into

the image a~.

Q.E.D.

~2

~(~i

~

to

= a

e 2 with

map

given by

~

+ d)

inseparable

t

defined by

over

is finite + d) =

k

by

In the latter

Let

P

system IP I

t I ~ (l,t,t2-a) A2

_ G

k] < 2.

e ~ 0, 2

a ~ k, ~ k 2 linear

~I

is excluded.)

Since

in

w it h

[k(~):

(This case

scheme

This automor-

(as + b)/(ca

Therefore

is a curve

The transla-

such that

Then

of the complete

_ p )

pl

= b.

has a k-

the point

is k-rational. G a.

G

G.

+ b)/(ct

~ is purely

and

two and the rational of

of

G) = ~.

or

is isomorphic p = 2, d = a

t

- b = 0.

p = 2

Set again

of

i_~s 2.

of the u n d e r l y i n g ~i

> ( at

G

a~k

point.

ad - bc ~ 0, and leaves

a parameter

Hence

case,

t[

and

We shall prove

is rational,

a k-automorphism

On the other hand, 6.7.1.

k(G)

Let

field

a~k,

with

is straightforward•

is given by and

p = 2

other than the neutral

a k-normal

~I

a,b,c,d ~ k fixed•

P

induces

G, hence

p h i s m of

y2 = x + ax 2

Since

[Ii]).

Then the function

if and on!y if

The "if" part

the "only if" part.

[I0], R u s s e H

= b/c. = pl

IP I

G is

is an embedding

(see 6.8.1).

defined by

Then

y2 = x +

89

6.9.3.

When our theorems 6.7.9, 6.8.1 and 6.8.3 are reviewed in

conjunction with Rosenlicht-Russe~'s

6.9.1 and 6.9.2,

that at least over a separably closed field

~k1

s

all those k-forms of

have been completely determined which either possess infinitely many

k-automorphisms of

k = k

it may be said

1 Ak k

or are of arithmetic

of arithmetic

morphisms?

genus

> 1

genus <_ i.

possessing only finitely many k-auto-

The answer is in the affirmative,

example communicated

Example

Are there, then, k-forms

as seen from the following

to us by M. Rosenlicht:

Let

(M. Rosenlicht).

k = k

and

p > 3, and fix an element

s

a E k - k p.

Consider

y 2zp-2 = x p - az p

C:

a hyperelliptic points P2

(in

k-curve of arithmetic genus

P1 = (al/P' 0, i), P2 = (0, i, 0)

is not.

field k(C).

However, Let

P2

parameters of

and X

2, respectively,

at (0,0).

C, P1

Of the two singular is

k-normal and

2 A~ k

is

(~ = z/y, ~ = x/y).

P2 = (5 = 0, ~ = 0) of

t = ~q/~q-l, u = ~/~

1

of

whose equation on

~p-2 + a~p = ~p

At the unique singular point

are of order

(p-l)/2.

is dominated by only one place of the function

X: = C - {PI }

X:

~),

where

X, the functions

q = (p-l)/2

and may be taken as uniformizing

In fact, the correspondence

(5, ~) ~+ (t, u)

90

gives an anti-regular

birational

transformation

of

X

to

Y

given in

As

Y

is k-smooth

2 ~k

by Y:

with the inverse

t 2 - at2u p - u = 0

formulae

Q = (t = 0, u = 0)

and

~Q,y is the integral tion

X

point

of P1

clearly

X

X

closure of

ratios

field

of

leaving

for some extended

k(v)

~v + B ¥v + ~

=

~, $, y, ~ £ k action of

o,

'

holds. and

~

genus

~

over

k

It is therefore

be an arbitrary

by the invariant

k-automorphi@m

of its k-normal

a k-automorphism

invariant.

is

and w 2 = v p - a.

2 generated

to a k-automorphism

It then causes

the subfield

o*(v)

Let

X

as

of index

k(X).

singular

has only two k-automorphisms.

of the first kind.

X, which we extend uniquely ~ u {pl}"

X

genus of

of arithmetic

k'

k(i) = k(C)

is the unique subfield

that the k-normaliza-

the geometric ~i

at

the local ring

We conclude

with v = x/z, w = y/z

of the differentials

pletion

point,

curve with a unique k-normal

Let us now show that

under any k-automorphism of

affine

must be a k-form of

the hyperelliptic

k(v)

%2,X.

Since moreover

k(X) = k(v,w)

Then,

~ = tu q.

is a one-place

is a k-smooth

q = (p-l)/2 > i. Write

P2

at infinity.

zero,

~ = tu q-l,

o*

of

k(X)

com-

= k(v,w)

Thus,

- BY

= i

On the other hand,

PI = (v =

°Pl = P1

a I/p, w = 0), so that

by the

91

°PI = a(al/P'

Thus,

O) = (o*(al/P),

(~a I/p + B)/(ya I/p + 6) = a I/p

implies y = $ = 0 and

~ = 6 # O.

o,(w 2) = o*(v p - a) = w 2 .

Xo

follows.

Therefore,

Since

o*(v) = v

We have thus established

morphisms induced by (v,w) ~-~ (v, ! w) of

o*(0)) = P1 = (al/p' 0).

p # 2, this and hence

that the two k-auto-

are all and only k-automorphisms

92

6.10.

G = Spec(A) be a k - g r o u p of Russell type and n m al,''" , A = k[X,Y]/(Y p = X + alXP + -.. + a m X p ) w i t h

let

am ~ k

Let

and not all of

completion

of

the s t r u c t u r e

G

and

al,... ,a m ~ k p. let

~

Let

= C - G.

of the d i v i s o r

class

C

We are

group

C(A)

be a k-normal interested

in

= Pic

of

(G)

A.

6.10.1

THEOREM.

then the f o l l o w i n g

0 ÷ ~

(ii)

0 + Pic°c/k(k)

pr

P

in

sheaves

~(G~)

over

C

C, w h o s e on

C

C

is l i n e a r l y

f ~ k(G) divisor

G.

restriction

f.

G.

p

Let D

on

D

by r e s t r i c t i n g

is surjective. be the c l o s u r e

is

equivalent

Define

corresponding

invertible

In fact, of

D

is given by an invertible G

and an integer of

index of the place

is o b t a i n e d

down to

is k-normal,

0

pr ~ pn

p

on

We have

0

÷ C(A) ÷ ~/przz÷

and

The map

be a divisor

Since

Pic(C) ÷ C(A)÷

J

be as above.

sequences

is the r a m i f i c a t i o n

Proof.

D

two exact

(i)

where to

Let the n o t a t i o n s

s

~(D). to

0

Next, on

such that

the map

j

assigning

on

C.

sheaf on

if a divisor

G, there E

let

sP

E

is a f u n c t i o n

= (f), the I 6 ~

to

P .

93

It is then easy to see that the sequence the other hand, we have an exact

(i) is exact.

On

sequence

j 0 ÷ Pic°c/k(k)

where

the neutral

+ Pic(C)

point of

Pic(C)/Pic°c/k(k) ~j(ZZ)

= (0),

Pic°c/k(k)

6.11.

is a generator

of

~ =

equivalence.

Pic°c/k(k)

since the degree of every element Moreover,

j,(p~)

we get the second exact

= pr

in some special

of the divisor

cases

of

From these

sequence.

We shall now carry out concrete

calculations structure

G

7Z. ÷ 0

up to algebraic

is zero.

observations,

+

and explicit

in order to clarify the

class groups of R u s s e ~ t y p e

k-

groups. 6.11.1. closed

LE~D~.

Let

k

be a non-perfect,

field of c h a r a c t e r i s t i c

p ~ 0

separably

and let

A = k[X,Y]/

r

~YD = X -alXP kp

...

such that

with generators

when

Proof. (1)

class

Let

are

group

C(A)

and

C(A)

al,

..., a r ~ k

P-independent

indexed by elements

I__n particular,

p = 2

), where

al,.-.,a r

Then the divisor

Spec(A).

arXP

over

of

A

is a

of

G(k), where

and k p.

~/pg-module G =

is an infinite p-~roup,

except

r = i.

The proof consists X~ = a i

with

of several

~i ~ kl/p

steps.

for

i _< i _< r.

94

Let

k'

over

= k(Xl,---,Xr).

k,

there

A' = k'

u n iquely

DiD j

a

@

A.

extended

of

A'lDi(a')

= 0

k[x,y],

let

y = t

+ XltP

for t

Xl,...,X

k-derivations

A'

for

+ -..

+ XlX + Xr t p r

A', A

= t¢ .

and the quotient

L/L 0 -- C(A), noting

A'

(II) k'

(t)

that

X

f(t)

z =

and

that a monic tp

=

1 -< i,j -< r. can be

and

= k'[t].

of

For

Z 6

with

N

tp 1 < i

<

A'

Thus we have

O.

polynomial

with

A = x =

L.

f(t),

Let

, d 6 k'

X

and

= such

Note here should be of

k' [t]

in

z e K'

g(t) 6 k'[t]

have no common divisors.

irreducible

A = {a'e

writing then

the group

f(t)/g(t)

g(t)

d

k'

C(A')

We shall calculate Write

the form

field

k'

<_a~d C. 4.27 we can apply 6 . 4 _ i ~

Therefore

that

< r --

+ krxpr-1;

and

in

D i ~ 0, D~1 = 0

Moreover,

i-I r, Di(t ) = x p

for

DI,...,D r

1 -< i,j

+ ...

p-independent

(D1,...,Dr)

and satisfy:

= gij

are

DiD J. = D.D. 3 1

and

1 <_ i <_ r } .

= y

r

The derivations

on

DjDi, D i(kj)

=

set

Di( Xj) = 6ij

such that Let

is

Since

d ~ k 'p

if

S

Z = C g j=l

> 0. Hence z can he w r i t t e n in the form P ) ~j , where c ~ k', aj 6 2 + and (t (~j - dj ) d_. ~ J

k'

'

(C)

*

d.j 's

satisfying

dj ~ dj

I

To avoid triple

power of the entity

if

the c o n di ti on

Z 3 = £j .

!

and

superfix we write X.

dj ~ k 'p

X p(~)

if

for the

Z] •

>

0°

p~-th

~:~

m

L-~

('D

II

,.._~.

ii

F-h

",-,J •

~

~

~

0

L.~.

h~ ~

~ o

~

~"

:...,.

0

CI 0

0 ~

~ t-'~, :'E~

L.J.

~-'

II

b~

~'~:~

i

@

0

@

~n

?

II

rl"

LJ.

II

I

!

0

t--4.

L.a.

II

M

o

,.~ E: I.-~. <:

t~ (~

~

0

0

~'~

~

N

~e

I~.

-

~

I

t..a.

~,

II I, ~ ~ "-~" ~ "~

0

t_.a0

F-~0

rf

c'--d

0

i'D

~,.J,

"d

II

M

+

+

II

l~

~.

~

I

tr'

~,

+

II

~:~

CD

~ N

~-'

t:::J

~

C~

~

t..J.

0

LC) U1

96

(D 1 (t

d) / (t

...,D r(d)

d) ,... ,Dr(t

- d)/(t

with

- d))

D 1 (d) = d p,

= d pr.

(iii)

Let

d

be an e l e m e n t

of

k'

such

that

Di(d )

i = dp

i < i < r.

for

written k

in the

such

form

It is easy

d = c0+ClXl+...+Crk

r

that

d

can be

with

c O ,..-,c r

that 2 2 2 c2 = cpO +aPcPl± +'''+aPcPrr '

Cl = cP0+~61cp+l "''+arcPr'

• ..

to s h o w

r r-i pr r-i r c r = c p +aP1 c I + . . . +a pr cp .

'

r-i Then

r

Hence

c2 = c~,...,l Cr = c~I

Therefore

(y = Co,

of

A,

i.e.~

Co,

x = Cl)" 6 G(k),

x = Cl)

(y = Co,

cI

satisfies

x = Cl)

the

= cP+a Iclp+...+arcp1 defining

equation

~ G(k).

C o n v e r s e l y if (y = r-I d = c 0 + c l X 1 + ...+ c p kr satisfies

1

Di(d)

= dp

for

generator

of

(D l(t

1 <_ i j_ r . - d)/(t

Hence it

- d),

--.,

gives

D r(t

rise

to a

d)/(t

- d))

L. i (IV)

+ d pl-1,

where

the o t h e r

i

D i ( t _ d ) / ( t _ d ) = tp

Now

t pI-I

hand,

= Di(t)/t

-i +.tP

and

we h a v e A s u r j e c t i v e

d pI-I

-2 d +...+

i td p -2

= Di(d)/d.

On

homomorphism

~

: L +~:

r

= the last

group

generated

coordinate

coordinate ~l~ -~ G(k) .

by the

of e l e m e n t s

group

of

L.

Therefore

is an i n f i n i t e

p-group

coefficients of

L.

It is clear

C(A)

We call that

is g e n e r a t e d

except

when

of

p = 2

tp ~

-2 the

in the last

~(L 0) = 0 by

G(k) and

and

and

C(A)

r = I.

When

97

p = 2

and

r = I, it is easy to show that

6.11.2 over

Remarks.

(i)

If

k p, the c o m p u t a t i o n (2)

wiht

Consider

a ~ k

and

al,...,a r

of

C(A)

If

0 ÷ G(k) ÷ C(A) ÷ ~ / 3 ~ ÷

0

is exact

completion

totheplane

y3 = XZ 2 - aX 3

are

cubic

inflection

is i n d u c e d

points.

caused

on

is the n e u t r a l

point.

