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Introduction to Affine Algebraic Groups in Positive Characteristic, Lecture notes of Algebra

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INTRODUCTION TO AFFINE ALGEBRAIC GROUPS IN
POSITIVE CHARACTERISTIC
In construction, Lyon 2014
P. GILLE
Contents
1. Introduction 2 1.1. Over an algebraically closed field of characteristic p > 0 2
1.2. Over a non-perfect field of characteristic p > 0 3 2. Sorites 4 2.1.
R-Functors 4 2.2. Definition 5 2.3. Examples 6 3. Basic results on algebraic
groups 8 3.1. Reduced subschemes of affine algebraic groups 9 3.2.
Smoothness 9 3.3. Algebraic subgroups generated by images, derived group
of a smooth
group 10 3.4. Morphisms, I 11 4. Sequences of group functors 13
5. Actions, transporters, centralizers, normalizers 15 5.1. Actions 15 5.2.
Transporters, normalizers, centralizers 15 6. Weil restriction 16 6.1. The case
of a purely inseparable field extension 20 7. Tangent spaces and Lie algebras
20 7.1. Tangent spaces 20 7.2. Lie algebras 22 8. Descent 26 8.1. Embedded
descent, field of definition 26 8.2. Faithfully flat descent 27 9. Flat sheaves 27
9.1. Covers 27 9.2. Definition 27 9.3. Monomorphisms and epimorphisms 29
9.4. Sheafification 29 9.5. Group actions, quotients sheaves 30 9.6. k-orbits 33
9.7. Frobeniuseries (see [SGA3, VIIA.4] 34 10. Solvable and unipotent groups
35 10.1. Unipotent groups 35 10.2. Structure of commutative algebraic groups
36
Date: June 10, 2014.
1
2
10.3. Solvable k–groups 36 11. Unipotent radicals and Levi subgroups 37 12.
List of structure results 38 12.1. Tits structure theorems for smooth solvable
k–groups 38 12.2. On the field of definition of the solvable radical 39 13.
Greenberg’s functor 39 References 40
1. Introduction
The theory of affine algebraic groups over an algebraically closed field
of characteristic zero is well spread out and is rather close of the theory of
complex Lie groups. When dealing with semisimple real Lie groups, we
deal real algebraic groups and that theory extends well to a base field of
characteristic zero. When dealing in the positive characteristic case, new
objects (as non smooth groups for examples) and new phenomenons (as
failure of reducibility for linear representations of GL2) occur. Technically
speaking, it is also harder since the language of varieties is not anymore
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pf9
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pff
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pf13
pf14
pf15
pf16
pf17
pf18
pf19
pf1a
pf1b
pf1c
pf1d
pf1e
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pf20
pf21
pf22
pf23
pf24
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pf29
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INTRODUCTION TO AFFINE ALGEBRAIC GROUPS IN

POSITIVE CHARACTERISTIC

In construction, Lyon 2014

P. GILLE

Contents

  1. Introduction 2 1.1. Over an algebraically closed field of characteristic p > 0 2

1.2. Over a non-perfect field of characteristic p > 0 3 2. Sorites 4 2.1.

R-Functors 4 2.2. Definition 5 2.3. Examples 6 3. Basic results on algebraic

groups 8 3.1. Reduced subschemes of affine algebraic groups 9 3.2.

Smoothness 9 3.3. Algebraic subgroups generated by images, derived group

of a smooth

group 10 3.4. Morphisms, I 11 4. Sequences of group functors 13

  1. Actions, transporters, centralizers, normalizers 15 5.1. Actions 15 5.2.

Transporters, normalizers, centralizers 15 6. Weil restriction 16 6.1. The case

of a purely inseparable field extension 20 7. Tangent spaces and Lie algebras

20 7.1. Tangent spaces 20 7.2. Lie algebras 22 8. Descent 26 8.1. Embedded

descent, field of definition 26 8.2. Faithfully flat descent 27 9. Flat sheaves 27

9.1. Covers 27 9.2. Definition 27 9.3. Monomorphisms and epimorphisms 29

9.4. Sheafification 29 9.5. Group actions, quotients sheaves 30 9.6. k-orbits 33

9.7. Frobeniuseries (see [SGA3, VII A .4] 34 10. Solvable and unipotent groups

35 10.1. Unipotent groups 35 10.2. Structure of commutative algebraic groups

36

Date : June 10, 2014.

