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Complete lecture notes combined for Affine Algebraic Groups
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In construction, Lyon 2014
P. GILLE
Contents
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
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
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
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:
m×id
m −−−−→ G
can