If

6.10.1).

G = Spec(A)

of the group

in

Ga × Ga

cubic.

If

of w h i c h

G(k)

which

coincides

in the usual

Here

Moreover,

p = 3,

is isomorphic

~2k' all points

(l,-al/3,0)

a plane

complicated.

= ~ /2~.

(cf.

in

addition

C(k)

to d e f i n e the addition on Z = i)

of

The a d d i t i o n

from the usual

the a d d i t i o n

C

more

A = k [ x , y ] / ( y p = x - a x p)

p = 2, C(A)

p = 3~ a k - n o r m a l

= Z / 2~.

are not p - i n d e p e n d e n t

becomes

the case w h e r e a ~ k p.

C(A)

with fashion

(X = 0, Y = 0,

i: G - ~ P i c ° c / k

is

an isomorphism.

6.12.

In 6.11,

a non-perfect late

C(A)

we need the

separably

over

a

closed

of

field

not n e c e s s a r i l y

C(A)

k.

was made over

In order

separably

to calcu-

closed

field,

following

6.12.1 p ~ 0

the c a l c u l a t i o n

LEM~IA.

and let

with

Galois

that

A× = kx

k'

group

Let

k

be a field of c h a r a c t e r i s t i c

be a s e p a r a b l e F.

and let

= the group c o n s i s t i n g

Let A'

of

A = k'

algebraic

closure

of

be a s m o o t h k - a l g e b r a ~

A.

F-invariant

Then

C(A)

elements

of

k

such

= C(A')

C(A').

F

98

Proof.

Since

A'

is a smooth k'-algebra,

C(A')

is

isomorphic to the group of isomorphism classes of projective A'-modules

of rank i.

Let P be a projective A'-module

such that the corresponding invariant.

For any

class

{P}

o ~ F, denote by

p ÷ ap = (k',o)

go

multiple

A 'x = k 'x

and

~ P. k' k 'x since

in

°gz • go

.

= a(ez,~)a(e,~)

Thus

in

.

k 'x

For

~,~,v ~

g'oz

= og'~-g'.

In other

data

on

k'/k.

P

to

For any

of rank

~'~ ~

P '~z

a(o,T) ~ k 'x ,

i.e.,

~a(~,,~)a(~,Tv)

is a 2-cocycle of

= Ob(T)b(oT)-lb(o).

P

F-

the isomorphism

F , we have

for all

b(~)

a(o,z)

for

is

F

with

Hl(F,k 'x) = (0) (Hilbert's Theorem

and

P

C(A')

is determined up to a scalor

a(-,-)

Since

90), there exists

A-module

g~

differ by a scalor multiple

ga~ = a(~,z)~¢~.g

values

in

of rank 1

o ~ P Let

words,

{g'

Therefore 1

P, we h a v e a n i n v e r s e

g ~

exists

P = k'

map t o

b(o)g

Io ~ F}

there

such that

such that

the

@

b(o) ~ k 'x .

Then

defines

a descent

a projective P.

canonical

Assigning homomorphism

C(A) + C ( A ' )

6.12.2

LEbfi~A. Let

A = k[X,Y]/(Y p = X - alXP

.....

r

arXP ) ar

with

are p-independent

C(A)

of

where

A

A

k

where

is a

= k S

of

a l,.--,a r ~ k

with

@

over

and k p.

¢ kp

such that

a I,---,

Then the divisor class group

2Z/pTZ-module which is isomorphic t__0_o C(As)F, A, k

s

being a separable algebraic closure S

Galois group

G = Spec(A) .

F.

C(A s)

i__ssgenerated by

G(ks),

99

Proof.

This follows

immediately

from Lemmas

6.11.1

and 6.12.1. 6.12.3 a simple let

transcendental

G = Spec(A).

closure that

of

Let

®

Let

a ~ k p.

C

- aX p

of

group.

C

x = X/Z 0)

~3

and

separable

implies

while

domain.

with into

a E k ~2

and

and take

is d e fi ne d by the equation

and

coordinates

y = Y/Z.

(but not g e o m e t r i c a l l y

and

Pic___°C/k is a smooth affine

(D

2)/2.

normal

(X,Y,Z)

The point at

is not k-rational.

As indicated

This

domain,

canonically

be

= {(X = 0, Y = 0))

factorization

in the homogeneous

(i, -a I/p,

k-normal

G.

field

is an algebraic

factorization

G = Spec(A)

such that

infinity

G(k)

A = k[x,y]/(y p = x - ax p)

the c o m p l e t i o n

~2

ks

is not a u n i q u e

~mbed

YP = XZ p-I

Then

if

k = ~3(t)

of the prime

is an infinite

is a unique

All/x]

p = 3, let

extension

However

k, G(ks)

A[i/x]

(2)

of

(I)

A = k[X,Y]/(y 3 = X-tX3).

with

ks

Remarks.

M or eo ve r

except when

if

is

p = 2),

group of d i m e n s i o n

in 6.11.2,

C

(p - i).

p = 3, i: G ÷ Pic°c/k

is an isomorphism. 6.13.

Finally we shall make one remark on

for a o n e - d i m e n s i o n a l k

is an a l g e b r a i c a l l y

and if

G

unipotent closed

is a c o n n e c t e d

EXtcent(G,Gm)

group of Russel~type

field of ar bi tr ar y

algebraic

is isomorphic

to

EXtcent(G,Gm) G.

If

characteristic

linear k-group,

Pic(G).

However

this result

I00

no longer example

holds

is given

Russell=P~,type" # C ( A ~ and

k

by a o n e - d i m e n s i o n a l

= Spec(A)

unipotent

k-group.

of

G

G

when

by k

Hence

unipotent

see below

# 0.

Let

We shall

that

prove

G = Spec(A)

A counter-

group

of

EXtcent(G,Gm)

A = k[x,y]/y p = x

ax p, a ~ k first

a

be a c o m m u t a t i v e

Then the canonical

First

of all,

are commutative.

m

closed.

map

Extcent(G,Gm)

i_~s injective

(I)

Proof. by

C(A)

LEPTA.

= HI(G,Gm)

we shall with

a ~ k p, while

C(A)

is not a l g e b r a i c a l l y

In fact,

6.13.1.

G

if

is a l g e b r a i c a l l y E = G × G

to show

closed.

(a direct

m

that

all extensions

let

E

be an e x t e n s i o n

that

E

is c o m m u t a t i v e

Indeed,

It suffices

G m.

note

E

Then

product).

of

is k-solvable.

In particular,

E

is

commutative. (II)

Let

= Spec(B).

E

Since

= the symmetric p-i

with

× Gm

f(x,x')

on the left,

acts on

E

algebra

over

pmp-n

A-module

÷ C(A)

of

g, g' ~

G

= pm-n of rank

sends

as a scheme,

by a 2-cocycle and

Gm

relations

EXtcent(G,Gm)

G

of

tensor

is a p r o j e c t i v e

E = G

be an extension

E

is a 2-cocycle

for

(g,x)(g',x') such that

Let

E

B = A [ P , P -I] by

P

The canonical -I

If

lhe m u l t i p l i c a t i o n in

G m.

and

m, n E X, where

i. P

by

generated

A

to

w i t h values

Gm,

G

Gm;

map

P = A, then of

i.e.,

E

is given

for

x, x'

= (gg'f(x,x'),xx'), f(x,e)

P

= f(e,x)

= i.

where Since

I01

(A ® A)x

= k x,

f (x, x')

is a constant.

However since

f(x, e)

k = f(e, x) = i, f(x, x') = 1 (g, x ) ( g ' ,

x')

6.13.2

= (gg',

unipotent a e k

p > 2

Thus

Let

k

x, x' ¢ G. E = G x G

a ~ k p.

Proof. E

Let

Then

E

be a separably closed field

and let

G = Spec(A)

EXtcent(G,

Gm) ~ C(A) = HI(G, Gm). G

AlP, p - l ] ,

i.

q2 : A

be the comultiplication

and

q2(b)

:= l ~ ) b ,

the A @ A - m o d u l e PQ

(A~)A,

by

A : A ....~ A @ A ,

respectively.

G . m

where

A-module of rank ~ A@A

Let

be a one-dimensional

A = k[x, y]/(yP = x - axP),

be an extension of

is isomorphic to

Therefore as a g r o u p .

m

k-group of Russell type with

and

ring of

xx').

PROPOSITION.

of characteristic

for all

P

The affine

is a projective

ql : A ~ of

A,

Denote by

A@

ql(b)

A*(p)

A

and

:= b C ) l

(or

q.~(P))

&*(P)

:= P Q (A @ A , A) (or q~(P) := A Then P satisfies the relation A*(P) =

qi)).

A q~(P) Q q ~ ( P ) , Let D

on

0

and

where

k' = k(k)

k'

such that

P

@

is taken over

with

Xp = a.

D(X) = i.

Let

corresponds to an element

AQA.

There exists a A' = k' Q A . ~ ~i

k-derivation Then

C(A') =

D(t-di)/(t - di)

of

i L dp 1

with

~i ¢ Z

for all

i

and

d i ¢ k'

(cf. 6.3.1).

such that

D(t) = tp

and

D(di) =

It is easy to see that the relation

102

A*(p) = q~(p) @ q ~ ( p )

implies

~i D(t + t'

di)/(t * t' - d i)

~i D(t - di)/(t

- di)

i

i

Y

~i D(t' - di)/(t'

- di) c L0(A' G A ' ) ,

i where

i

L0(A' @ A ' )

is

L0

for

A' ~ A ' .

Hence we have

~i[{(t + t') p-I + (t + t')P-2d. + ... + d p-I} i i

_{t p-I + tP-2d.

+ ... + dp-I} _ {t,P -I + t,P-2d. i

i

+ ... + dP-l}[

I

I

¢ L0(A' @ A ' ) .

Hence we have Since with

Z i

~' = 0, i

~ i

C(A) # 0, there exists d ¢ k'

H1 (G, Gm).

such that

~idi = 0 . . . . . an element

D(d) = d p.

~ i

a.d p-2 = 0. x x

D(t - d)/(t

Therefore

- d)

of

Extlcent(G, Gin) #

L

103

6.13.3.

THEOREM.

Then,

the

w i t h the

Let

abelian

following

(the a d d i t i o n

Proof. Ga

group

additive

{(c0,...,Cp_ 2) E k P - I I

is made

be

G

the same as in

EXtcent(G

6.13.2.

,Gm)

is i s o m o r p h i c

group:

Cp_ 2 = c 0 P + c l P a + . . . + C p _ 2 P a P

-2

}

component-wise).

As is w e l l - k n o w n ,

are r e s p e c t i v e l y

the Hopf

k[u,u -I]

and

algebras

k[z]

of

Gm

and

w i t h the comulti-

plications

fl(u) = u @ u

If we put form of k'-Hopf

k' = k(k) Ga

and

A(z)

with

= z ~ i + I @ z.

k = a I/p,

and the c o r r e s p o n d i n g

algebra

k'[z]

(in the

then

G

is a

k-derivation

sense

of Lemma

Dl(k)

= i.

y

-

(k'/k)-

DI

6.1)

of the is d e f i n e d

by

Dl(Z) (The i d e n t i f i c a t i o n Let

DO

denote

= zp is

= 0

For each element

k'[z,u,u

-1] D[(X)

Lemma

zp

=

the trivial

D0(u)

of

x

and

6.13.4 b e l o w

=

z

~z p

k-derivation

and

D0(X)

@k' k ' [ u ' u - 1 ]

D[(z)

shows

: that

zp

and

of

and

O(G)

= k[x,y]). i. e . ,

k'[u,u-l],

= 1.

~ ~ k', we d e f i n e

= k'[z] = 1,

,

a k-derivation

by D~(u)

(D~) p = 0

= ~zu.

if and only

if

D~

104

~P = dP-2(t), where

d

such that

(see 6.1).

group of put of

d(X) = i ~ ~ k'

denotes the unique k-derivation of

such that

A~ = k'[z,u,u-l] Dr.

We denote by

~P = dP-2(~)

k'[z,u,u -I]

the additive

For each

This is a Hopf algebra

k[z,u,u-l], because the k-derivation

the comultiplication of

Z

DE

k'

t ~ ~

(k'/k)-form

commutes with

in the following sense:

A(Dt(c)) : (c[)D~(c(1)) ® c(2) + (c)[c(1) @ D~(c(2)) for all

c E k'[z,u,u-l].

Observing the following commutative diagram: k'[z

i ~ k'[z,u,u -I]

P ~ k'[u,u -I]

i > k'[z,u,u -I]

P > k'[u,u -I]

DI k'[z where

i

and

p

denote the canonical inclusion and projection

respectively, one concludes that the homomorphisms

i

and

induce the following maps of k-Hopf algebras: O(a)

it

-- k[x,y]

-- k'[z]

DI

i}

At

and

P> k,[u,u-l] D° = k[u,u-l].

The corresponding sequence of commutative k-group schemes (E t)

i --9 G m

Spec(p) > Spec(At)

is clearly exact, because we obtain a map

(E t) ® k'

S pec(i)~ G --~ i

is split exact.

Thus

p

105

¢:

H --9 E X t c e n t ( G , G m ) , ~I--~ ~(~)

It is an easy e x e r c i s e morphism

of a b e l i a n

to check

groups.