1

2

10.3. Solvable k –groups 36 11. Unipotent radicals and Levi subgroups 37 12.

List of structure results 38 12.1. Tits structure theorems for smooth solvable

k –groups 38 12.2. On the field of definition of the solvable radical 39 13.

Greenberg’s functor 39 References 40

  1. Introduction

The theory of affine algebraic groups over an algebraically closed field

of characteristic zero is well spread out and is rather close of the theory of

complex Lie groups. When dealing with semisimple real Lie groups, we

deal real algebraic groups and that theory extends well to a base field of

characteristic zero. When dealing in the positive characteristic case, new

objects (as non smooth groups for examples) and new phenomenons (as

failure of reducibility for linear representations of GL 2 ) occur. Technically

speaking, it is also harder since the language of varieties is not anymore

adapted (and quite dangerous) and the natural framework is that of group

schemes.

The theory of affine algebraic groups over a field of positive

characteristic was significantly extended recently by the theory of

pseudo-reductive groups of Conrad-Gabber-Prasad; it will be presented in

the course of B. Conrad.

To start with this topic, I recommend “Basic theory of affine group

schemes” by Milne [M1] before reading the Demazure-Gabriel’s book.

Let me discuss basic examples for motivating the lectures. We denote

by k a base field and by p ≥ 1 its characteristic exponent.

1.1. Over an algebraically closed field of characteristic p > 0.

1.1.1. There are non trivial commutative extensions of G a by itself. Such

an example is provided by the k –group W 2 of Witt vectors of length 2. As

k -variety, we have W 2 = A

2 k and the (commutative) rule law is given on^ k

2

by ( x 0 , x 1 ) + ( y 0 , y 1 )

=

x 0 + y 0 , x 1 + y 1

S 1 ( x 0 , y 0 ) p− 1

where

S (^1) ( x, y ) = X

i =

i !( p − 1)!

( p − i )! x

i y

p−i .

The projection map W 2 G a is a group homomorphism and we have an

exact sequence (a precise sense will be provided, see § 4) of k –groups

0 G a → W 2 G a → 0 3

This sequence is not split. For simplicity we consider the case p = 2. It the

sequence splits, we would have a k –group morphism G a → W 2 , t 7→ ( t,

f ( t )) where fk [ t ]. Such a f satisfies the rule f ( x + y ) = f ( x ) + f ( y ) + xy and

xy cannot be written as a difference f ( x + y ) − f ( x ) − f ( y ). Another way for k

= F p is to see that W 2 (F p )

= Z /p

2 Z.

In constrast, recall that in characteristic zero, any commutative k –group

which is extension of G a by itself is trivial [DG, 2.4.2].

1.1.2. Intersection of smooth subgroups are not necessarily smooth. In

the additive group G = G

2 a , we consider the closed^ k –subgroup^ H 1 given

by the equation x

p

  • x = y. Then H 1 is isomorphic to G a , hence is smooth.

In the other hand, we consider the additive subgroup H 2 of G defined by

the equation x = y. Then the intersection H = H 1 ∩ H 2 is given by the

equations x = y and x

p = 0, so is isomorphic to the infinitesimal group αp

whose coordinate ring is k [ t ] /t

p

. It is not smooth. In the language of

h X → F is of this shape for a unique ζF ( R [X]): ζ is the image of IdR [ X ] by

the mapping ϕ : hX ( R [X]) → F ( R [X]). In particular, each morphism of

functors h Y → h X is of the shape hv for a unique R –morphism v : Y X.

A R -functor F is representable by an affine scheme if there exists an

affine scheme X and an isomorphism of functors hX → F. We say that X

represents F. The isomorphism hX

−→ F comes from an element ζ

F ( R [X]) which is called the universal element of F ( R [X]). The pair (X , ζ )

satisfies the following universal property:

For each affine R -scheme T and for each ηF ( R [T]), there exists a

unique morphism u : T X such that F ( u

∗ )( ζ ) = η.

