Colle du 2 décembre : Algèbre linéaire

G g ∈G g2=e
G
E K f1, ..., fn
E f1◦... ◦fn= 0
A1, ..., AnK
A1→A2→... →Anfi:Ai→Ai+1 i= 1, ..., n −1
i(fi+1) = (fi)
A1
f1//
g1
A2
f2//
g2
A3
f3//
g3
A4
f4//
g4
A5
g5
B1h1
//B2h2 //B3h3 //B4h4 //B5
i hi◦gi=gi+1 ◦fig1, g2, g4, g5
g3
Fqq SLn(Fq)
A B Mn(K)A+B=AB A
B
A= (aij )∈Mn(C)i6=j aij aji = 0 i, j, k
aik = 0 aij ajk = 0 (A)
KA∈K[X]P∈K[X]
B∈K[X]AB ∈K[P]
A, B ∈Mn(RAB =BA (A+B)≥0
p > 0 (Ap+Bp)≥0
Mn(K)
V X, Y ∈V T r(XY )=0
n∈N∗
n×n1,2, ..., n2

2n+ 1
2n n
ECn≥1
G GL(E)F E G
F G
p E p G
p p G
q F f =Pg∈Gg◦q◦g−1
a b X3+aX +b
P Q
Γt∈C7→ (P(t), Q(t)) ∈C2R∈C[X, Y ]
Γ
p A B Mn(Z)T r((A+
B)p)≡T r(Ap) + T r(Bp) mod p
A∈Mn(Z)T r(Ap)≡T r(A) mod p
p(un)u0= 3 u1= 0 u2= 2
un+3 =un+1 +unp up