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
n≥1p:Cn−1×{0} → Cn−1
Cn−1× {0}
CnCn−1

n≥1Cn
Cn
Γ1=Z C
Γ2={(t, t2, t3)|t∈C}C3
Γ3={(n, 2n,3n)|n∈N}C3
Cn
V W V
V W
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
x
P∈Z[X]P(x)=0 Z
Z
Z
Z
Z
k k k[X, Y ]
XnY n > 0
X X
XR
A M A n ∈N
A u :M→An(u)
A
R
M R m1, ..., mn
M f M

A= (aij )∈ Mn(R)f(mi) = Pn
j=1 aij mj1≤i≤n
P(X) = det(XIn−A)P(f) = 0
M R
I R IM =M a ∈I
(1 −a)M= 0
M
m n g :Rm→Rn
R m ≤n
A
M N P A 0→
M→N→P→0N∼
=M⊕P
M N P A
•N∼
=M⊕P
•A0//Mf //Ng //P//0
A s :P→N g ◦s=P
•A0//Mf //Ng //P//0
A t :N→M t ◦f=M
A P
M N A
0→M→N→P→0N∼
=M⊕P
A
A
A A
A=ZA
A
A=B×C B C
B A
A2π
R R
P A R R
f(x+ 2π) = −f(x)x∈R
A P ⊕P∼
=A⊕A P
N A f R