Corrigé

k
(P),(R)⊂A=C[x1, . . . , xn]Q
V(P), V (R) Spec(A){Q6= 0}An
CV(P)⊂V(R)
p(R)⊂p(P)
A QmRd=P P0C[x1, . . . , xn]P0∈C[x1, . . . , xn]
m > 0C[x1, . . . , xn]P R
X x, y ∈X x 6=y
x∈U⊂X y ∈V⊂X U ∩V=∅X
X U ⊂X U ⊂X
U Y =X−U Y ∪U=X U =X
U, V ⊂X U ∩V6=∅
X U ⊂X U
U1, U2⊂U U1∩U2=∅Ui
X U
X Y ⊂X Y
Y1⊂Y2=Y Y
Yi∩Y Y Y i = 1 2
Y⊂YiY⊂Yi
X xy ∈√0p⊂A
x∈p y ∈p X =V(x)∪V(y)X V (x) = X V (y) = X
V(x) = X x x ∈ ∩p=√0
√0Ar´ed
X=V(I)∪V(J)p IJ IJ ⊂√0
√0I⊂√0J⊂√0V(I) = X
V(J) = X X
A=k[x, y, z]/(y2−xz, z2−y3)Y
Y A p
y2=xz z2=y3z2=xyz z(z−xy) = 0 z∈p z −xy ∈p
y3=z2p y ∈p p1= (z, y)
Y1= Spec(A/p1) =
Spec(k[x]) xA3
k
z=xy A/p y2=x2y y(y−x2) = 0 A/p
y∈p y −x2∈p p =p1
p=p2= (y−x2, z −x3)

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

A=C[X]X
A
(X−x0)x0∈C
A=C[X, Y ]X
f(X, Y )∈C[X, Y ]
ASpec(A)
{0}