Download Introduction to Affine Algebraic Groups in Positive Characteristic and more Lecture notes Algebra in PDF only on Docsity!
INTRODUCTION TO AFFINE ALGEBRAIC GROUPS IN
POSITIVE CHARACTERISTIC
In construction, Lyon 2014
P. GILLE
Contents
- 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
- 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
- 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 f ∈ k [ 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 ζ ∈ F ( 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 η ∈ F ( 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]
id ⊗ m ∗
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 X ∈ N 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 ∈ ( A ⊗ R 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 ( N ⊗ R 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^ a ∈ A
R ea
and the multiplication is given by ea eb = ea + b for all a, b ∈ A. 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 ) = ea ⊗ ea ,
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( A ⊕ B )
∼ −→ 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 J ⊗ OG,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
0 ⊂ U ⊂ G 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 ) i ∈ I 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 ) i ∈ I ) of G such that each fi factorizes within that
k–group. Furthermore Γ G (( fi ) i ∈ I ) is smooth.
3.3.2. Remarks. (a) The k –group Γ G (( fi ) i ∈ I ) 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 ) j ∈ J 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 : X ⊂ G 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 g ∈ G ( k ), we have then gG
† ⊂ G
† , hence g ∈ G
† ( 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
- 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
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
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
- 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 s ∈ F 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
- 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.
- 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 ) =
g ∈ G ( 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 ) =
g ∈ G ( 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 ) =
g ∈ G ( S ) , | θ ( g )( f ) = f for all S –rings S
0
and for all f ∈ F 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 u± : 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 x ∈ α 2 ( R ) and b ∈ R ; 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 J ⊆ NG ( U ).
- 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
0 ⊗ RS ) =
N ⊗ S ( R
0 ⊗ RS ) = j ∗ N ⊗ RR
0
W ( j ∗ N )( R
0 ).
(2) The assumptions implies that j ∗ N is f.g. over R , hence W ( j ∗ N ) is
representable by the vector R –group scheme W( j ∗ N ).
If F is a R -functor, we have for each R
0 /R a natural map
ηF ( R
0 ) : F ( R
0 ) → F ( R
0 ⊗ R S ) =
FS ( R
0 ⊗ R 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 → S ⊗ RS 2 is
split by the codiagonal map ∇ : S ⊗ R 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
S ⊗ RS 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 ( Pα ) α ∈ I
with Pα ∈ 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
Pα
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.
