Endomorphismes semi

E k n
u∈L(E)u F ⊂E
u k
k0
p > 0
0
p > 0Fp(X)
u
u
u k
⇐⇒ ⇐ k
u µu=Pα1
1. . . P αr
r
Ei= ker(Pαi
i(u))
F⊂E u F =⊕F∩Ei
F∩Ei
F x ∈F x =x1+· · · +xrxi∈Ei
πi:E→Eix7→ xiu F u
πixi=πi(x)∈F
⇒u µu
µu=P2Q(P Q)(u)=0
u P 2Q F = ker(P(u)) S u F
a= (P Q)(u)
•a F a = (QP )(u) = Q(u)◦P(u)
•a S y ∈S a(y)∈F P (u)[a(y)] = (P2Q)(u)(y)=0
a(y)∈S S u a u
a(y)∈F∩S a(y) = 0 F∩S= 0
a= 0 µu
⇐µu=P1. . . PrPi
F Ei=
ker(Pi(u)) F=⊕F∩Eii
F∩EiEiF
E µu|Ei=Piµu=P

F=E G = 0 x∈E−F
k ϕ:k[X]→E
Q7→ Q(u)(x)
Gx={Q(u)(x), Q ∈k[X]}Px
F∩Gx= 0 Gx0, Gx00
Gx⊕Gx0⊕Gx00 ...
µu=P Px|P P
y∈F∩Gxy=Q(u)(x)P|Q
y= 0 F∩Gx= 0 P Q
UP +V Q = 1 ϕ
E
U(u)[P(u)(x)
| {z }
=0
] + V(u)[Q(u)(x)
| {z }
y
] = x
V(u)(y)∈F F u x ∈E−F
⇐u
k k µuk
k
⇐k µ0
Xpµu(X) = F(Xp)k
P p F (Xp)=(G(X))p
µuµ0
u6= 0
µuµ0
uk1
µu
k[X]
k=F2(T)
TF2E=k2
u=1T+ 1
1 1 .
u χu(X) = X2+T1 = −1k
T k u
u k k α
χuk χu(X) = (X+α)2u k α Id
u=αId
u k 
A k Endk(A)
k a ∈A Ga:A→A
a Ga(x) = ax

a Gak
A →Endk(A)A n Endk(A)
(n, n)
k=Fp(T)A=k[U]/(Up−T)u
U A u ∈Endk(A)'Mp(k)Xp−T
u
k α k u
u=d+n
k u ∈L(E)
(s, n)
u=s+n
s n
sn =ns
k=R
Mn(R) Mn(C)a
a
u=d+nCu=u u =d+n
d=d n =n d n
Rd
K
µuk k
G K k E L(E⊗kK)
(n, n)K G
L(E⊗kK)
k=KGL(E⊗kK)GL(E)
K u =d+n d n L(E⊗kK)
σ∈G uσ=u u ∈L(E)uσ=dσ+nσ
dσnσd+n
dσ=d nσ=n d n G L(E)
s=d K k
u=s+n
k=Fp(T)
A=k[U, V ]/(Up−T, V p)u, v U, V
A u Xp−T u +v
Xp−T v Xp
u= (u+v)−v s +n