Notes de travail pour l`agrégation

k X
SLnC
SOnR
SL E
ZnZ
nGLn
3

PGL2C

k X
A A X
A X1, . . . , Xn
k k X
A
A X
P A X c P
A
A P, Q
A c P Q c P c Q
c P 1P
A appP
A X
appP
A X
K X
ZX
QXZX
k X
k X
A K
P A X c P 1P A X
K X
P K X P QR
A X K X Q R
K Q
1c P c Q c R ac R Q a
A P A X
P A X
P QR Q R K X
λ, µ K Q λQ R µR Q , R
A X 1P λµQ R
1λµ P Q R P
A X Q R
A Q R K
GL2kAutkk x
a b
c d XaX b
cX d
PGL2k k
k X
2,2

k X
PGL2kAutkk x
ϕ:k X k X F
ϕ X U X V X ϕ k F
k X :k F 1F U X V X
nmax deg U , deg V
k X :k F n n 1
k X :k F n
X k F
n
P T d´ef V T F U T .
P
k F T
k X :k k X :k F
fini
k F :k
F k k F
P T V T F U T
k T F
•
1F
•P1
F k X F
k X F
k T F k F T P T
k F T k F T
ZnZ
A1
M n A
f:AnAnA
f f det f A
fdet f A
AZfdet f
A k X f k
deg det f
coker f Anf An
f
coker f0
e1, . . . , enAn

f i if i
eig eiii
g:AnAnf g Id
ei
det fdet g1 det f
Mt Mt MM det MId
f
ddet f d
f x 0Mx 0
dx 0
x d x 0f
d f
u A ud 0
µ M uµ 0
1
M f ue10ue10f
N M u det N0
r r n
ud 0
M
N r
i1, . . . , n
r M i
Pir1
Pi
m1,1. . . m1,r 1
mr,1. . . mr,r 1
mi,1. . . mi,r 1
.
i r Pi
i r 1M r 1
u
r u det Pi0
ur1
j11jmi,j µj0µj
r1, j i
M ux 0x
µ1,..., 1r1µr1,0,...,0uµr1udet N0
ux 0f
R, S
ZD:RM S
d1, . . . , dn
didi1i
coker fZnDZnZd1Z Z dnZ
coker f d1. . . dndet f
R, S
k X D :RM S
P1, . . . , Pn
PiPi1i
coker fk X
P1
k X
Pn
dimkcoker fdeg P1deg Pndeg P1. . . Pn
deg det f