The injectivity. that

the

sequence

(E~)

that the map

We claim

Suppose

= class(E~).

that

splits.

is a homo-

that ¢(<)

is an isomorphism. = 0

This means

or e q u i v a l e n t l y that

the

cano-

nical map p: A~ ----) k[u,u -I]

induces hence

in p a r t i c u l a r

~zu, of

a bijection

it follows

of the sets of g r o u p - l i k e that

that

elements

and

A~ = k'[z,u,u-1] D~. Since

u ~ ~ = 0.

This

proves

D~(u)

the i n j e c t i v i t y

¢. The

surjectivity.

(E)

i

be a central = 0

by

~ G

extension

1 --9 (Gm)k,

is split

exact.

D

denote

D

should

G

it follows

~ i

Since

m that

---~ Gk,

EXtcent(Ga,Gm)

the sequence

= (Ga)k,

--9 1

we can identify

~

O(E)

O(E).

D(~)

> @

by

=

k'[z,u,u-1].

the k - d e r i v a t i o n

w i t h the k - f o r m

Since

G

--9 Ek,

Hence

k'

9 E

m

of

DG-III,~6,5.2,

(E)k,

Let

Let

of

k ' [ z , u , u -I]

Then we have

= i

commute

and

with

D(z)

the

associated

clearly

= z p.

comultiplication

A

of

106 k'[z,u,u-l],

it f o l l o w s

that

£(D(u))

= D(u)

or e q u i v a l e n t l y k'[z,u,u-l].

that

Since

u-iD(u)

primitive

with

a { k'[F],

form

~z

= ~z

some

~ 6 k'[F].

that

the d e g r e e

of

is a p r i m i t i v e

every

is of the for

@ u + u @ D(U)

~

is

Since zero,

element

element

of

of

k ' [ z , u , u -I]

it f o l l o w s

that

u-iD(u)

D p = 0, it f o l l o w s

i.e.

~ ~ k'.

Hence

easily we h a v e

D = D

Since

( D ) p = 0, it f o l l o w s

This means

that

tivity

@.

of

Thus

class(E)

we h a v e

then --

c

~P =

p-2

-

c.Pa i

i=0

Cp_ I = 0

This

proves

and p r o v e s

an i s o m o r p h i s m

~ e

the

E.

surjec-

of a b e l i a n

groups

k'

as

d p-2

{ ~

Z

({) if

~ =

:

p-I ~ i=O

ki ci

(P-2)!Cp-2

and

only

Cp_ 2 = coP + c l P a +

with

c. ~ i

+ (P-1)!Cp-ll

if

... + C p _ 2 P a P - 2

Theorem.

LEMMA. of

D[(X) T h e n we h a v e

Hence and

6.13. 4. k-derivation

~ ~ and

1

Cp_lX.

)

6.13.4 that

H --~> E X t c e n t ( G , G m ) .

an e l e m e n t p-I ~

= class(E

constructed

@: If we w r i t e

from Lemma

For

each

k ' [ z , u , u -I]

= 1,

D[(u)

~ ~ k' defined

= Szu

and

,

let

D

by

D[(z)

= z p.

denote

the

k,

107

(D~)P(u) where = 0

the

d

is the

if and

only

Proof.

Put

= dP-l(~)zu canonical

if

k-derivation

D = D~. A

Since of

k'[z]

that

above).

for all

n

degz(u-iDn(u))

particular

we h a v e

must

of the

be

for

On the > 0

and

Hence

clearly

some

(D~) p

one

it f o l l o w s

hand

u-IDn(u)

can p r o v e + i

degz(u-iDP(u))

commutes

with

that

~ Ei k'[F].

other

~ (p-l)(n-l)

belongs

by i n d u c t i o n

for all

n

> 0.

~ (p-l) 2 + i < p2

In Hence

form

= ~ + ~'F

Next,

Dp

k'[z,u,u-l],

u - i D P ( u ~) = ~z

to

o f k'

~P = d P - 2 ( ~ ) .

comultiplication

(See the p r o o f

+ (~P - dP-2(~))zPu.

for e a c h

integer

u-iDi(u)

= di-l(~)z

with

~, n' E

0 < i < p, one

k'.

can p r o v e

by i n d u c t i o n

that

where

the

putting and

zPu

+ ~izi

+

degree

is t a k e n

i = p-l, in

= ~P - d P - 2 ( ~ )

+

(the

terms

of

I < degree

( i - l ) d i - 2 ( ~ ) z p + (the t e r m s

we

DP(u)

with

respect

can d e t e r m i n e = D(DP-I(u))

respectively.

the to be

This

to

z.

of d e g r e e

> p),

In p a r t i c u l a r

coefficients ~ = dP-l(~)

proves

< i)

Lemma.

of

zu and

~'

7.

Actions of unipotent

In this section 7.0.

group schemes

(§7), the ground field

We consider the following

k

is arbitrary.

standard situation,

for

which the notations will be fixed all through §7: 7.0.1.

An affine algebraic k-group scheme

an affine k-scheme

X = Spec(A) u: X x

G

The coaction determining

u

p: A

to

7.0.2.

>X is written as

> A~)O(G)

The underlying denote by

cohomology group of

7.0.3.

O(G)-comodule

G

k-module Hn(G,A)

of

A

can be viewed as

the n-th Hochschild

with coefficients

in

A ~

see 7.2

DG -II, §3. Let

(fpqc)-sheaves faisceau dur in

~

on

denote the category of all set-valued ~k"

(This is what is referred to as

DG -III, §2, 1.3.

DG, one would write the quotient sheaf of

x/G.

giving a right

A.

a left k-G-module;

below and

acts on

on the right through

which is a k-algebra homomorphism structure

G

(~-~ X

Thus,

in the fashion of

in lieu of by the action of

.)

We consider

G, denoted by

109

7.0.4.

~ ~,

In a like manner, we denote by

~~ ~

respectively

the category of all set-valued presheaves

(covariant functors)

and that of all set-valued

on

~

= ~

and

~

(fppf)-sheaves

= ~ ~ ~ .

c

~k"

Thus,

Note that the inclusions

c

7.1.

The

purpose

of

the

present

section

is

to prove

the

following 7.1.1.

THEOREM.

With the notations

7.0, assume further that

G

statements

to each other:

(i)

are e~uivalent The action of

definition), (ii)

X/G

Hn(G,A)

(iii)

i_~s unipotent.

on

X

is free

i__ssrepresentable

= 0

H I(G,A)

7.1.2. a result

and

G

and assumptions

for all

of

Then, the following

(see 7.4 for the

and affine;

n > I; and

= 0

Comments.

Earlier,

one of the authors asserted

[4; Theorem i] roughly the same as 7.1.1 here,

somewhat stronger in some ways and weaker

in others.

His

proofs of the assertion and a key lemma for it (ibid., Th.l and Lemma 3) tended to ground too fast.

go

A careful re-examination

in [4] has revealed that unfortunately

over the

of the arguments

certain of them

contain gaps that are hard to fill; as a result, both Theorem 1 and Lemma 3 of [4] remain unsupported by proof in the stated form.

Rather than patching up and making

ii0

corrections

here and there on the basis

red to rebuild

the theorem

remain

out, however,

in the present

that the basic

the same as those of

ed as follows:

[4].

The questions

is free and whether are indifferent

extensions.

commutes

Therefore,

in proving

7.1.1 one may assume

either

Ga

or

~p.

through higher directly. of

G

7.2.

derivations,

G

each

of

G, denoted

gm,

all

R6~-~k,

module

gCG(R),

we mean

p: G-->GL(V) nothing

(g,m) J ~

else

a pair a linear

R { ~

of

g

V~R

by

Suppose

that

and

is R-linear G

G 'a s

quotient

V

functor

is

3) takes care and

~ p 's.

M

with

g(m+m')

an operation

and

a k-G-module

of the form the induced

for

By a k-G-

is a k-module Thus

func-

= gm+gm '

(DG-II,§3).

where

g eG(R)

to be k-solvable

By a G-module

such that

representation.

than a G-module

that for any

of

functor

m, m ' 6 M ( R ) (V,p)

G

Lemma

functor.

k-group

field extensions.

7.1.1 for this case

(ibid±,

tor we mean a commutative

representable

the G a- or ~p-aCtion

extension

be a k-group

of 7.1.1

The Demazure-

successive

one proves

sequence

is a multiple

Let

scalar

By interpreting

The spectral

which

with

in which

§8 below).

or not the action

is affine

cohomology

series

(cf.

The said ideas may be summariz-

Hochschild

with a central

work

as to whether

field

in the framework

ideas of proof

or not the quotient

to scalar

[4], we have prefer-

from the ground up,

best suited for our purpose We point

of

Va

is

such

endomorphism

(DG-II,§2).

is an affine

k-group

scheme.

As usual,

IIi

the Hopf algebra of known

G

will be denoted by

O(G).

(DG -II, §2, 2.1), the linear representations

correspond bijectively with the right comodule V ÷ V~O(G).

Hence we may and shall identify the category with that of right O(G)-comodules,

~O(G)'

G

if

Let groups

M

denoted

is affine.

be a G-module

H~(G,M)

as follows

G + GL(V)

structures

of k-G-modules

of

G

functor.

The Hochschild cohomology

with coefficients

(DG -II, §3):

in

M

are defined

Put

Cn(G,M) := L~(Gn,M)

for

Define the coboundary homomorphism by

As is well-

vn+l(_l)i ~i' n where for all sn:= Li=0

n >__%0.

$n: Cn(G,M)

~ cn+I(G,M)

gl,...,gn+ 1 ~G(R)

and

f ~ cn(G,M)

gl ( ~ f R ) (gl, • • • ,gn+l ):=

($nfR)(g I,

,gn+l ):

fR(g2,. • .,gn+l ) ,

fR(gl ,-- ,gigi+l,-..,gn+l ), l
($n+ifR)(g I, ..,gn+l ):= fR(gl ,

Then

8n8 n-I = 0

the complex

for

C'(G,M):=

n > 0

and the n-th cohomology group of

((Cn(G,M)),(sn))

and called the n-t_~h Hochschild

,gn ) •

is denoted by

cohomology group o_~f G

H~(G,M) with

112

coefficients Let

V

in

M.

be a k-G-module.

be denoted by

Hn(G,V).

G = Spec O(G).

Let

Then the group

Suppose that

p: V ÷ V @ O ( G )

comodule structure map.

Let

G

H~(G'Va)v

will

is affine:

be the associated

C'(G,V)

be the following

complex:

cn(G,V) := V~)O(G)~ . . . ®O(G)

(n copies of

O(G)),

~n:= vn+l (_l)i~n Li=0

where

'i

~n: V ~ O ( G ) ® n 1 f

'

÷ V®O(G)®(n+I )

~n(v~al0 ~''" ® a n ) : =

~n

i(v~al®

is defined by

p(v) ®a 1 ®...

... ~ a n):= v ~ a l ~ . . . ~ A a

®an

i®...~an,

~n

n + l ( V ® a l ®...~)a n ) : = v ® a l ® - . . ® a n ® l

where

A: O(G) ÷ O(G) ® O ( G )

Then we have Hence

C'(G,Va)

7.3.

Let

G

group functor of normal subgroup of G-module functor. element

x 6G(k),

defined by

,

DG-II, §3, 3.1).

(see

in case

G

is affine.

be a k-group functor and G.

By G(R)

K

l<__i<=n ,

denotes the comultiplication.

= C'(G,V)

Hn(G,V) -~ Hn(c "(G,V))

,

K

normal we mean that

for all

For an n-cochain let the n-cochain

R E~.

Let

f ~Cn(K,M)

a normal subK(R) M

be a

and an

x---~f 6Cn(K,M)

is a

be

113

[x--~ f] (gl'''''gn)

for

Then

all

= X f ( x - l g l x ' ' ' ' ' x - l g n x)'

gl'''''gn

one can easily

verify

the

action

then

of

action

G(k) f: M ÷ N

the

induced are

of

Suppose

on

C'(K,M)

is a h o m o m o r p h i s m homomorphisms

easily

commutes

G(k)

seen

of G-module

of groups

to commute

with

of complexes

with

that

induces

a left

H0(K,M).

the h o m o m o r p h i s m

already

~nf

n

on

If

H n(K,N) 0 since

left

R~ ........

that

~ n ( x _ ~ f) -- x - X

Hence

~ K(R),

functors,

f) : H n(K, 0

H~(K,M)

the G(k)-action,

C'(K,f):

C'(K,M)

÷ C'(K,N)

the G(k)-action.

K

is affine.

Then

a short

exact

sequence

0 of G-module

o

~ M'

~M

functors

induces

> c" ( K , M ' )

Since

each

element

exact

sequences

> M"

x

) 0

an exact

~ C" (K,M)

of

G(k)

sequence

~ C" (K,M")

induces

of complexes

> 0

a homomorphism

of

114

o ~

C'(K,M')

0

that

H ~ + I ( K , M ')

also

7.3.1.

o._n_n H~(K,M)

If

is e f f a c e a b l e

H n(K,M) 0

such that H n - I ( K , M ") commutes

~ Hn(K,M)

with

by the

7.4.

= 0

exact

> M"

for all

induction

We place

exact

for all

Therefore

§3,

1.3). functors

the a s s e r t i o n

in the

situation

on

X = Spec(A).

situation

A

be referred

to as a right

of

the functor

argument.

ourselves

action

that

is obvious

n > 0 , which

G-action

The

lemma

M.

