Y2= Spec(A/p2) = Spec(k[x])
x7→ (x, x2, x3)A3
k
A/I A
A/I
f=g I f =g A/I
A=k[x, y]/(y2−x2−x3)X f =y/x
X−{0}A[1/x]
X A A →A[1/x]
g:= y2−x2−x3k[x, y]k[x, y]
A A →A[1/x]
f6∈ A f ∈A y =fx A
k[x, y]y=fx +gh h ∈k[x, y]x= 0 y=y2h(0, y)
k[y]
X
U⊂X V ⊂Pn
k
XA1
k
t=y/x X x =t2−1y=tx =t(t2−1)
A1
k→X t 7→ (t2−1, t(t2−1))
A1
k\ {±1}
⇒Γ(X, OX)e
U0={x∈X , e(x)6= 0 k(x)}U1={x∈X , e(x)6= 1 k(x)}
X
⇒Γ(X, OX)'A1×A2e= (1,0)
⇒X X =YqZΓ(X, OX) = Γ(Y, OY)×Γ(Z, OZ)
X r
Γ(X, OX)r2r
k[x, y, z]/(x2−yz, xz −x) (x, y) (x, z)
(z−1, x2−y)Y Y
n A
A A m n
m=n n ⊂m
A m m =n x ∈n x
A A
m n =m
n m A
m=n x ∈n x m m A
m n =m A n
A=k A =k[]/(2)X