TD13 : Topologie de Zariski, dimension, lissité, anneaux de

(C),(C[X]/(X2)),A1
C= (C[X]),
A2
C= (C[X, Y ]),A1
Z= (Z[X]).
(C[X]/(X2(X8+ 1))),(R[X]/(X3+ 2X2+ 2X)),
Gm,C= (C[T, T −1]),(C[X, Y ]/(X2−Y2)),
(C[X, Y ]/(X2−X)),(C[X, Y ]/(X2)).
k n ≥1kn
(k[x1, . . . , xn])
kn⊆(k[x1, . . . , xn]) k
k2k×k k
A(A)⊆(A)
(A)
(A)
(A) (A)A

Γ = {(n, 2n,3n)|n≥1}
Q3F3
pp
ΓC3
pFpFp
ΓFp
3
ΓF2
3F3 3
V((I:J))
A J ⊆I A (I:J) = {a∈A|aJ ⊆I}
V((I:J)) = V(J)\V(I)
XRnC(X, R)
XR
Uf={x∈X|f(x)6= 0}f∈C(X, R)
X
C(X, R)C(X, R)
x7→ Mx={f∈C(X, R)|f(x)=0}
ϕ:X∼ −→ C(X, R).
C(X, R)ϕ
f:A→B
f f]:B→A
I A p :A→A/I
p]:A/I →A
A/I V (I)A
A A
AZ
mZ∩m
mZ→A
Z/(Z∩m)Z

A/m
Z/(Z∩m)
Z∩m= 0
A/m
f∈An
AfAf/n
A/(A∩n)
p(0)
(A)
Cn
V W V
V W
V(Y2−X)⊆A2
C, V (XY )⊆A2 C, V (X2+Y2)⊆A2 C,
V(Y2−X3−X)⊆A2
C, V (X2, XY )⊆A2 C, V (XY, Y Z, ZX)⊆A3 C,
V(X2Y3−X3Y Z, Y 2Z−XZ2)⊆A3
C,
V(ZX2−2ZY 2, Y 3−Z3)⊆A3
C, V (ZX2−2ZY 2, Y 3−Z3)⊆A3 Q.
X
Y X dim Y≤dim X
X Y X
dim Y= dim X X =Y
X X
(Ui)Xdim X= supidim Ui

k A k X = (A)U
X
Xdim U= dim X
U X
X X
dim U= dim X
f:A→B f]:B→
Adim A > dim B
f:A→B f
f]:B→Adim A= dim B
k f :A→B k
f∗({0}) = {0}f
dim (f])−1({0}) = dim B−dim A
k A ⊆B k
p1p2Ap1⊆p2
q1q2B
q1∩A=p1q2∩A=p2q1⊆q2
pAp1 p p2
qBq1 q q2
q∩A=p
k l k k[[t]]
k[[t]] l
k A =k[X1, . . . , Xn, . . . ]
(un)N
u1= 0 pn(Xun+1, . . . , Xun+1 )S
pn
S−1A
(un)
S−1A
(un)
R

x∈Rr{0}
R
S−1A
k n ∈N Pn(k)
kn+1 x= (x0, ..., xn)∈Kn+1 \ {0}[x0:... :xn]∈
Pn(k)x
I A =k[X0, ..., Xn]
V(I) = {[x0:... :xn]∈Pn(k)|∀f∈I , f(x0, ..., xn)=0}.
Pn(k)
V(I)I A
I A V (I) = ∅
0≤i≤n d ∈NXd
i∈I
Pn(k)
Pn(k)Sn
i=0 UiUi
Pn(k)kn⊆(k[Y1, ..., Yn])
Pn(k)n
k n ∈NS=k[X0, ..., Xn]Pn
k
S S+= (X0, ..., Xn)I
S V+(I)Pn
k
I f ∈S+D+(f) = {p∈Pn
k|f6∈ p}
V+(I)Pn
k
Pn
k
D+(f)Pn
k
Pn
kD+(Xi) 0 ≤i≤n
0≤i≤n D+(Xi)An
k
Pn
k
n
k
V:x2−xz −y2−1 = yz −x2−z2= 0
A3
k