Then we have

a right

algebra.

of

functor

of K-module

possessing

will

action

.......> 0

n > 0.

the K(k)-action.

The

of DG-II,

sequence

~ M

induced

Notice

the proof

~> 0

,

H n(K,M") 0

G-module

G = K.

= MK(k)

(cf.

we can form a short

the

for every

that

H 0(K,M) 0

~ 0

the G(k)-action.

is affine,

is trivial

0 ---~M

holds

K

~ 0

) c" ( K , M " )

homomorphism

with

We can assume

n = 0, since

H~(K,-) Hence

commutes

LEMMA.

Proof. for

the c o n n e c t i n g

(K,M")

~, c"

> C" (K,M') ----> C'(K,M)

it follows

K(k)

- C'(K,M)

G

on

X

is said

of 7.0 above, In this

O(G)-comodule

to be free

if

115

the morphism (x,xg)

(Prl,u): X x G

for all

k-functors,

xEX(R)

where

first projection.

> X x X, given by

, g (G(R),

(x,g) I

is a monomorphism of

xg = u(x,g)

and

Prl: X × G

The quotient

sheaf

X/G

defined by the following exact sequence

in

)X

is the

(see 7.0.3) ~

is

(cf. DG-III,

§2, 1.3):

X x G

Prl ~X

z ~X/G

u

If the action is free we have clearly

X×G

>Xx~

X X/G

Suppose that Spec(B). DG-III, that

Then

X/G

is representable

A

is a faithfully

affine and put flat B-algebra by

§I, 2.11 and also III~ §3, 2.5.

B = AG

structure on (i) (ii)

A ~AGA

It is easy to see

(taken with respect to the induced k-G-module A).

Therefore

The action A

X/G =

u

the following are equivalent:

is free and

is a faithfully flat

A®O(G),

is an isomorphism.

a®b

I

X/G

is affine.

AG-algebra and the map

) (a®l)p(b)

116

7.4.1.

LEMMA.

above holds, Proof.

then

If one of the equivalent conditions Hn(G,A)

= 0

for all

n > 0.

Indeed we have an isomorphism of k-functors

N_

X x Gn

~X x z

--- x

x/G (x,gl,...,gn) ~

~

X

(n+l

copies of

X)

x/G (x,xgl,xglg2,...,xgl...gn).

Let

A®O(G)®n 4=

A~AG . . . .

be the associated

® G A

(n+l copies of

A)

A

isomorphism of k-algebras.

These homomor-

phisms form an isomorphism of complexes

C" (G,A) = C'(A/AG,Ga )

the right-hand side of which denoting the Amitsur complex relative

to

Ga

(cf. DG-III, for

n > 0 7.5.

and the canonical projection

§5, 5.2 and §4,6.4). by

Let

DG-I, G

§1,2.7,

Hn(A/AG,Ga ) = 0

the assertion follows.

be an affine k-group scheme and

closed normal subgroup scheme of G. canonical Hopf algebra surjection. space.

Since

Spec(A) ÷ Spec(A G)

The composite

Let Let

K

a

~: @(G) + O(K) V

be a k-vector

be the

117

l®A

makes

. . . . . .

V~O(G)

into a right O(K)-comodule(i.e.)a

7.5.1. on

I®I~-

V~O(G)

)

COROLLARY.

V~O(G),

V~)O(G)~)O(G)

With

the above

Proof.

action

Since

to show

k-K-module

(which

is affine

7.6.

Let

closed normal the right

= 0

C'(K,V®O(G)) Hn(K,O(G))

structure

the assertion G/K

>V®O(G)®O(K)

k-K-module).

k-K-module

structure

we have

Hn(K,V~O(G))

suffices

,-

on

n > 0

= V®C'(K,O(G)),

= 0

O(G)

for all

for

n > 0.

is induced

it

Since

the

from the right

is free)

G x K

~ G,

follows

from 7.4.1,

(DG-III,

G

(g,h) ~-

in view of the fact that

§3,7.2).

be an affine

subgroup

> gh

scheme

derived

functors

~ C o m o ~ (G)

,-

k-group

of

G.

scheme

and

K

a

We shall now determine

of

.~

, given by

V I

> VK ,

O(G/K) where

one should notice

for any k-G-module

V.

that

VK

is naturally

a k-G/K-module

118

7.6.1.

Our task begins with making

k-G/K-modules module.

for each k-G-module

For each

k-algebra

Hence a natural G(R)-action is defined as in 7.3. map i ~ :

V~R

on

Let

R, V ~ R

V

be a k-G-

(from the left) on X ~ R

Hn(K®R,V~R) the induced

is an R - ( G ® R ) - m o d u l e

Hn(K®R,

V~R)

into

is an R - ( G ~ R ) - m o d u l e .

Since for each

> V®R

the G(R)-action

V.

Hn(K,V)

homomorphism,

is R-linear.

Note that

Hn(K@R, V~R) = Hn(K,V)~R

(DG-II,

§3, 3.6).

Hence

left R-linearly. to

G(R)

acts on

Since this action

R, we have defined a k-G-module

Since the induced K(R)-action 7.3.1, Hn(K,V) 7.6.2. fR: V ® R

becomes

Let

+ Hn(K,V ')

V ÷ V" ÷ 0 0 ÷ V'®R it follows

Hn(K,V)~R

Hn(K~R,fR):

÷ V@R

Hn(K,V).

is trivial by

~ 0

that the connecting

map.

maps.

Hn(K,£): 0 ÷ V' ÷

of k-G-modules.

is exact for every homomorphism

R, the

÷ Hn(K,V ' ) ~ R

Let

commutes with the G(R)-action.

long exact sequence

Since

map for all

Hence the maps

are k-G/K-module

-+V"®R

on

Hn(K,V)~R

be a short exact sequence

Hn+I(K,V ' ) ~ R resulting

structure

be a k-G-module

commute with the G(R)~action. Hn(K,V)

is natural with respect

is an R - ( G ~ R ) - m o d u l e

induced homomorphisms

from the

a k-G~module.

f: V ÷ V'

÷ V'~R

on

Hn(K,V)®R

Since

R 6 ~ ,.........

Hn(K,V '')~R Therefore

> the

119

• .--> Hn(K,V)

> Hn(K,V '')

~ Hn+I(K,V ') ___>Hn+I(K,V)

--~ ..

z

consists

of k-G/K-module maps.

~O(G)

--> C~°m-~O(G~K)

(i.e.,

Thus the functors

constitute

Hn(K,-):

a cohomological

an exact connected sequence of f u n c t o r s ) ,

functor

which we

shall denote by H'(K,-). 7.6.3.

PROPOSITION.

The

(cohomological)

H" (K,-3: [ C - ~ m - ~ ( G ) - - - > ~ O ( G / K= ) (cohomological) [Com---m~

~

is the right derived

functor of the functor

given by

functor

~ ( G )

--~

V I > V K.

O(G/K) Proof.

Let

V

structure map of

be a k-G-module.

V: V ÷ V ® O ( G )

where the k-G-module one.

Since

follows

that the functor

= 0

Let

closed normal

G

where

~(V) = V K

for

and k-G-linear,

is the canonical

n > 0

by 7.5.1,

is effaceable

H0(K,_)

=

scheme of

G.

it

and hence

K

be an affine k-group scheme and

subgroup

~ ( G ) - - - ~

V®O(G)

H'(K,-)

the right derived functor of 7.7.

is injective

structure on

Hn(K,V®O(G))

Then the O(G)-comodule

The functor

K

a

VL ~vG:

factors as

and

~(W) = W G~K.

and

~

are clearly

120

left exact.

The right derived functors of

respectively 7.7.1.

H'(G/K,-),

H'(K,-)

PROPOSITION

(cf.

DG-III,§6,3.3).

for every k-G-module

~ ( G ) functor

~

structure W~O(G)

7.1.2 above and

sequence

),, HP+q(G,V)

to show that the abelian categories

them (cf. [17;Th.2.4.1]).

W, W ® O ( G )

I~A:

W®O(G)

comodule structure map

V.

For any k-

--~ W ~ O ( G ) ® O ( G ) .

With this structure

since

Since for any k-G-module V, the O(G)p: V ÷ V ® O ( G )

is k-G-linear,

abelian category OC~ o ( G

) has enough injectives.

~(W®0(G))

W~O(G) K

= (W@0(G)) K

any vector space

and the

has a natural right O(G)-comodule

is an injective k-G-module,

for every k-G-module

are

by 7.6.3.

~C~-°m--~°~dn~(G~K) have enough injectives

preserves

vector space

[4;Lemma3],

~

V.

It suffices and

H'(G,-)

There is a spectral

HP(G~K,Hq(K,V))

Proof.

and

~, ~ and

W, the functor

This proves our proposition.

W®O(G/K) ~

preserves

the

Since

(see 7.4) for injectives.

121

7.7.2. and

K

COROLLARY.

Let

a closed normal

k-G-module

V, we have

G

be an affine

subgroup

scheme

an exact

of

sequence

k-group

G.

scheme

For every

of k-vector

spaces

0 ---e HI(G/K,V K) ~ HI(G,V) ~ HI(K,V) G --~H2(GTK,vK)--~H2(G,V)

For a proof Th.5.12,

consult

Let

G

be an affine

shceme.

(Thus,

finitely

generated.)

suppose

given. right

Let

that a right

comodule

the purpose

let

(cf.

(esp.

(i) k-sheaf

Hn(G,A)

(iii)

H I (G,A)

Proof. is clear.

of

= 0 =

As announced

is to prove

G

is free

is

in 7.1,

7.1.1:

above).

In the situation

and the

(fpqc)-quotient

representable. for all

n > 0

0

(i) ~=> (ii) It remains

G

k-scheme,

are equivalent:

is affine

(ii)

of

and

be the associated

map.

and 7.1.2

k-group

be an affine

u: X x G + X

structure

[4; Th.l]

The action X/G

action

unipotent

is irreducible

X = Spec(A)

section

the following

O(G)

p: A ÷ A ® O ( G )

algebra

of this

THEOREM

algebraic

the Hopf algebra

As in 7.0,

of 7.8,

textbook.

p.328).

7.8.

and

Cartan-Eilenberg's

follows

to prove

from 7.4.1,

(iii) ~

(i).

and

(ii) ~

Since

G

(iii) is

122

algebraic

unipotent,

• .. D G n = (I) Gi_I/G i Ga

e.g.,

can assume

is a central

subgroup

exact

of

G = G 0DG 1 G

subgroup

such that schemes

By the induction

that the theorem holds Suppose

series

schemes

to some closed

DG-IV,§2,2.5).

K = Gn_ I.

following

0 ~

of closed

are isomorphic

(see,

where

there

that

argument,

true for the group

HI(G,A)

= 0.

of we

G/K,

Consider

the

sequence:

HI(G/K,A K) ~

HI(G,A) ---> HI(K,A) G --> H2(G/K,A K) ,

0 where sheaf. (Apply

G/K

denotes

Since (iii)

---> (ii) to

The element

l®x

is central acts on

isomo r p h i s m in

G, fx

ka

is G-invariant. a EA x

of

G/K.) of

c an on ic al ly

O(K)

factors

A(x)

is easily

with

= C'(K,Aa) through

it follows

ax®l

in

x

+ l®x).

seen to be a I-

l®x

Let

fx:

through

in 7.2.

the

Since

that the l-cocycle

HI(K,A) G = 0 , there

= l®x

= xQl

Let

K

k a -+ A a , 1 ~* i, and

such that

P(ax)

(fppf)-

H 2 ( G / K , A K) = 0.

(see 7.9 below).

associated

C'(K,A)

that

as an

HI(K,A) G = 0.

(viz.,

C'(K,A)

trivially, Since

Hence

A®O(K)

be the l-cocycle

canonical

G

element

of the complex

K -+ A a

considered

HI(G/K,A K) = 0, it follows

be a primitive

cocycle

G/K

AGO(K).

is an element

fx

123

Since

K

is isomorphic

Hopf algebra Proposition

O(K)

to some subgroup

scheme of

is generated by the primitive

7.9.1 below implies that the action

is free and that

X/K

is affine.

Since

Ga, the elements.

Xx K ÷ X

X/K = Spec(A K)

and

HI(G/K,A K) = 0, it follows from the induction hypothesis

that

the induced action

X/K x G/K

"

is free and that the quotient Consider the map X x G ~ xh

for some

G/K on

acts on X

feely,

x~X(R), z X/K

(iii) ~

X x X,

freely.

on

X

we have

gh "I = I.

is free and completes

It remains

xg = since

K

acts that

the proof of

to prove our Theorem in the case where

commutative Hopf algebra.

A

and

This proves

elements.

this case using the theory of Hopf algebras.

Let

If

gh-l~K(R),

x(gh_ 1 ) = x

is generated by the primitive

elements

is affine.

(x,g)~---~ (x,xg).

Since

that

-~ X/G

(i).

7.9. O(G)

G

(X/K)/(G/K)

g, h ~ G ( R ) ,

it follows

the action of

X/K

)

in

H.

Thus

Let

We shall treat Let

H

be a

P(H) be the set of primitive

P(H) = {x~ H i A(x) = x ® l

be a right commutative H-comodule

algebra.