We can also deal with R –functors in groups. A basic example is the R

functor Aut( X ) of automorphisms of a given R -scheme X , that is defined by

Aut( X )( S ) = Aut S ( XS ). for each R –ring S. We can add of course additional

structures (groups,...).

2.1.1. Example. Automorphisms of the additive group. In characteristic

zero, it is well-known that the R –functor Aut gr (G a ) is representable by G m.

That is for each k –ring R , the R -group schemes automorphisms of G a,R

are homotheties by R

× [DG, II.3.4.4]. In characteristic p > 0, the k –functor

Aut gr (G a ) is not representable by an algebraic k –group since for each

k –ring R , Aut gr (G a )( R ) consists in the morphisms X → a 0 X + a 1 X

p

  • · ·

· + arX

pr where a 0 ∈ R

× and the ai are nilpotent elements of R.

5

2.2. Definition. An affine R –group scheme G is a group object in the cat

egory of affine R -schemes. It means that G /R is an affine scheme

equipped with a section : Spec( R ) G, an inverse σ : G G and a

multiplication m : G × G G such that the three following diagrams

commute:

Associativity:

(G ×R G) ×R G

m×id

−−−−→ G ×R G

m −−−−→ G

can

y

∼ = % (^) m

G ×R (G ×R G)

id×m −−−−→ G ×R G

Unit:

G ×R Spec( R )

id× −−−−→ G ×R G

×id

←−−−− Spec( R ) × G

y .p 1

G

Symetry:

p 2 & m

s G

G

id×σ

−−−−→ G

×R G.

y

m

y

Spec( R ) −−−−→ G

We say that G is commutative if furthermore the following diagram com

mutes

G ×R Spec( R )

switch

−−−−→ G ×R G

m

y

m

y G =

G.

Let R [G] be the coordinate ring of G. We call

∗ : R [G] → R the counit

(augmentation), σ

∗ : R [G] → R [G] the coinverse (antipode), and denote by

∆ = m

∗ : R [G] → R [G] ⊗ R R [G] the comultiplication. They satisfies the

following rules:

Co-associativity:

R [G]

m

−−−−→ R [G] ⊗ R R [G]

m ∗ ⊗ id

−−−−→ ( R [G] ⊗ R R [G]) ⊗ R R [G]

m

&^ can^ R [G] ⊗ R

R [G]

idm

x

−−−−→ R [G] ⊗ R ( R [G] ⊗ R R [G]).

Counit:

−−−−→ R [G] ⊗ R R [G]

×id

R [G]

id ⊗ (^) ∗

id -^ m

←−−−− R [G]

x %id

R [G]

6

Cosymmetry:

R [G] ⊗ R [G]

σ ∗ ⊗ id

finite presentation.

1 This is Waterhouse definition [Wa, § I.4], other people talk about cocommutative coas

sociative Hopf algebra.

7

Proof. It follows readily of the universal property of the symmetric algebra

Hom R

0 −mod ( N^ ⊗ R R

0 , R

0 )

←− Hom R−mod ( N, R

0 )

−→ Hom R−alg ( S

∗ ( N ) , R

0 ) for

each R -algebra R

0 .

The commutative group scheme V( N ) is called the vector

group-scheme associated to N. We note that N = V( N )( R ).

Its group law on the R –group scheme V( N ) is given by m

∗ : S

∗ ( N )

S

∗ ( N ) ⊗ R S

∗ ( N ), applying each XN to X ⊗ 1 + 1 ⊗ X. The counit is σ

∗ :

S

∗ ( N ) → S

∗ ( N ), X 7→ −X.

2.3.2. Remarks. (1) If N = R , we get the affine line over R. Given a map f :

N → N

0 of R –modules, there is a natural map f

∗ : V( N

0 ) V( N ). (2) If N is

projective and finitely generated, we have W ( N ) = V ( N

∨ ) so that W( N ) is

representable by an affine group scheme.

(3) If R is noetherian, Nitsure showed the converse holds [Ni04]. If N is

finitely generated projective, then W( N ) is representable iff N is locally

free.

2.3.3. Lemma. The construction of (1) provides an antiequivalence of cate

gories between the category of R-modules and that of vector group

R-schemes.

