TD11 : Algèbre et géométrie, Nullstellensatz

A S A
S−1A
A S
S−1A
A S
A S−1A
A S 0
S−1A
A S 0
S−1A
B A B A
S−1B S−1A A
B S−1A
S−1B
pA
M A MpM
A\pA
ApA
AppAp
k(p)Ap/pAp
A→k(p)ϕp:A/p→k(p)
k(p)A/p
ϕpp
A
f∈Arpk(p)k(pA[f−1])

A
A Am
m∈(A)
A A
Amm∈(A)
A Am
m∈(A)
A M N P A
M→Qm∈AMm
M= 0 Mm= 0 m∈(A)
A f :M→N g :N→
P0→M→N→P→0
0→Mm→Nm→Pm→0
m∈(A)
M Mm
m∈(A)
M
(Mp)p∈AAp
A A
A A
A
A
A
m
n > 0mn= 0
A
A

P∈C[X, Y ]C[X, Y ]/(P)
C[T]C[T, T −1]P
C
K(x, y)∈K2
5x2+ 3y2= 2
(x2+y2)2+ 3x2y−y3= 0
K f1, ...fr∈K[X1, ..., Xn]g1, ..., gs∈K[X1, ..., Xm]
B=K[X1, ..., Xn]/(f1, ..., fr)C=K[X1, ..., Xm]/(g1, ..., gs)VB
f1=... =fr= 0 VCg1=... =
gs= 0 f:B→C
f∗
X:VC(X)→VB(X)K X
X→Y K
VC(X)//
VB(X)
VC(Y)//VB(Y).
f∗:VC→VB
C[X, Y ]/(Y3−X5)
C[T]
V y3=x5A2
C
f∗:A1
C→V
V
f∗
C:C→V(C)
g∗:V→A1
Cg∗ C
f∗
Cf∗
VA1
C
C[X, Y ]/(Y3−X5)[X−1]C[T, T −1]
V x 6= 0
A1
Ct6= 0 VA1 C

C[X, Y ]/(Y2−X3−X2)
C[X, Y ]/(Y2−X4−X5)
C[X, Y ]/(Y2+ 2iXY −X3+X−1)
P Q R C[X1, ..., Xn]P
x∈CnP(x) = 0 Q(x)6= 0 R(x) = 0
P|Q P |R
k A B k
A B A ⊗kB
V →Am
QV(C) = ∅
V(Q) = ∅
f1, ..., fn∈Z[X1, ...., Xm]
f1(x) = ... =fn(x) = 0 Cm
Fpp
A=Z[X, Y ]/(X2+Y2−1) Fp∈Fp[X]
ζ(A, s) = Y
p
Fp(p−s).
ζ(A, s) = ζ(s−1) ∞
X
n=1
(−1)n−1
(2n−1)s!.
A=Z[X, Y ]/(XY −1),
B=Z[X1, X2, X3, X4, Y ]/((X1X4−X2X3)Y−1),
C=Z[X]/(X21 −1),
D=Z[X, Y ]/(X2−Y3),
E=Z[X, Y ]/(Y2−X2(X+ 1)).