+ l~x} This means

that an algebra map p: A

~ A~H

which is also a right H-comodule

structure map is given.

Put

124

AH =

[

(a~A

p(a)

= a®t}

It follows from 7.4 that the following are equivalent: (i)

The right action Spec(p):

is free and (ii)

Spec(A)/Spec(H) A

Spec(A)x Spec(H) ÷ Spec(A)

is affine.

is a faithfully flat AH-algebra and we have an

isomorphism of k-algebras

A~

H A

-~> A ® H

,

a@bl----~

(a~l)p(b)

A Recall that the Hochschild complex

C'(Spec(H),A)

goes as

follows in lower dimensions.

3 0 ~ C 1 (Spec (H) ,A)

C O (Spec (H) ,A)

II

I

A 30(a)

= p(a)

~l(a®h)

7.9.1.

I

A®H a®l

= p(a)®h

It follows that

31 ~ C 2 (Spec (H) ,A)

A®H~H

,

,

a®A(h)

~l(l®x)

PROPOSITION.

= 0

+ a@h®l.

for all

I__ff H

x~P(H).

i_~s ~enerated by

P(H)

as

a_~n algebra, the following are equivalent: (i)

The right action

Spec(p): Spec(A)x Spec(H) ÷ Spec(A)

125

is free

and the quotient

(ii) aX

A

For each

such

that

(iii) (Notice

element

~0(ax)

that we do not

of

A

For each

such

Hence

that

AH

ax

Let

and all

have

if

x

of

P(H)

generated.)

We show take

that

an element

- ax ®

1 = l®x

determined

modulo

ax® 1 6A of

A H = Ker(~0).

® H A is wellA A generated by

Thus

= AH[ax ; x CP(H)].

H A', x ~ l ® a - a ®I is clearly A x x If the c h a r a c t e r i s t i c p of k is positive, we

that

the u n i v e r s a l P(H)

n > 0.

are clear.

be the s u b - a l g e b r a

+ A' ®

~(x p) = ~(x) p, since

well-known

x.

to be finitely

= l®a x

ax, x 6 P ( H ) .

~: P(H)

k-linear.

= P(ax)

~(x)

A'

A'

The map

element

is an element

= 1®

for all

(ii)

is uniquely

the element

defined.

~

there

- ax ® l

H

is affine.

that

~0(ax)

Notice

= 0

assume

(i) --~ (iii)

(ii) ---> (i).

x £ P(H),

= P(ax)

Hn(spec(H),A)

Proof.

ax

Spec(A)]Spec(H)

is c a n o n i c a l l y

enveloping

p > 0

Prop.13.2.3]),

H

~0(axP)

(cf.

algebra

[13;page

and the usual

= ~0(ax)P isomorphic

= l ® x p. to

of the abelian

274,

Th.13.0.1

universal

It is

U(P(H)), p-Lie

algebra

and page

284,

enveloping

algebra

of

126

P(H)

if

p = 0.

Hence

to an algebra map

can be u n i q u e l y

the map A'.

¢: H + A' ® H A A T

¢: A' ® H---* A' ~ H

Let

a®h ~

'

extended

(a®l)~(h)

-A be the

induced A ' - a l g e b r a

is c l e a r l y

map.

a sub-H-comodule,

On the other hand,

we can well

define

since

A'

an a l g e b r a

map

~: A' ®At t A ' - +

We claim ~

that = I:

a x ® l ) = P(ax)

=

~(ax®l

+

Let

Indeed l®x)

if

x~P(H),

m:

the set of

2)

m(~)

< p m

+

l®a

x

be a k-basis

I)

~(~(l®x))

= ~(l~a x

then

¢(~(l®ax)

) = ~(P(ax))

l®a

>__ 0)

X ~A

~ 0

of m

such

H.

of

=

A ÷ ¢~ (= the integers that

~ ~A

m(X)

if char(k)

x

[A]

be the

such that is f i n i t e ~ a n d

= p > 0.

[A], put re(X)

= ~A

ex

since

H is c o m m u t a t i v e . )

Poincar6-Birkhoff-Witt of

- ax®l

Let

is w e l l - d e f i n e d

a k-basis

then

P(H).

for all

e

celebrated

x~P(H),

-- a x ® l

(ex)x~ A

For an element

forms

if

- ax® 1 = l®x.

set of functions

(This

~-> ( a ® l ) p ( b )

¢~ -- 1 = ~¢ Indeed

¢~ -- I:

a~b

A'®H,

We call

theorem,

(e m

Iml = ~m(X)

By the I m ~[A]}

the degree

of

127

e

m

X~A

For each

, let

a0(a.k) = p(aA)

It is easy to see that AH

as an algebra.

be an element

aAEA

- ak~l

A'

= l®e k

is generated

For an element

7.9.2.

LEMMA.

Proof.

That

The set

such that

by

(a~x£ A

m ~ [A], put

(am~m~[A]

over

am = ~a~ m(X)

form_~s a_n_nAH-basis

of

A,.

char(k)

= 0.

Then

ex p

a~ p

~

Suppose

is generated by that

is of the form cx a

AH + ~X~A AHax implies

A'

char(k) ~

c~ e

~ A H , it follows

~am) is clear if

= p > 0. with

cx~ ~k.

X EA Since then

that the AH-submodule

is closed under the p-power

immediately

Let

that the AH~module

operation.

This

A'

is generated by

of

~am~

am~ It remains

to see the independence

given an AH-linear and the set of n

relation

m~[A]

be the highest

~m~[A]

such that

degree of

em

~m a

m

Suppose

= 0, where

~m ~ 0 such that

~m ~ A H

is finite. ~m ~ 0

Let It is

easy to see that we can write

p(a m) = a m @ l

where

the term

(~)

+ (~) + l ® e m

is an A-linear

combination

of

1 ® e m'

128

such

that

~m am = 0 such

that

0 <

Im'l

< Iml

Applying

and considering Iml

the map

p to

the A - c o e f f i c i e n t s

of

m

l®e

= n, one obtains

Xlml=n Cm ® em a contradiction.

Therefore

0

=

a

the

,

m

, m ~ [A],

are A H - l i n e a r l y

independent.

We resume AH-module, Since

the proof

it is clearly

we have

it follows

of 7.9.1:

A' ~ A H

from

(i)

Since

a faithfully

A'

=> A ' ® H

> (iii)

A'

flat , a~b

is a free

AH-algebra. I

> (a~l)p(b)

of our p r o p o s i t i o n

that

the

sequence

a0 A'

is exact. right

al

.......

We have

H-comodule,

only

p(a)

since

such

that

that

that

A = A'

a®leA'®H

,

- a®l

a®l)

if

of

A/A'

is such

a'®l

Since is zero.

H

is a

of

is an element

= p(a')

+ A H = A'.

A/A'

the socle

a6A

= 0, there

= ~0(a')

the socle

Since

to see that Indeed

a = a' + (a-a') E A '

it follows Done.

to show

is zero.

~l(p(a) p(a)

> A'®H®H

it suffices

(as an H-comodule)

then

A'~H

~

A/A' that

a' ~ A '

This means is irreducible, Therefore

A = A'

129

7.10.

COROLLARY

X = Spec(A)

(to Theorem 7.1.1 = 7.8).

be an affine k-scheme

affine unipotent k-group scheme p: A + A ~ O ( G ) map.

Let

and put and

Y/G

is affine,

X/G

is affine.

= 0.

Let

that

O(K)

G

be a subalgebra of

acts from the right.

p(B) ~ ~®O(G)

such that on

Y

is free

then the action of

G

on

X

is free and

to show that

be a closed central is generated by

that the assertion

algebra structure

G

It suffices K

A

Let

If the action of

Y = Spec(B).

Proof.

on wNich an algebraic

be the associated comodule

B

Let

subgroup

P(O(K))

is true for

HI(G,B)

G]K.

= 0 ~

scheme of

as an algebra. If

HI(G,B)

H I (G,A) G

such

Suppose

= 0, then

the following groups all vanish in view of the exact sequence of 7.7.2

(and by the induction hypothesis):

HI(G/K,BK),

H2(G/K,BK),

and hence we have

HI(K,B) G, HI(G/K,A K)

HI(G,A)

=> HI(K,A) G

Let

was seen in the proof of 7.8, the l-cocycle G-invariant. such that HI(K,A)

p(b)

= 0.

7.10.1. affine,

Since

Remark.

= l~x.

HI(G,A)

If

l®x

EB®O(K)

As is

b EB

It follows from 7.9.1 that = HI(K,A) G = 0.

G

acts freely on

then we have an isomorphism A G ®BG B

H2(G/K,A K)

x ~ P(O(K)).

HI(K,B) G = 0, there is an element

b ®i

Hence

and

-- >

A.

Y

and

Y/G

is

130

This follows from the fact that the composite

A®H

= A~B

(B~H)

- A ~B (B ~ G B

>A

is the identity and that

A

B) -- A ~ G (AG ~ G B) A B

® G A = A~tt A

is a faithfully flat

AG-algebra.

8.

The underlying

scheme of a unipotent

algebraic

group In this section has a positive G of

~38), after 8.1 th e ground field

characteristic

and a closed subgroup

For affine k-group scheme

p. H,

scheme

k

we write

G/H

inn place

G/H. 8.0.

a field see

Let

G

be an affine algebraic group scheme over

If

k

is perfect,

k.

DG-IV,

32, 3.9

and

the following are equivalent

34, 4.1(Thm.

of Lazard);

also

Lazard's original proof in [3]: (i)

G

is connected k-smooth unipotent.

(ii)

G

has a central

quotients (iii)

isomorphic to

series of closed subgroups with

G a"

The underlying k-scheme of

~n=Spec(k[Xl,...,Xn])

G

is isomorphic to

n~0°

for some

In this section we shall extend the above result to the case of non-perfect

ground field.

that the characteristic of 8.1.

PROPOSITION.

Hence we may and shall assume k

is

p>0.

For an algebraic k-group scheme

the following are equivalent: (i)

G

is connected,

(ii)

G

has a central series of closed subgroups

G = GO ~

G1 ~

k-smooth an d unipotent.

... ~

G n = (I)

G,

132

such that for all

l~i~n, Gi_I/G i

for some integer Proof. Let

G

is obvious.

We prove

be a unipotent algebraic k-group. for all

central series

i>l.

Let

GDHID...DHq=(I

)

xI

Gp

(see SGAD-VIB,

Hi_I/H i

(pV factors)

are connected

§7, for instance).

denote the image of the

~ x---x

HI=[G , G],

in which all subgroups

therefore assume from the beginning that Let

(i)---~(ii):

We then obtain a standard

and their successive quotients

and k-smooth

(Ga)m(i)

m(i)>0,

(ii)~(i)

Hi=[G,Hi_I]

Hi

is a k-form of

G

We may

is commutative.

p -th power operation

inside the group

G.

We then

obtain a central series N G

Z)

Gp

:D

where again all

'..

Gp

D

Gp

Z:) ' ' '

:D Gp

=

(1)

and all successive quotients

are connected k-smooth.

Furthermore,

these last quotients

are killed by the p-th power operation l.g,~ their Verschiebungs

GpV-I/G p

p-id,

are all zero.

so that by

Therefore, by 1.7~

~,~-1

each quotient 8.2. n~0, m>0, an

Gp

Let

/G p B

be a commutative

m×m matrix

a column vector

is a k-form of

(aij)

m 1 Z=(zi)i=

with

(Ga)m(~), q.e.d.

k-algebra.

with entries in z I.~B,

For integers k[F]

we define a

and

133

commutative

B-algebra

..Xm,Yl,...,Ym] indeterminates

B(n,(~ij),Z )

be the B-algebra

as follows: of polynomials

XI,...,Xm,YI,...,Y m.

B[XI,...,Xm,YI,...,Ym]

Let

I

Let in

B[XI,.. 2m

be the ideal in

generated by

Fny i - ~j ~ijXj

z i,

i=l,...,m, r

(where one should recall that B-algebra

B[XI,...,Ym]/I

images of

Xi' Yi

in

Frx.=x. p (§I)). The quotient J J is denoted B(n,(~ij),Z). The

B(n,(~ij),Z)

and called the canonical

generators of

Fny i = ~aijxj

It is an easy exercise

eI

{Yl

"''Ym

emxlfl

B(n,(aij),Z).

Let

be the left

+ z i.

(cf. The P-B-W theorem). k[F]-module

defined by the set of

m

on generators

relations

Fnvi = ~ ~ijuj

as in 2.4.

Thus

f = . pn , 0 < f.} "''Xm m I 0 < e I < : 1

B(n,(~ij),Z )

ui, vi, l
xi' Yi

to show that

form a B-basis of M=M(n,(~ij))

are denoted by

It is easy to see that the algebra map

134

P:

. B(n,(aij),Z)

B(n, (aij) ,Z)

@ U(M)

xi I

>

Yi

>~ Yi @ 1 + 1 @ v i

I

is well-defined and makes comodule algebra. Spec(B(n,(~ij),Z))

x. @ 1 + 1 @ u. 1

1

B(n,(~ij),Z)

into a right

By 7.9.1 the action of

~(M)

U(M)-

on

is free and the (fpqc)-quotient k-sheaf

Spec(B(n,(~ij ),z))7~(M)

is affine.