2.3.3. Group of invertible elements, linear groups. Let A/R be an algebra

(unital, associative). We consider the R -functor

S 7→ GL 1 ( A )( S ) = ( A ⊗ R S )

× .

2.3.4. Lemma. If A/R is finitely generated projective, then GL 1 ( A ) is rep

resentable by an affine group scheme. Furthermore, GL 1 ( A ) is of finite

presentation.

Proof. We shall use the norm map N : A → R defined by a 7→ det( La )

constructed by glueing. We have A

× = N

1 ( R

× ) since the inverse of La can

be written Lb by using the characteristic polynomial of La. The same is true

after tensoring by S , so that

n

GL 1 ( A )( S ) =

a ∈ ( AR S ) = W( A )( S ) |

N ( a ) ∈ R

× o

We conclude that GL 1 ( A ) is representable by the fibered product

G −−−−→ W( A )

y N

y

G m,R −−−−→ W( R ).

Given a R –module N , we consider the R –group functor

S 7→ GL 1 ( N )( S ) = Aut S−mod ( NR S ).

8

So if N is finitely generated projective. then GL 1 ( N ) is representable by an

affine R –group scheme. Furthermore GL 1 ( N ) is of finite presentation.

2.3.5. Remark. If R is noetherian, Nitsure has proven that GL 1 ( N ) is rep

resentable if and only if N is projective [Ni04].

2.3.4. Diagonalizable group schemes. Let A be a commutative abelian (ab

stract) group. We denote by R [ A ] the group R –algebra of A. As R -module,

we have

R [ A ] = M^ aA

R ea

and the multiplication is given by ea eb = ea + b for all a, bA. For A = Z,

R [Z] = R [ T, T

1 ] is the Laurent polynomial ring over R. We have an

isomorphism R [ A ] ⊗ R R [ B ]

−→ R [ A × B ]. The R -algebra R [ A ] carries the

following Hopf algebra structure:

Comultiplication: ∆ : R [ A ] → R [ A ] ⊗ R [ A ], ∆( ea ) = eaea ,

Antipode: σ

∗ : R [ A ] → R [ A ], σ

∗ ( ea ) = e−a ;

Augmentation:

∗ : R [ A ] → R , ( ea ) = 1.

2.3.6. Definition. We denote by D( A ) /R (or A

b ) the affine commutative

group scheme Spec( R [ A ]). It is called the diagonalizable R–group scheme

of base A. An affine R–group scheme is diagonalizable if it is isomorphic

to some D( B ).

We denote by G m = D(Z) = Spec( R [ T, T

1 ]), it is called the multi

plicative group scheme. We note also that there is a natural group

scheme isomorphism D( AB )

−→ D( A ) ×R D( B ).

2.3.7. Proposition. Assume that R is connected. The above construction

induces an anti-equivalence of categories between the category of abelian

groups and that of diagonalizable R–group schemes.

Proof. It is enough to contruct the inverse map Hom R−gp (D( A ) , D( B ))

Hom( A, B ) for abelian groups A, B. We are given a group homomorphism

f : D( A ) D( B ). It induces a map

We recall that for an affine algebraic k -variety X that we say that X is

geometrically reduced if the algebra k [ X ] is reduced (that is k [ X ] is

separable in the Bourbaki sense [Bbk1, V.2]). If this condition is satisfied,

it implies that the smooth locus of X is dense [GW, th. 6.20.(ii)] and also

that the set X ( ks ) is dense in X ( ibid , prop. 6.21).

Proof. Without lost of generality, one can assume than k is algebraically

closed. The implications (1) =⇒ (2) =⇒ (3) are trivial.

(3) =⇒ (4): We assume that the ring OG,e is reduced. Let J be the nilradi

cal of k [ G ]; it is a basic fact that the formation of nilradical commutes with

localization. Hence JOG,e is the nilradical of OG,e so is zero and since the

k [ G ]–module J is finitely generated, there exists an affine open neighbour

hood U of e in G such that k [ U ] is reduced. Then U is generically smooth,

so that there exists a nom-empty open U

0UG which is smooth.

