Rapport de M2 Groupes algébriques unipotents

A
(As, Au)AsAuAu−Id
AsAuA=Au.AsG
G GLn
Gs={gs|g∈G}Gu={gu|g∈G}
G
G
GsGuG
Gs×Gu→G
G
kRu,k(G)G
G k G
Ru,k(G) = {1}
{1}→Ru,k(G)→G→G/Ru,k(G)→ {1}.
Ru,k(G)G/Ru,k(G)

K
V=KnS=K[T1, . . . , Tn]S
V I V V(I)
I X V I(X)
S X
I(V(I)) = √I
V(I)I S
K K
K0
X K[X] := S/I(X)
X K[X]A
r X ⊂Kr
K[X] = A
K
X⊂Knf U
X x g, h ∈K[X]V x
V⊂U∩ {z|h(z)6= 0} ∀y∈V f(y) = g(y)
h(y)

f U U OX(U)
K U
K
(X, OX)K X
OX
k K k
F k
F k A F
R7→ Homk(A, R)F
E F k A B
E→F
k B →A
E→F B →A
k
X K A =K[X] Max(A)
A
x∈X7→ I({x})∈Max(A)
Max(A) HomK−alg(A, K)X0
A X X0(K)
X
k
k K k
V K k V k
Vk⊂V V ∼
=Vk⊗kK f :V→W
K k f k k
f(Vk)⊂Wk
A K k A k Ak⊂A
A∼
=Ak⊗kK f :A→B K K
k f k k
k
X K X k
I(X)kI(X)
k k A0

K[X]K⊗kA0→K[X]
A0=k[X]X k
X(k)k k
A0→k
X Y k α :X→Y k
K α0:K[X]→K[Y]k
ksk
X k X(ks)X
Γ = Gal(ks/k)ksk
V K k VkV∼
=Vk⊗kK
ΓVks=ks⊗kVkVk=VΓ
ks
V k ΓV(ks)
V
V(ks) Homks−alg(ks[V], ks)x∈V(ks)
exγ∈Γ
eγ(x)=γ◦ex◦γ−1
γ ksks[V] = ks⊗kk[V]
V k Z
Z k
Z ksZ(ks) Γ
E⊂Z∩V(ks)EΓZ
k K k
(G, +) K
G
e∈G
µ:G×G→G
i:G→G
G k µ i k e ∈G(k)
G k
G G0f
G G0
G G0k f k
f k