Since

{Yl el...ym em Xl fl "''Xm fm I 0 =< e.i < pn ' 0 =< f.} 1

are also a B(n,(~ij),Z)~(M)-basis

of

B(n,(~ij),Z)

as is

seen in the proof of 7.9.1, it follows that

B = B(n, ( c ~ i j ) , Z ) D~(M)

Conversely let algebra with

p: A

that the action of Spec(A)~D(M) i=l,...,m, Then

A

be a commutative right

>A @ U(M) D(M)

is affine. such that

U(M)-comodule

the structure map.

on Spec(A)

is free and that

Then by 7.9.1, there are

P(si)=si@l+l@ui

c i = F n t i - ~ i j s j, l
Suppose

belong to

and

si, tieA,

P(ti)=ti@l+l®v i.

A ~D(M)

clearly.

Since

{tl

el

ems I " " "tin

fl

"-.s

m

f m I 0 < e. < pn, 0 < f.} = 1 = i

135

form an

A~(M)-basis of

that the

A

A~(M)-algebras

A

canonically isomorphic. A=A~(M)(n,(~ij),(ci))

by the proof of 7.9.1, it follows and

A~(M) (n, (~ij) ' (ci))

are

In fact the canonical isomorphism commutes with the action of

~(M).

This observation implies immediately in particular the following 8.2.1. Let

LEMMA.

n, n' _>_ 0

M~(k[F])

and

and

and

i__nn k[F].

with

be integers.

Let

(~ij)6

be matrices with entries k[F]-modules

M(n,(~ij))

are isomorphic.

Then for each vector m' zl.~B, there is a vector Z'=(z i')i= 1 with B-algebras

B(n,(~ij ),z)

and

B(n',

are i somorphi c.

8.2.2. suppose that

COROLLARY.

With the notations as above,

(~ij(0))~(k),

constant term of

m

m, m' > 0

such that the

(~ij')'Z')

in

be a commutative k-algebra.

Suppose that the left

Z=(zi)i= Im

(~ij),Z)

B

(~ij')~,(k[F])

M(n',(~ij') )

zi'6B

Let

~ 13 ...

Then the

where

~ij(0)

denotes the

(k @ B)-algebra

~ @ B(n,

is isomorphic to the p plynomial ring over

~ @ B

indeterminates. Proof.

B(O, (aij) ,Z)

Because

~ @ M(n,(~ij))=~ @ M(0,(~ij))

is a polynomial ring over

B

in

m

and

indeter-

minates. 8.3.

Definition.

A commutative k-algebra

A

is

136

said to be of type (~) if the following fied: let

For integers k[X~,Y~]

N>0, n(i)=>0

and

be the polynomial

conditions re(i)>0

k-algebra

are satis-

for

l_
in the indetermi-

i IF] , ajkek

nates

X i yi l<=j<m(i) l
(~ij)~(k[F])

(zi)mi=l

type

(~), the

to the quotient of

k[X~,

generated by

be a commutative

and

(~).

zi~k[X~ ','Yi''l_<__j'<m(i') j j,, = ,l
such that

Fn(i)y i J

and there

1<_j_<m(i)

k-algebra,

l
n~0

a matrix such that

a column vector with k-algebra

z I.~B .

B(n,(~ij),(zi))

and

m>0

(~ij(0))~n(k)

Then, if

B

is of

is clearly of type

This remark and 8.2.2 prove by induction on

N

the

following 8.3.1. of type

PROPOSITION.

If

(~), the k-algebra

A

~ @ A

is a commutative is a polynomial

k-algebra algebra i__n_n

finitely many indeterminates.

8.3.2.

THEOREM.

group scheme. (i)

G

Let

G

be an affine algebraic

The following are equivalent: is connected

k-smooth unipotent.

k_

137

(ii)

G

has a central series of cl~sed subgroup

schemes with successive quotients isomorphic to_o k-forms of some products of finitely man Y copies o_ff G a(iii)

O(G)

Proof. @ O(G)

is a k-algebra of type (*).

(i)~(ii)

is 8.1 above.

= O(G @ ~) = ~[Xl,...,Xr] ,

of Lazard that This proves

G ~ ~

(i).

(iii) ~ ( i ) :

Since

it follows from Theorem

is connected k-smooth unipotent.

(ii)---%(iii):

Let

G=G0~GI~...~GN=(1)

be a central series of closed subgroups such that for each l~i~N

Gi_I/G i

is a k-form of

(Ga)m(i)

for some

By the induction hypothesis we may assume that is of type (*). matrix

(~ij)~(k

[F])

with

o (G/GN_ 1 ) n>O

By 2.5, there are an integer (~ij ( 0 ) ) E @ m (k) '

m(i)>0.

and a

where

m=

m(N-l), such that

GN_ 1 = ~ ( M ( n , ( ~ i j ) ) ) .

Since the action of

GN_ 1

is free and the quotient

on

G

G/GN_ 1

by the right translation is affine, it follows from

8.2. that

O(G) = O(G/GN_I) ( n , ( c ~ i j ) , ( z i ) )

for some type

Zl,...,Zm~O(G/GN_l ).

(~), so is

O(G).

Since

This proves

O(G/GN_ I)

(iii).

Q.E.D.

is of

138

8.4.

We offer a remark supplementary

to Lazard's

Theorem 8.0: 8.4.1.

PROPOSITION.

scheme such that subgroup

G=A\n,

scheme of

Proof. tl,...,t n.

G

Write

Let

G

and let

be an affine k-group H

be a k-closed normal

H=G a .

such that

Then,

G=Spec k[tl,...,tn]

By the Splitting

Lemma

~-/-~n-I

with indeterminates

(see Appendix),

U=~G-/~x~ I,

whence we can write

k[tl,...,tn]

where

~7-~=Spec

may clearly dimension,

B

contains

and

u

there exists

B[u]

polynomial

r~B

which is mapped under the iso-

Suppose as we may that

total degree

the constant

Since we

to have a positive

f(tl,...,tn)

> k[tl,...,tn].

tl,

B

an element

a term of minimal

ti) involving

= B[u]

is an indeterminate.

assume the k-algebra

onto a nonconstant morphism

= B @ k[u]

(with respect

term being excepted.

is to say, suppose

~I ~2 ~ ~I. ~n f = a + bt I t 2 '''tnn +.--+ ct I ''t n + (terms of higher total degree,

where

[~i .... =[ui =N(say)'

bg0

and

if any),

~i>0.

f

to all That

139

Let us take the automorphism

of

k[tl,...,t n]

t I } > tl, t2 I > t2+~2tl,... , tn [ for

l~j~n.

~ tn+~nt I

Under the automorphism "02 '~ (bc~ 2 ...~ n n+...

g = a +

f

V2

+c~2

terms of total degree

We may assume

k

to be infinite,

tion is trivially true.

Thus,

defined by with

is transformed

~n-

N

N) + (higher degree terms).

for otherwise

the proposi-

it is possible to fix up

in such a manner that the coefficient

is nonzero.

Consider now the homomorphism

obtained by composing the inclusion B[u]----> k[tl,...,tn],

.... tn]

fixed as above,

...,tn]

> k[tl]

j>l.

r

given by

is mapped onto

if any) with

d~0.

B----> k[tl]

B C-->B[u], the iso-

the automorphism

tl I > t I, tj I > 0 B

> k[tl]

a+dt~+(higher

of

~l=spec k[tl] ......~. ~GT-~=Spec B.

unipotent

algebraic k-group,

for some unipotent k-group k[tl,...,tn]=C[Ul,U2 ]

N

kit I,

for all

just defined,

the

degree terms in tl,

Since

G/H

kis a

this means that it is not k-

wound by Tits' Theorem 4.3.1, whence follows that

find that

k[t I,

We have thus obtained a nonconstant

morphism

indeterminates.

N t1

of

and finally the homomorphism

Under the homomorphism

element

to

...a n )t I + (mixed

~2,...,~nCk

morphism

~j ~ k

N.

where

Then, N=Spec C

(G~=N×~

~-/-~×~i=~×~2, and

Ul, u 2

1

or are

One can simply repeat the above argument to is not

k-wound.

And so on.

Thus,

~U7-~=& n-I

140

is shown. 8.4.2.

COROLLARY=[(iii)~(ii)

proof of this fact is essentially DG-loc.

cit.

8.4.3

then

different

COROLLARY.

Let

X

scheme of some

from the one in

be an affine k-group

k-scheme which

scheme.

I__~f X×~ 1

X=~ n-I

The proof is obvious.

This corollary

special case of the unsolved question: X

Note that our

and is quite elementary.

i_zs the underlying =A\n,

of 8.0].

is such that

xx~l=~ n,

settles

If an affine

is it then true that

a (very) k-scheme

x=~n-l?

9.

Th%hyperalgebra

of

In this section

a unipotent

2].

Let

G

H

k

p.

For a Hopf algebra

Hopf algebra of

scheme

(§9), after 9.1 the ground field

has a positive characteristic 9.0.

~roup

H

over a field

will be denoted

H0

k, the dual

[13; page 122, §6.

be an affine algebraic k-group scheme.

The irreducible component of called the hyperal~ebra off The Lie algebra Lie(G) of

O(G)

containing

1

G, denoted by by(G) G

is

[16; 3.2.2].

is then isomorphic to P(hy(G)),

the Lie algebra of primitive elements in hy(G)

[16; Prop.3.

1.8]. Suppose G

that

char(k)

is unipotent

algebra

if

Lie(G)

(the

latter

from

the

Hopf

algebra

: H

> k.

analogy

in

result

(cf.

if

It

is well-known

G

is

consists

each

element

as

a locally

nilpotent

H

we

by

denote

following

case

of

DG-IV,

THEOREM. connected.

that

The

the

only

= P(hy(G))

means

left

and

= 0.

§2,

can

positive

H+ be

connected of

of

G

and

nilpotent Lie(G)

that the

the

kernel

considered

acts

on

O(G) For

of

the

to

be

characteristic

Lie

elements

endomorphism).

of

a

counity an

the

above

2.13):

Suppose that char(k) = p > 0

Then

then

and that

G

i__~s

i ss unipotent if and only if the ideal

÷

hy(G)

o_~f hy(G) consists of nilpotent elements

of ring theory).

(in the sense

142

9.1.

Recall that a Hopf algebra

a hyperalgebra

H

if it is cocommutative

1.3.5].

A hyperalgebra

the set

P(H)

H

over

k

is called

and irreducible

is said to be of finite

of primitive

elements

in

H

[16;

type if

is finite

dimensional. 9.1.1. type

LEMMA.

Let

H

be a hyperalgebra

(over a field of characteristic

a union of a directed

~/

p > 0).

dimensional

of finite Then

H

is

subfhyperalgebras~

family of finite Proof.

Let

x ~ H.

Then

x

finite dimensional

sub-coalgebra

Th.2.2.1].

C

algebra

Since

C

I<X,I> pN

[13; page 46,

This means

iterated Frobenius map

,

>

k, ~ I

:

C

> XX p

> lpN) is identical with large

iterated Verschiebung

VN

N

,,> C , X ® X ,

for sufficiently

the N-times

H

[13; page 160, Lem.8.0.2].

,

fN : k

of

the dual

F N : (k,f N) ® C

(where

C

in some

is pointed and irreducible,

is local

that the N-times

is contained

N.

~ ® X 1

Or, equivalently,

map

~ (k, f N ) ®

C

(which is the dual coalgebra map of

F N) is equal to c I

~(c) ® I.

is denoted

Prop.

(Notice that the map

1.9.1].)

* It follows

Let

H'

VN

~.

in [16;

be the union of subcoalgebras

that the finite dimensional

°~ H ÷o~m ~ d i ~ t e ~

TN

>

D

sub-hyperalgebras

143

of

H

vN(d)

such that

the map

VN : H

follows that killed by

= e(d) ~ 1

> (k, fN) ® H

Since

d E D.

Since

is a Hopf algebra map,

is a sub-hyperalgebra

H'

V N.

for all

of

H

it

which is

x E H', the assertion follows from

the following: 9.1.2•

LEMMA.

cocommutative

coalgebra

(= the primitive group-like

Let

C

[13; page 157].

elements of

element)

n, then

Proof. Let

M

{C i}

Let

C gc

with respect to the unique

Vn : C

If the n-times

> (k, fn) ® C

be the unique group-like {X E C

be the coradical C 1 = kg C + P(C)

finite dimensional

P(C)

is trivial

is finite dimensional.

be the ideal

Since

C

Suppose that

is finite dimensional.

iterated Verschiebung map for some

be a pointed irreducible

by

] <X, gc > = 0}

filtration of

C

of

C

C.I

C.

Let

[13; page 185,

is finite dimensional, [16; Prop.l.4.1].

element in

§9.1].

are all

We know that

C i = {x ~ C I <M i+l, x> = O}

[13; page 220, Prop.ll.0.5.]. is

c I

~ E(c) ® gc

Since

> (k, fn) ® C

Vn : C

' it follows that

n

<X p

Since

C

x> = 0

for all

is noetherian by

X E M

and

x E C

[16; Prop.l.4.1],

it follows

144

$¢

that the ideal that

M

of

C

M N+I = 0, then

9.2.

is nilpotent.

C = CN

PROPOSITION.

N > 0

is such

is finite dimensional.

Let

unipotent k-group scheme.

If

G

be an affine algebraic

Then all elements

in

hy(G) + are

nilpotent. +

Proof.

Let

xE

sub-hyperalgebra

by(G)

Let

containing

x.