10

(4) =⇒ (1): We denote by U the smooth locus of G , it is an non empty

open subset of G which is stable by left translations under G ( k ). We have

then U = G ( k ) U = G , thus G is smooth.

3.2.2. Remarks. (a) In characteristic zero, algebraic groups are smooth

(Cartier, [DG, II.6.11]), see [Oo] for another proof.

(b) The algebraic subgroup x

p = ty

p of G a, F p ( t )

is reduced but not geo metrically reduced. Hence

reduceness is not enough to detect smoothness.

We can come back to the K = F p ( t )–group G defined by the equation x

2 p

= tx

p

. Then G red is not smooth and O G red,e = K , so G red cannot be equipped

with a K –group structure. In particular, G red is not a k –subgroup of G a,K.

3.3. Algebraic subgroups generated by images, derived group of a smooth

group.

3.3.1. Proposition. [SGA3, VI B .7.1] Let G/k be an affine algebraic k–

group. Let ( fi : Vi → G ) iI be a family of k-morphisms where the Vi are

geometrically reduced k–schemes. Then there exists a unique smallest k–

closed subgroup Γ G (( fi ) iI ) of G such that each fi factorizes within that

k–group. Furthermore Γ G (( fi ) iI ) is smooth.

3.3.2. Remarks. (a) The k –group Γ G (( fi ) iI ) is called the k –subgroup gen

erated by the fi. Its formation commutes with arbitrary base field exten

sions.

(b) If we are given a morphism of k –groups f : H → G where H is an

affine smooth k –group. The above statement provides a closed k –group

G

0 such that f factorizes within G

0

. By construction, it is the reduced

subscheme of G whose topological space consists in the reunion of the

images of H

2 n → G , ( h 1 , h 2 ,... , h 2 n− 1 , h 2 n ) 7→ f ( h 1 ) f ( h 2 )

1

...

f ( h 2 n− 1 ) f ( h 2 n )

1 , hence Γ G ( f ) is the reduced subscheme of G with

underlying topological space f ( H ).

(c) If ( Hj ) jJ are smooth k –subgroups of G , then Γ Gij : Hj (^) → G is called

the closed k –subgroup generated by the Hj. It is smooth. Further

more, if the Hi are connected, then Γ Gij : Hj → G

is connected [SGA3,

VI B .7.2.1].

3.3.3. Corollary. Let G/k be an affine algebraic k–group. Then G admits a

maximal smooth closed k-subgroup which is denoted by G

. It satisfies the

following properties:

(i) G

† is the maximal smooth closed k-subscheme of G and the maximal

geometrically reduced closed k–subscheme of G.

(ii) We have G

( k ) = G ( k ) ; if k is separably closed, and G

† is the

schematic closure of G ( k ) in G;

(iii) If k is perfect, then G red = G

† ;

(iv) If K/k is a separable field extension, then G

† ×k K = ( GK )

† . 11

Proof. We apply the above remark c ) to the family of all closed

k -subgroups of G.

(ii) Let i : XG be a geometrically reduced closed k –subscheme of G.

Then Γ G ( i ) is a smooth closed k –subgroup of G , hence X ⊂ Γ G ( i ) ⊂ G

;

(iii) If gG ( k ), we have then gG

G

, hence gG

( k ). Thus G

( k ) =

G ( k ). If k is separably closed, G

( k ) is schematically dense in G

, so we

conclude that G

is the schematic closure of G ( k ) in G.

(iv) Since G

is smooth, it is reduced so we have G

G red. If k is perfect,

then G red is geometrically reduced, hence G red ⊂ G

, thus G red = G

. (v)

See [CGP, lemma C.4.1].

3.3.4. Remarks. (a) Consider again the case G = μ 3 oZ / 2Z in characteristic

  1. Then G

= Z / 2Z and is not normal in G.

(b) The formation of G

does not commute with inseparable extensions.

Put k = F p ( t ) and consider the k –subgroup x

p

  • ty

p = 0 of G

2 a,k. Since^ t 6

( ks )

p ,

G ( ks ) = 0 hence G

= 0. We put k

0 = k (

p t ) = k ( t

0 ). Then we write x

p

  • ty

p

( x + t

0 y )

p = 0 and G

† k

0 is isomorphic to G a,k 0.

