TD3 : Ensembles algébriques, topologie de Zariski, anneaux

R[X]/(X5(X4−1))
Z[1+√13
2] (3)
IC[X1, ..., Xn]
V(I) = {(z1, ..., zn)∈Cn|∀f∈I, f(z1, ..., zn)=0}.
V(I)Cn
Cn
Cn
CnV
CnV
Cn
Cn
f∈C[X1, ..., Xn]D(f) (x1, ...., xn)∈Cn
f(x1, ..., xn)6= 0 D(f)
Cn
C
C2
C×C C

∆ = {(x, x)|x∈C}
∆ = {(x, y)∈C2|x−y= 0},
∆C2
C
C C2\∆
U×V U V C
∆C×C
n≥1p:Cn−1×{0} → Cn−1
Cn−1×{0}
CnCn−1
IC[X1, ..., Xn]J I
C[X1, ..., Xn]→C[X1, ..., Xn−1]
p(V(I)∩(Cn−1× {0})) = V(J).
p
JC[X1, ..., Xn−1]I
JC[X1, ..., Xn]
p−1(V(J)) = V(I)∩(Cn−1× {0}).
p
p
n≥1Cn
C
C
Cn
Γ1=Z C
Γ2={(t, t2, t3)|t∈C}C3
Γ3={(n, 2n,3n)|n∈N}C3
Cn

n= 1 X=ZX
C
C C X=F1∪F2
F1F2X X
xy = 0 C2
X F1F2
X=F1∪F2X F1=X F2=X
X
f:X→Y
X F1F2f(X)f(X) =
F1∪F2X=f−1(F1)∪f−1(F2)f−1(F1)f−1(F2)
X f−1(F1) = X f−1(F2) = X F1=f(X)
F2=f(X)f(X)
V W V
V W
V F1F2W
W=F1∪F2V V =F1∪F2
V F1=V F2=V
F1=V W =F1∩W=F1W
W F1F2V
V=F1∪F2W= (F1∩W)∪(F2∩W)W
F1∩W=W F2∩W=W
F1∩W=W F1=W W V V
VCn
I
O(V)V(I)6=∅
eO(V)
Ve={(z1, ..., zn)∈V|e(z1, ..., zn)=1}
V
V
W V
eO(V)W=Ve

C
xy(xy −1) = 0
C2O(C) = C[X, Y ]/(XY (XY −1))
C
O(C)
A=C[X1, ..., Xn]
x= (x1, ..., xn)∈Cnmx={f∈A|f(x1, ..., xn) =
0}A
X1, ..., XnX1, ..., XnA/m2
x
mx/m2
xCX1−x1, ..., Xn−xn
φ:A/m2
x→A/mx⊕mx/m2 x
f7→ f(x),
n
X
i=1
∂f
∂Xi
(x)(Xi−xi)!
C
CnxC
mx/m2
xC(mx/m2 x,C)
VCn
x= (x1, ..., xn)∈VmV,x ={f∈ O(V)|f(x1, ..., xn)=0}
O(V)
V x C
T∗
V,x =mV,x/m2 V,x TV,x =C(mV,x/m2 V,x,C)
I(V)ψ:A→ O(V)
ψ ψ∗:T∗
Cn,x →T∗
V,x
dx:I(V)→T∗
Cn,x
f7→ ∂f
∂Xi
(x)(Xi−xi).
dxX∗
1, ..., dxX∗ nTCn,x X1−
x1, ..., Xn−xnTV,x
TCn,x Pn
i=1 λidxX∗
i
Pn
i=1 λi∂f
∂Xi(x) = 0

(0,0)
x7+xy =y7+y x2=y3
C2
VCn X
C[X]/(X2)C[X]/(X2) = C[]θ
C[]→C
θ∗:C−(O(V),C[]) →C−(O(V),C).
x∈Vx:O(V)→Cxx∈
C−(O(V),C) (θ∗)−1({x})
V x
C−(O(V),C)
xx∈V
A={P∈Q[X]|P(0) ∈Z}
A
A
A
A
A
a+XP b +XQ a, b ∈ZP, Q ∈Q[X]
a6= 0 b6= 0 R∈Q[X]
(a+XP, b +XQ) = (1 + XR)Q[X]c
a b Z(a+XP, b +XQ)=(c+cXR)A
⊆c R
u+XU v +XV ∈Q[X]
(a+XP )(u+XU)+(b+XQ)(v+XV ) = c(1 + XR).
au +bv =c
Zα, β ∈Zaα +bβ =c a(α−u) + b(β−v)=0
(a+XP )(u+XU+α−u
b(b+XQ))+(b+XQ)(v+XV +β−v a(a+XP )) = c+cXR.
⊇ab 6= 0