Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
414
T. Kambayashi M. Miyanishi M. Takeuchi
Unipotent Al...
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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
are polynomials l
by the ideal
Let
B
integers,
A
is isomorphic I
i i ~k C~jkXk - z ji'
(~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,