(c) If G is connected, G

can be disconnected, see [CGP, C.4.3].

→ H. Now

f is a strict epimorphism ([M1, 2.17, 2.18]), so that there exists a morphism

of k –schemes s : G → H making the factorization as follows

H

f id H

!

G

s / H.

We conclude that f is an isomorphism.

3.4.3. Remark. One could conclude also by using the more general result

that a finitely presented faithfully flat monomorphism is an open immersion

[EGA4, 17.9.1] so that the previous proposition yields that f is an isomor

phism.

13

Lecture II: Using functors

  1. Sequences of group functors

We say that a sequence of R –group functors

1 → F 1

u → F 2

v → F 3 1

is exact if for each R –algebra S , the sequence of abstract

groups 1 → F 1 ( S )

u ( S ) → F 2 ( S )

v ( S ) → F 3 ( S ) 1

is exact.

If w : F → F

0 is a map of R –group functors, we denote by ker( w ) the

R –group functor defined by ker( w )( S ) = ker( F ( S ) → F

0 ( S )) for each R

algebra S. If w ( S ) is onto for each R –algebra S/R , it follows that 1

ker( w ) → F

w → F

0 → 1

is an exact sequence of R –group functors.

4.0.4. Lemma. Let f : G G

0 be a morphism of R–group schemes. (1)

Then the R–functor ker( f ) is representable by a closed subgroup scheme

of G_._

(2) The sequence of R–functors 1 ker( f ) G G

0 → 1 is exact iff

there exists a k–map s : G

0 → G such that f ◦ s = id G 3_._

Proof. (1) Indeed the carthesian product

N −−−−→ G

y (^) f

y

Spec( R )

0

−−−−→ G

0

does the job.

(2) Assume that the sequence is exact and take id G 3 ∈ G 3 ( R [G 3 ]). It lifts to

an element s ∈ G 2 ( R [G 3 ]) = Hom R (G 3 , G 2 ) which satisfies by construction

f ◦ s = id G 3. The converse is obvious.

4.0.5. Lemma. Let 1 → F 1

i → F 2

f → F 3 1 be an exact sequence of R–

group functors. If F 1 and F 3 are representable by affine R–schemes (and

then by affine R–group schemes), so is F 2_._

Proof. denote by G 1 (resp. G 3 ) the affine R –group scheme which

represents F 1 (resp. F 3 ). Again we lift the identity id G 3 ∈ G 3 ( R [G 3 ]) =

F 3 ( R [G 3 ]) to an element sF 2 ( R [G 3 ]) which satisfies by construction f ◦ s

= id G 3. We consider the R –map ρ : F 1 × F 3 → F 2 defined for each R –ring S

by F 1 ( S ) × F 3 ( S ) → F 2 ( S ), ( α 1 , α 3 ) 7→ i ( α 1 ) s ( α 3 ). We claim that ρ is an

isomorphism of R –functors. For the injectivity of ρ ( S ), assume we have

i ( α 1 ) s ( α 3 ) = i ( α

0 1 ) s ( α

0 3 ). By pushing by^ f , we get that^ α 3 =^ α

0 3 , so that^ α 1 =

α

0

  1. For the surjectivity of^ ρ ( S ), we are given^ α 2 ∈^ F 2 ( S ) and put 14

α 3 = f ( α 2 ). Then f applies α 2 s ( α 3 )

1 to 1, so that by exactness, there exists

α 1 ∈ F 1 ( S ) such that i ( α 1 ) = α 2 s ( α 3 )

1

. This ρ is aan isomorphism of

R –functors and we conclude that F 2 is representable by an affine R

scheme.

We can define also the cokernel of a R –group functor. But it is very

rarely representable. The simplest example is the Kummer morphism fn :

G m,R → G m,R , x 7→ x

n for n ≥ 2 for R = C, the field of complex numbers.

Assume that there exists an affine C–group scheme G such that there is a

four terms exact sequence of C–functors

1 → hμn → h G m

h →

fn h G m → h G 1

We denote by T

0 the parameter for the first G m and by T = ( T

0 )

n the pa

rameter of the second one. Then T ∈ G m ( R [ T, T

1 ]) defines a non trivial

element of G( R [ T, T

1 ]) which is trivial in G( R [ T

0 , T

]) It is a contradic

tion.