H

be a finite dimensional

Since

H C hy(G) ~ O(G)

0

,

a natural Hopf algebra map : 0 (G)

is induced.

Since

surjective.

In particular

because so is Therefore 9.3. type.

x ~ H+

the Hopf algebra

Hence

H

is local by

is nilpotent

PROPOSITION.

Let

H

H0

H'

is irreducible,

[13; Lem.8.0.2].

H

~_~ ~ hyperalgebra o_! finite then the dual

i__ssirreducible.

Let

surjecteve.

C

be a finite dimensional H

subcoalgebra of

> C

is clearly

Hence there is a finite dimensional of

H

surjective

is clearly

(in the sense of ring theory).

Then the induced algebra map

gebra

~

I_~f H + consists of nilpotent elements,

Proof.

is

is finite dimensional,

O(G).

Hopf algebra

H0

H

> H

such

that

by 9.1.1.

we h a v e o n l y

to prove

assume that

H

the

Since that

H '0

restriction then is

H' ~---~ H

C C H'0, irreducible.

is finite dimensional

sub-hyperal-

it

follows

~ C that

T h u s we c a n

from the beginning.

145

But, then, the ideal (see, e. g., Hence

H

W

is easily seen to be nilpotent

[N. Jacobson,

Lie Algebras,

is irreducible by

9.4. V

H+

Let

V

= Homk(V , k)

said

to

be

space.

dense

A subspace

if

the

induced

W

of

map V-

)

is injective. 9.4.1.

scheme.

LEMMA.

If

G

Proof. bra map

Let

be an affine algebraic k-group then

by(G)

is dense in

O(G)

O(G) ...... ~.. hy(G) 0

Since

9.4.2.

Then G

H = Spec(O(G)/I)

and

that

is connected,

G = H.

G

THEOREM.

Let

Then, G

G

is a

by(G) C hy(H) ~ by(G).

[16; Prop.3.3.6]

k-group scheme.

H

is an open subgroup Hence

I = 0.

be a connected affine algebraic

i__ssunipotent

if and only if the

+

ideal

hy(G) Proof.

of

hy(G)

Since

is unipotent.

consists of nilpotent

"Only if" part follows from 9.2.

consists of nilpotent by 9.3.

.

Let I be the kernel of the canonical Hopf alge-

It follows from G.

G

is connected,

closed subgroup scheme of

of

[13; Lem.8.0.2].

be a k-vector is

page 33, Th.l]).

elements,

O(G) C by(G) 0

then

hy(G) 0

by 9.4.1,

elements. If

hy(G) +

is irreducible

it follows that

G

APPENDIX

Central...extensions

In this Appendix, all group

schemes

A.0.

ground

k

field

here

is to outline

of affine

Results

group

summarized

(Cf.

others

are known to be true

[5; Appendix],

of the two exact though

[5; Prop.

A.I. H, E, G

the sequence

is exact,

and

The sequence, extension

2, page

(iii)

by

and if the image of

H. H

then we say the sequence

E

stated

of

or, plainly, (ii)

context. One

new,

result

k-group exact

by

for any k-algebra

flat

is referred

1 ÷ H ÷ E ÷ G ÷ 1

is contained is central

groups

epimorphism.

to as an is k-exact

in the center of k-exact

schemes

if (i)

of abstract

is a faithfully itself,

If

fields

Chap. VII].)

is probably

+ E(R) + G(R)

E ÷ G

ground

but

649].

k-homomorphisms,

or often

o__ff G

A.8,

1 ÷ H + E ÷ G ÷ 1

1 ÷ H(R)

known,

in a more general

an erroneously

is said to be k-exact represent

closed

[14;

theory

over an arbitrary

here are mostly

§6 and below,

after

A sequence

all arrows R

sequences

it is modeled

Miyanishi

DG-III,

an elementary

schemes

some are proven only over algebraically while

i__ssarbitrary , and

are affine.

extensions

field.

group schemes

the ground

Our purpose

of central

of affine

G,

or is a central

147

extension

of

G

by

H.

extension

of

G

by

H, by a slight

A.2. is said

Let

k-split, such

if there

by definition,

SPLITTING of k-grou~

LEMMA.

schemes

ida.

abuse

is a central

of language.

be k-exact.

The sequence

if there

(~ ÷ F ÷ ~)=

E

The sequence

is a k - h o m o m o r p h i s m

(G ÷ ~E ÷ G) = id G.

that

say that

i + H ÷ E ÷ G ÷ 1

to be k-split

such that

We also

exists

Then,

An exact

is g e o m e t r i c a l l y

a k-morphism

obviously,

sequence

is g e o m e t r i c a l l y

G ÷ E

G ÷

F = H x G.

1 + H ÷ E ÷ G ÷ 1

k-split

whenever

H

i_~s

k - i s o m o r p h i ¢ t__o_o G a This

is implied

cal framework, i, p.99].

by

DG-III,

this was

Let us,

first

however,

§4,

6.6.

proved

prove

Within

the classi-

by Rosenlicht

[9; Th.

it in a few lines

by another

method:

Proof. Since

Let

H = Ga

H = Spec acts

freely

we have

A = kiT] ~

p. 403],

and the ring

evidently

A.3. important.

splits

n

~ = Spec A E

with

scheme to

if

k

over

the fact

k-group

is perfect,

2, TI

K.

Lemma

B.

G,

given by

of this

derive

of a u n i p o t e n t

G = Spec

[4; Lemma

geometrically

immediately

~n

of

A + B

The c o n s e q u e n c e s we

and

the quotient

by virtue

homomorphism

the sequence

Firstly,

is k - i s o m o r p h i c

on

B = B[T]

Remarks.

the u n d e r l y i n g

k[T],

are that

of d i m e n s i o n or more

~0

148

generally and

if the group

[9; Cot.

to make 177]

2, p.101]).

Serre's

valid

groups.

(Analyzing

see directly merely

theorem

every c o m m u t a t i v e to a unique

assumed

set-up,

that

perfect

the extensions

to

possessing

a regular

cross

to the Lemma.

This

without

as done

in DG-V,

A.4. k-group E

resort §3,

schemes

given

thereby

This

k: is

k

to affirm, + 0

one can is

in that over

k

one can now ascertain,

section gives

cit],

in case

n

175-

of Witt vector

[14; loc.

+ E + W

a

field

k-group

product

Wn ÷ E

a proof

to the theory

defined

over

k,

of Chevalley's

of Dieudonn~

extension

modules,

0 + H + E ÷ G + i k-split,

by 2-coc)~cles

G x G + H

the group m u l t i p l i c a t i o n

the group

of law on

in the w e l l - k n o w n on

~ = ~ x ~

is

by

(gl,hl)(g2,h2)

for

fields,

algebraic

is g e o m e t r i c a l l y

To wit,

i0, pp.

6.11.)

If a central

is d e t e r m i n e d

manner.

0 ÷ G

is the key

a perfect

inability

§4, 4.1

No.

the argument

H2reg(Wn'Ga)s"

theorem

over

argument

is the

correspond

thanks

closed

(up to order)

fails

lemma

[14; VII,

unipotent

Serre's

that what

this

argument

non-algebraically

Chevalley's

k-isogenous

(see DG-IV,

Secondly,

induction

over

establishing To wit,

is k-solvable

gl,g 2 £ G(R),

= (glg2,

h l + h z + Y ( g l , g 2 ))

hl,h 2 ~ H(R),

where

y: G x G + H

is a

149

k-morphism

satisfying

~(gl,gz) + ~(glgz,g3) = ~(gl,gzg3) + y(gz,g 3) for all

gl,g2,g 3 ~ G~R).

Conversely,

y: G × G ÷ H, one can c o n s t r u c t + E ÷ G ÷ i

by d e f i n i n g

fashion by m a k i n g

a central

a group

use of

y.

given a 2-cocycle

law on

extension G × H

The e x t e n s i o n

0 + H

in the above

thus o b t a i n e d

w i l l be d e n o t e d b__~y G ×y H.

A.5.

Let

1 + H + E 1 ÷ G + I, 1 + H ÷ E 2 ÷ G + 1

be

G

are

extensions

of

by

equivalent

if there

H.

We say that

these

is a k - h o m o m o r p h i s m

extensions

E1 ÷ E 2

making

the d i a g r a m

1 +H÷

1

E l ÷G

÷H

+

E

2

÷

÷i

G +

1

commutative.

The

of

is d e n o t e d by

G

by

H

commutative, central (G,H),

we consider

extensions which

of

G

the by

is a subset of

is c o m m u t a t i v e , tative

set of e q u i v a l e n c e

classes

Ext(G,H).

In case

set of e q u i v a l e n c e H

and denote

Ext(G,H).

is r e p r e s e n t e d

as

H

is

classes

it by

Ext

If in a d d i t i o n

the set of all e q u i v a l e n c e

extensions

of e x t e n s i o n s

classes

EXtcom(G,H).

of

cent G

of commu-

150

A.6. group

Let

G

scheme.

and suppose schemes.

be a k-group

Let

given

Then,

0 4 H 4 E + G 4 1 a k-llomomorphism

one can construct

0 4 H 4 E' 4 G' 4 1 commutativity

scheme,

unique

H

a commutative

be a central

~: G' 4 G a central

k-

extension,

of k-group

extension

up to equivalence

subject

to the

of the d i a g r a m

0 4 H + E' 4 G' 4 1

0 +H4E

We write

%*E

+G

41.

in place

k-homomorphism

of

E'

~; H--+H',

up to e q u i v a l e n c e

0÷H

subject

÷

E

In a like manner,

one can construct to the c o m m u t a t i v i t y

for a given

9,E

uniquely

of the d i a g r a m

÷G+I

0 ÷ H' + ~,E ÷ G + i.

Possessing structure usual

~E

of additive

fashion,

and left

VII,

~E,

one can proceed

group

on the set

and EXtcent(G,H)

Endk_gr(H)-bimodule.

verifications Chap.

and

pertaining

§i],

EXtcent(G,H )

becomes

a right

The c o n s t r u c t i o n s

to the foregoing

[5; Appendix],

to introduce

DG-III,

are

a

in the

Endk_gr(G)and

found

in

§6 and SGAD-III,

[14; VIA,

151

XVII-App. I., though with various degrees of generality. A.7.

LEB~A.

group schemes

$: G + H

G, H.

(resp. central) $(G)

Let

in

If tile subgroup H(R)

for every

is a k-closed normal In fact, the image

the k-group functor and

h ~ H(R)

R' such that ÷ $(G)(R').

k

$(G(R)) R ~A!~,

(resp. central)

$(G)

x' E $(G(RV)), where

is a field,

is normal then the image

subgroup o_f_f If.

so that if

then there is a faithfully

By assumption,

of k-

is the (fpqc)-sheafication

R I ~ $(G(R)),

and then, by (fpqc)-descent, Because

be a k-homomorPhism

x[

of

x ~ $(G)(R)

flat R-algebra

) x'

under

$(G) (R)

(h')-ix'h ' ~ $(G(R')) C__ $(G)(R'), one gets

$(G)

h-lxh 6 $(G)(R).

is clearly

argument for the centrality of

$(G)

k-closed.

when each

Similar

$(G(R))

is

central. A.8.

THEOREM.

Let

0 ÷ fl ÷ E + G ÷ 1

extension of k-group schemes, H every commutative

be a central

being commutative.

k-group scheme

For

A, consider the sequence

of additive groups

0-+ H O m k _ g r ( G , A ) +

-+ EXtcent (G ,A )

where

y

sends

HOmk_gr (E ,A) -+ HOmk_gr (H ,A)

-+

EXtcent (E ,A )

~ ~ HOmk.gr(II~A )

--+

Extcent (H ,A )

(1)

t__0othe extension class of

152

~,E.

Then: (i)

The s e q u e n c e

EXtcent(E,A), (ii)

where

(1) only

Assume that

is

except

T*~* = 0

G, H

HOmks_gr(H,A ) = {0}.

exact

and

Then,

the

possibly

holds A

at

in general.

are

k-smooth

sequence

(1)

is

and

that

exact

througl~out. Proof.

(i)

We shall make q u i c k v e r i f i c a t i o n

ness

at each spot,

leaving

a)

The exactness

at

b)

Next,

through

out all routine

Hom(G,A)

at H o m k _ g r ( H , A ) ,

z

so that

with

E ÷ A × E

E(R)

it w i t h

÷ ¢,E. closed

The c o m p o s e d subgroup

G ÷ ~,E.

the

scheme

is trivial.

zero

H

in

with

the d i a g r a m

T

÷

E ÷

G ÷1

T!

0 + A ~ ~,E ÷ G ÷ 1 P

is trivial. factors

Then d e f i n e

x I ~ (-~x,x)

for all

homomorphism

E ÷ ~E

and t h e r e b y

gives

vanishes

x A × E

on the

a k-homomorphism

(G ÷ ~,E ÷ G) = id G, so

EXtcent(G,A).

the p r o p e r t y

Hom(E,A)

4: E ÷ A.

the c a n o n i c a l

that

procedures.

~ ~ Hom(H,A)

Then we have a c a n o n i c a l

A x G + A

0 +H

by

homomorphism

It is immediate

represents

suppose

~ = ~

a k-homomorphism and c o m p o s e

and at

of exact-

Suppose

now that

projection

(A + ~,E ÷ A) = id A .

