TD - Feuille 3

k
X→Z Y →Z
X, Y, Z X ×ZY
Spec(K)→X X OX,x
x∈X mxOX,x k(x) = OX,x/mx
Spec(K)→X K (x, i)
x X i :k(x)→K
X x ∈XSpec(A)
x px⊂A x x
px⊂A
X= Spec(A){x}V(px)
U= Spec(A)V= Spec(B)x
x U V f ∈A
x∈Spec(Af)⊂Spec(B)g∈B x ∈Spec(Bg)⊂Spec(Af)
x X
U= Spec(A)x px⊂A
U= Spec(A)x px⊂A⇐⇒
⇐⇒
x X
{x}X
X F ⊂X ηF
F=X= Spec(A)η1η2F
U1X η1U1∩F η2η1=η2
x7→ {x}X
X F 7→ ηF
X Y, Z X Y ∩Z
Y×XZ X Y ∩Z
Y Z
f:X→Y y ∈YSpec(k(y)) →Y
f y f−1(y)X×YSpec(k(y))
X Y f−1(y)
f:Y→X
X Ui= Spec(Ai)Vi=f−1(Ui) Spec(Bi)
BiAi

X Y
f Y
A⊂B B A
B A
p⊂A q ⊂B q ∩A=p
B A B A
X⊂A2
ky2=x3
f:A1
k→X f(t) = (t2, t3)f
A f ∈A
0 dim(A/f)≤dim(A)−1
A
Adim(A/f) = dim(A)−1
p⊂R Rp
p p0(p1(· · · (pr=p
pht(p)
m= dim(A/f)p0(p1(· · · (pmA/f
p0(p1(· · · (pmA p0
0
0
A f ∈A0
dim(A/f) = dim(A)−1
A4
k= Spec(k[r, s, t, u])
X rt =s2ru =st
Y rt =s2ru =st su =t2