4.0.1. An example : the semi-direct product. Let G /R be an affine group

scheme acting on another affine group scheme H /R , that is we are given a

morphism of R –functors

θ : h G Aut( h H).

The semi-direct product h H o

θ h G is well defined as R –functor.

  1. Actions, transporters, centralizers, normalizers

5.1. Actions. Let F be a R –functor. We denote by Aut( F ) the R –functor in groups

of automorphisms of X , that is defined by Aut( X )( S ) = Aut S−functor ( FS ). An action of

an affine group scheme G/R is a homomorphism of R –functors in groups θ : hG →

Aut( F ).

5.2. Transporters, normalizers, centralizers. If F 1 , F 2 ⊂ F are R

subfunctors, the transporter is

n

Transp( F 1 , F 2 )( S ) =

gG ( S ) , | θ ( g )( F 1 ( S

0 )) ⊂ F 2 ( S

0 ) for

all S –rings S

o

and the transporter strict is

n

Transpst( F 1 , F 2 )( S ) =

gG ( S ) , | θ ( g ) induces a bijection F 1 ( S

0 ))

−→

F 2 ( S

0 ) for all S –rings S

0

for each R –ring S. Both are R –subfunctors of hG. If F 1 = F 2 , Transpst( F 1 , F 1 )

is called also the normalizer (or stabilizer) of F 1 in F and is denoted by

Norm G ( F 1 ).

This is coherent with the usual terminology of group normalizers for the

action of G on itself by inner automorphisms. Also the centralizer of F 1 is

the R –subfunctor of G defined by

n

o

Cent G ( F 1 )( S ) =

gG ( S ) , | θ ( g )( f ) = f for all S –rings S

0

and for all fF 1 ( S

0 ))

5.2.1. Theorem. [DG, II.1.3.6] Assume that an affine algebraic k–group

acts on a separated k–scheme X of finite type. Let Y, Z be closed

k–subschemes of X.

(1)The k–functors Transp( Y, Z ) and Transpst( Y, Z ) are represented by

closed k–subgroups of G.

16

(2) The k–functors Cent G ( Y ) and Norm G ( Y ) are represented by closed k–

subgroups of G.

In particular, if H is a closed k -subgroup of G , then the centralizer and

the normalizer Cent G ( H ) and Norm G ( H ) are represented by closed

k –subgroups denoted respectively by Cent G ( H ) and Norm G ( H ). It is called

the scheme theoretical centralizer (resp. normalizer). Finally, Cent G ( G ) is

called the scheme-theoretical center of G and is denoted by C ( G ).

5.2.2. Example. Assume than k is algebraically closed. In the book of

Borel, centralizers and normalizers are considered in the setting of

varieties. This centralizer (resp. normalizer) is the reduced subgroup of

the scheme theoretical one. We present here an example where the

objects are distincts. We assume that k is of characteristic 2 and consider

the k –group G = PGL 2 = GL 2 / G m where GL 2 is equipped with the standard

coordinates

a b c d

. The equation c = 0 defines the

standard Borel subgroup B of

G. We denote by U its unipotent radical. The equation c

2 = 0 d´efines

another k –subgroup J of G and we observe that B = J red. We claim that J

NG ( U ) and in particular that B $ NG ( U ). In other words, the classical

result “ P = NG ( U )” ([Hu, § 30.4, exercice 4]) holds only in the setting of

reduced varieties.

We denote by : G a → G the standard root groups of G and by α 2 the

kernel of the groyp G a → G a , t 7→ t

2

. According to [W93, prop. 4], the

morphism of k -schemes

α 2 ×k B → J, ( x, b ) 7→ u− ( x ) b

is an isomorphism. It remains then to check that u− ( α 2 ) ⊆ NG ( U ). Let R be

a k –algebra, let xα 2 ( R ) and bR ; we compute in GL 2 ( R )

1 0 x 1 1 b^ 0 1^ 1 0^ −x^^1

1 − bx b x

− x ( bx + 1)

1 + bx (^) = (1+ bx )

1 b (1 +

bx ) 0 1

It yields that u− ( α 2 ) ⊆ NG ( U ) thus JNG ( U ).