~,E ~,E

p: ~,E = Thus,

in

153

we have via c)

(pX)z

= ~, w h e n c e

~

factors

through

z

pX • At

EXtcent(G,A),

Hom(H,A) ~,E I

= p(~'~)

first

it is c l e a r

÷ E X t c e n t (G ,A ) ÷ E X t c e n t (E ,A )

> ~*(%,E)

= %,(~*E)

0 + A ÷ X ÷ G ÷ 1 is t r i v i a l , %: H ÷ A.

and So,

0 ÷A+

and

~*E

be a c e n t r a l

let us

show

consider

X÷

that

is zero,

as

is t r i v i a l .

extension

that

the c o m p o s i t i o n

such

X = %,E

~I

Next that

> let

~*X

for an a p p r o p r i a t e

the d i a g r a m

G ÷ 1

0 ÷ A + ~*X ÷ E ÷ i

-~, '\ ~

~

H =

H

in w h i c h

~X

to be the

said p r o j e c t i o n

H ÷ ~X,

-~ A x E

as s h o w n

of ~ X

with

d)

(~)*

Finally,

under

the

image at

and of

H.

X

is the This

EXtcent(E,A),

to

A.

-~

k-monomorphism identification

H ÷~X

is g i v e n

quotient

shows

Set

that

of

~X

hence

Assume E)

are

now

X = ~E.

it is o b v i o u s

HOmks_gr(H,A)

k-smooth.

Let

=

{0}

by

=

that

z*~ ~ =

0.

(ii) (and

Then,

by the

A x E, the m o n o m o r p h i s m

by the

=

a projection

preceded

above.

h ~ H(R) I > (-~h,zh) A x E

admits

and

G,H,A

0 + A ÷ X ÷ E ÷ 1

be

154

a central

extension

k-homomorphism on

H

~ : H ÷ X

where

morphism. as

p: X ÷ E

Let H

R

T*X

homomorphism

in

relation

ah ~ A(R).

E, we have

homomorphism

a: H k

÷ Ak S

Consequently,

in

X, whence

Now,

It is routine by

A

x(~h)x -I

right away the

is constant

with

a k

S

by assumption.

X(ks).

is k-smooth

that

is central

to check that

such that

so that we may write

which

so that

~(H(R))

since the image functor

the image itself

h ¢ If(R).

and thereby obtains

commutes

X, too,

follows

and

S

~(H(R))

being k-smooth,

flat homo-

p[x(~h)x-l(~h) -I] =

One verifies

a(h + h') = ah + ah'

the section

~*X + X, and clearly

be a ks-algebra , x E X(k s)

(px) (zh)(px)-l(Th) -I = e E E(R), with

Then we have a

is the given faithfully

is central

= (ah)(~h)

splits.

obtained by composing

with the canonical

p~ = m

Then,

such that

X(ks)

is central

R i } ~(H(R)) by virtue of Y

But, A

and

is dense

in

X(R).

is central A.6.

is a central

E

Let

in

X,

Y = X/Imp.

extension

of

G

~*Y = X, as desired. Q.E.D. T

A.9. extension

TIIEOREM.

o_ff commutative

group scheme.

Then,

7T

Le___Kt 0 ÷ B ÷ C ÷ A ÷ 0 k-group

schemes

the following

be a commutative

and let

G

b__eea k-

sequence of additive

groups

0 ÷ H O m k _ g r ( G ~ B ) ÷ H o m k _ g r ( G , C ) ÷ HOmk_gr (G ,A) T~

÷ EXtcent (G ,B) ÷

~

EXtcent(G,C)

÷

EXtcent (G ,A)

(2)

155

i_~s exact, where class of

T

sends

~ E HOmk_gr(G,A )

~C.

Proof of this theorem is omitted, to that of

A.8.

-- cf. SGAD, A.10.

to the extension

Besides,

as it is routine,

similar

the theorem is essentially k n o ~

loc. cit. and DG, loc. cit. Example.

the three group schemes with as in A.8 and A.9.

F 0 ÷ ~p ÷ G a ÷ G a + 0.

Consider

Coupling

Ga, one obtains two complexes

The second is exact.

As for the first,

the part

F~

EXtcent(Ga,Ga)

is not exact. EXtcent(Ga,Ga) basis

÷

EXtcent(Ga,Ga)

Indeed, note that is a free left

u o, u I, ''-, Un,--.

0 ~ i < ~.

( See 3.6.1.)

÷ Extcent(~p,G a)

EXtcent(~p,Ga)

k[F]-module with a countable

such that

F~(ui) = Fu i

The non-exactness

above is therefore evident.

~ k, while

for all

of the sequence

I N D E X OF T E R M I N O L O G Y

(free)

admissible

affine

k-scheme

affine

space

algebraic

arithmetic

3,

5 I 1

of type

(*)

genus

change

theorem

135 3~ 77 77 29

biadditive C

6

scheme

group

(k-)algebra

base

25

pair

affine k-group

B

63, 114

action

divisor

78

central

extension

2, 146

central

series

canonical

(divisor)

class g r o u p

2 59 35, 148

2-cocycle

108

coaction coalgebra

143

cohomology Demazure-Hochschild

3

first d e r i v e d

73

(fpqc)-

73

commutative

extension

108

comodule (k-normal)

completion

comultiplication connected

34

72 14 1

157 section

148

rational

40

k-normal

72

cross curve

Demazure-Hochschild derived

cohomology

119

functor

(Galois)

42

descent module

148

algorithm

8

Dieudonn@ division

7

divisor canonical

78

Weil

75

divisor dual

3

class

Hopf

group

algebra

59 141 119

effaceable

14

exponent extension commutative central

2, 146

modular

69 146

of G by H F

faisceau

108

dur

faithfully

12

flat

finite

k-group

finite

morphism

first

34

derived

(k-)form (fppf)-sheaf

scheme

cohomology

63 4O 73 2, 15 109

158

73

(fpqc)-cohomology (fpqc)-sheaf

108

(fpqc)-sheafification

151

free

Frobenius

Galois

6, 10

homomorphism field

7

descent

42

function G

63, 114

action

generalized

Jacobian

generically

separable

variety

71 3

genus arithmetic

77

geometric

90

geometrically

k-split

31, 147

(k-)group

4,6

(k-)group

functor

(k-)group

of R u s s e l l

(k-)group

scheme

5 type

2 4

finite

63 ring

78

height Hochschild Hopf

2, 27

affine

Gorenstein H

81

25, 46:, 63 cohomology

algebra

108 6

dual

141

hyperalgebra (k-closed)

group

4, 141

immersion

(p-)independent (k-)integral

scheme

12 93 7

159

invertible

sheaf

58

irreducible

121

(k-)isogeneous

148

J

(generalized)

K

Krull domain

61

L

(p-)Lie a l g e b r a

13

linear

78

variety

system

logarithmic M

Jacobian

derivative

minimum

splitting

modular

extension

81

62

field

51 69

module Dieudonn~

148

k[F]-

3, 8, 17

(G-)module functor N

ii0

N a k a y a m a ' s Lemma

18

(k-)normal

40

(k-)normal

curve

72

normalization PBW

4O

(Poincar@-Birkhoff-Witt)

Picard group PID

(principal

ideal domain)

17 4O

cubic

pointed

irreducible

p-polynomial presheaf proper

14 3, 58

place plane

Theorem

coalgebra

143 41 109

scheme

quotient

97

ring

73 9

160 R

ramification

rank

76

curve

40

(abelian) rational

62

index

3

(k-)rational

ii

reduced Riemann-Roch

(k-group of) R u s s e l l S

76

Theorem type

2, 27 3

scheme affine

3,5

k-

3,5

k-

7

k-integral

73

proper

2

k-group (generically)

separable

74

sheaf 77

dualizing

(fppf)-

109

(fpqc)-

108

invertible

58

((fpqc)-)sheafification

12

shift

4

(k-)smooth spectral

151

73

sequence

(geometrically

k-)split

31, 147

split t o r u s

6

T

torsion-free

9

U

unipotent

algebraic

group

(k-wound)

unipotent

k-group

6

3, 45

161

unique

factorization

universal V

enveloping

domain

61

algebra

14

variety

i

v e c t o r group

2

Verschiebung

homomorphism

Weil

divisor

Witt

vector group

(k-)wound unipotent

ii, 142 75 148

k-group

3, 45

TABLE

~in

OF N O T A T I O N

;

I, 8

109

5

dim

2

6

deg

8

G

12 , i01

x

, )

A ~p

6

D(M)

i~

Ax

7

d(K)

51

A 'D

7

Dg

AG

7

n

® M

9

E x t c e n t (B,A)

37 P AUtK_g r

B

51

Endk_g r

Aut G

52

Ext

n

2

iii

Cn(

(A,~)

Ga

(B,A)

6

m

15, 149

¥

2O

GL(V)

ii0

103

GYK

122

iii

H°mk_g r

13

ht(G)

46

HO( , )

73

HP( , )

73

, )

iii

2, 149

8 30, 149

At

104

E2

33

HO( Ho

~ )

lo8

(~)

,,7

hy(G)

141

47

H+

141

73

H2 reg ( , ) S

148

~

)

~

com

109

141

fj~(S, A)

29

E p 'q

60(B,A)

29

E6

i04

id

i0

133

Ext (G ,H)

l~g

Im

35

iii

F

6

J in

75

B(n, (eij) ,Z)

card

7

t~'p

70

k[F]

FM

1O

k ~ P

C (v)

20

FX

Ii

k

C(A)

59

cv

12

k(x)

!08

¢,

31, 150

109

¢

[~

iso

n

S

kl/P Lie(G)

3, $

M

i0 5 7 i0 141

O~

co 0

~ Co

I"-6

CO co

~

¢

I,

~-7

I--'

~

Ix) CO

Z:>PO

"-~

---.1

"If

C~

IX)

x

®

~

0"I

~

0"1 u3

C~

~

CO

co

~

tO

0 CO

x

~

~0 4=

~

CO

OO ~

x

~ t--'

x

~ Oo

~

,~ 07

<

I- ~ 0

<

CO

Cr~ ~

<

t--'

~ CO

~r~

H CO

p CO

<

~ C,"t

~ ~

~

t--' CO

o7 ~

~

t-~ CO

~

I-G]

~ 0

~-~ ~

~

--4

cn

o~

t--~ h3 (53

GO

-.-,1 06

0'~

REFERENCES

DG:

M. Demazure-P. 1970, Masson

Gabriel,

"Groupes

Alg~briques,

& Cie, Editeur-Paris;

tome I,"

North-Holland

Publ.

Co.-Amsterdam. SGAD:

"Schemas

en gr~upes,

et A. Grothendieck," Heidelberg,

I, II, III, dirig~s par M. Demazure 1970,

New York.

FGA: A. Grothendieck,

"Fondements

-- extraits du s6minaire Math6matique

[i]

[2]

-

I.H.E.S.

"Alg~bre,

T. Kambayashi,

-- Paris.

Chapitres

A note on groups

93(1966),

M. Lazard, C. R. Acad.

[4]

Secr6tariat

6 et 7," 1964, Hermann

Paris.

Zeitschr. [3]

Bourbaki,1957-62~"

"Elements de G6om6trie Alg6brique,"

Publ. Math.

N. Bourbaki, -

de la G6om~trie Alg~briques

-- Paris.

EGA: A. Grothendieck, 1960ff,

Springer-Verlag--Berlin,

Sci. Paris

de certains

241(1955),

On the vanishing

groupes

[5]

11(1971),

[6]

connexe,

"

J. Math.

399-414.

Ill. J

, Some remarks appear

•

Igebrlques

of the Demazure cohomolo-

, Une caract~risation simplement

a

1687-1689.

gies and the existence of quotient preschemes, Kyoto Univ.

Math.

289-293.

Sur nilpotence

M. Miyanishi,

in a category,

in Osaka J. Math.

Math.

d'un groupe alg~brique 16(1972),

on the polynomial

639-650. rings,

to

165

[7]

, "Introduction ~ la th~orie des sites et son application ~ la construction des pr~schemas quo~ients~' No.47, Publ. du S~minaire de Math. Sup., 1971, Les Presses de l'Univ, de Montr~al-Montr~al.

[8]

M. Rosenlicht, Some rationality questions on algebraic groups, A~nali Mat. Pura Appl.

[9]

(4) 43(1957), 25-50.

, Questions of rationality for solvable algebraic groups over nonperfect fields, Annali Mat. Pura Appl.

[i0]

(4) 61(1963), 97-120. , Automorphisms of function fields, Trans.

Amer. Math. Soc. 79 (1955), i-II. [ii] P. Russel, Forms of the affine line and its additive group, Pacific J. Math. 32(1970), 527-539. [12] P. Samuel, "On unique factorization domain," Lectures on Math. and Physics 28(1967), Tata Inst. Fund. ResearchBombay. [13] M. Sweedler, "Hopf Algebras," 1969, W. A. Benjamin & Co. --New

York.

[14] J. ~P. Serre, "Groupes alg~briques et corps de classes," 1959, Hermann - Paris. [15] J. Tits, "Lectures on Algebraic Groups," 1966/67, Dept. of Math., Yale Univ.

New Haven.

[16] M, Takeuchi, Tangent coalgebras and hyperalgebras, to appear in J. Math. Soc. Japan. [17] A. Grothendieck, Sur quelques points d'alg~bre homologique, T6hoku Math. J. 9(1957), 119-221.

I,