Endomorphismes cycliques

µ µx
k[X]
sln(C)
∴
E k u ∈L(E)P∈k[X]
P u P (u)
ϕ:k[X]→L(E)
P7→ P u
x∈E k
ϕx:k[X]→E
P7→ P u(x)
µ µxϕ ϕxk[u]Ex
ϕ ϕxx∈E µx|µ
u x ∈E Ex=E
u∈L(E)µ χ
u
µ=χ
u k[u]
u u
Z(u)k[u]⇔
⇔
a∈E µa=µ

µ=Pα1
1. . . P αr
rEi= ker(Pαi
iu)
xi∈EiPαi−1
iu(x)6= 0 µxi=Pαi ia=
x1+· · · +xrµa
0 = µau(a) = µau(x1)
| {z }
∈E1
+· · · +µau(xr)
| {z }
∈Er
Eiµau(xi) = 0 i µxi
µxi|µaµxiµa
µ|µaµa|µ
⇔µaEaϕa
k[X]/(µa)'Eaa µ =χ
µa=χdeg(µa) = ndim(Ea) = n
a∈E µa=µ Eau
u
k= deg(µa) = deg(µ)e1=a e2=u(a)
ek=uk−1(a)Eaek+1, . . . , en
E{e∗
i}
G={x∈E, ui(x)eki≥0}
=T
i≥0
ker(e∗
k◦ui)
=
k−1
T
i=0
ker(e∗
k◦ui)ukuii≤k−1
G u Ea∩G={0}dim(G) = n−k
E=Ea⊕G k
G
Pk−1
j=0 λj(e∗
k◦uj) = 0 ⇒Pk−1
j=0 λj(e∗
k◦uj+i) = 0 (∀i)
⇒e∗
k(ui(P(u)) = 0 (∀i)P(X) = PλjXj
⇒P(u)(a)∈G
P(u)(a)∈EaP(u)(a) = 0
µa|P P k −1λi
dim(G) = n−k
⇔
v u u x Ex=E
v(x) = P u(x)P v =P u
y∈E=Exy=Qu(x)v u
u
v(y) = v(Qu(x)) = Qu(v(x)) = Qu(P u(x)) = P u(Qu(x)) = P u(y)

⇒µ=Pα1
1. . . P αr
rEi=
ker(Pαi
iu)a=x1+· · · +xrµa=µ E =Ea⊕G
π G EaEaG u
π u π =P(u)P
P(u|Ea) = π|Ea= 0 µu=µu|EaP π =P(u)=0 G= 0
Ea=E u