  1. Weil restriction

We are given the following equation z

2 = 1 + 2 i in C. A standard way to

solve it is to write z = x + iy with x, y ∈ R. It provides then two real

equations x

2 − y

2 = 1 and xy = 1. We can abstract this method for affine

schemes and for functors.

We are given a ring extension S/R or j : R → S. Since a S -algebra is a

R –algebra, a R -functor F defines a S -functor denoted by FS and called the

scalar extension of F to S. For each S –algebra S

0 , we have FS ( S

0 ) =

F ( S

0 ). If X is a R -scheme, we have ( hX ) S = hX×RS. 17

Now we consider a S –functor E and define its Weil restriction to S/R

For a more general statement, see [SGA3, I.6.6]. , finite locally free is

equivalent to finite flat of finite presentation [GW, § 12.6]; in particular if R

is noetherian, finite locally free is equivalent to finite flat.

Proof. (1) For each R -algebra R

0 , we have

Y S/R

W ( N )

( R

0 ) = W ( N )( R

0RS ) =

N ⊗ S ( R

0RS ) = jNRR

0

W ( jN )( R

0 ).

(2) The assumptions implies that jN is f.g. over R , hence W ( jN ) is

representable by the vector R –group scheme W( jN ).

If F is a R -functor, we have for each R

0 /R a natural map

ηF ( R

0 ) : F ( R

0 ) → F ( R

0R S ) =

FS ( R

0R S ) =

Y

FS

S/R

( R

0 );

18

it defines a natural mapping of

R –functor ηF (^) : F → Q S/R

S –functor E , it permits to

defines a map

φ : Hom S−functor ( FS, E )

Hom R−functorF, Y S/R

FS. For each (^) E

by applying a S –functor map g : FS → E to the composition

Q g

F →

η F

Y S/R

FS

−→ Y

S/R

S/R

E.

Proof. We apply the compatibility with R

0 = S 2 = S. The map S → SRS 2 is

split by the codiagonal map ∇ : SR S 2 → S, s 1 ⊗ s 2 → s 1 s 2. Then we can

consider the map

θE :

Y S/R

E S^2

−→

Y

SRS 2 /S 2

ES ⊗ RS 2

∗ Y

S/S

E = E.

This map permits to construct the inverse map ψ of φ as follows

ψ ( h ) : FS

l S Y^ S/R

E S 2 →

θE E

for each l ∈ Hom R−functor F,

Q S/R

E

. By construction, the maps φ and

ψ

are inverse of each other.

In conclusion, the Weil restriction from S to R is then right adjoint to the

functor of scalar extension from R to S.

6.0.4. Proposition. Let Y /S be an affine scheme of finite type (resp. of finite

presentation). Assume than S is finite and locally free over R. Then

the functor

Q S/R h Y is representable by an affine scheme of finite type

(resp. of finite presentation).

Again, it is a special case of a much more general statement, see [BLR,

§ 7.6].

Proof. Up to localize for the Zariski topology, we can assume that S is free

over R , namely S = ⊕ i =1 ,...,d R ωi. We see Y as a closed subscheme of

some affine space A

n S , that is given by a system of equations ( ) αI

with S [ t 1 ,... , tn ].

Then Q^ S/R

h Y is a subfunctor of

Q

S/R

W ( S

n )

−→

W ( j ∗( S

n ))

−→ W ( R

nd ) by Lemma 6.0.3. For each I , we write

X i =1 ,..,d (^) y 1 ,iωi, X i =1 ,..,d

y 2 ,iωi,... ,

X i =1 ,..,d

yn,i

= Qα, 1 ω 1 +

· · · + Qα,r

ωr

19

with Qα,i (^) ∈ R yk,i ; i = 1 , .., d ; k = 1 , ..., n. Then for each R

0 /R , Q

S/R h Y ( R

0 )

inside R

nd is the locus of the zeros of the polynomials Qα,j. Hence this

func tor is representable by an affine R -scheme X of finite type.

Furthermore, if Y is of finite presentation, we can take I finite so that X is

of finite presentation too.