Corrigé

f:X= Spec(A)→Z= Spec(C)g:Y= Spec(B)→Z= Spec(C)
X×ZY
pr1:X×ZY→Xpr2:X×ZY→Y W
u:W→X v :W→Y f ◦u=g◦v W →X×ZY
W
u
v
X×ZY
pr1
zzv
v
v
v
v
v
v
v
vpr2
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H
H
H
H
H
H
H
H
H
X
f$$
I
I
I
I
I
I
I
I
I
IY
g
zzu
u
u
u
u
u
u
u
u
u
Z
X, Y, X X ×ZY u v
ϕ:A→Γ(W, OW)ψ:B→Γ(W, OW)
u◦f]=v◦g]f]:C→A g]:C→B
t:A⊗CB→Γ(W, OW)
X×ZY= Spec(A⊗CB)t W →X×ZY
f: Spec(K)→XSpec(K)
{z}x∈XOX,x → OSpec(K),z =K
f]:OX→f∗OSpec(K)
mxK0
k(x)→K
x∈X i :k(x)→K f : Spec(K)→X
f(z) = x f]:OX→f∗OSpec(K)
OX(U)→ OSpec(K)(f−1(U)) U
OX(U)→ {0}x6∈ UOX(U)→K x ∈U x ∈UOX(U)→K
OX(U)germe −→ OX,x →k(x)i −→ K .
V(I)p V (p)
p⊃I q ⊃p q q ⊃I
f g r :B→Afs:Af→Bg
AfBg
r Bgr(g)
a/fn=r(g)r r0:Bg→Aaf s◦r:B→Bg
g/1 = sr(g) = s(a)/s(f)ns(af) = gs(f)n+1 Bgs

s0:Aaf →Bgr0Spec(Aaf ) = Spec(Bg)
Z X Z
U Z U Z ∩U
UF(X)X
Z∩U=\
F∈F (X)
F⊃Z
F∩U=\
F∈F (X)
F⊃Z
F∩U∩U=\
G∈F (X)
G∩U⊃Z∩U
G∩U=\
H∈F (U)
H⊃Z∩U
H=Z∩U U ,
G=F∩U H =G∩U{x}X
U x {x}U
x U ⇐⇒
U= Spec(A)V= Spec(B)x px⊂A qx⊂B
x AfBg
Spec(Af)⊂Spec(A)Af
A pxA Af'Bg
qxB⇐⇒
F=X= Spec(A){p}X
V(p)p A
A p A F =X
η1, η2F U1= Spec(A)
X η1η2F η1F
η1η2U1∩F
U1F1
U1∩F η1=η2
X= Spec(A)Y=V(I)Z=V(J)Y∩Z=V(I+J)W=Y×XZ
X
W= Spec(B)B
A/I A/J A
B=A/I ⊗AA/J 'A/(I+J)
X=A2
kY y =x2Z y = 0
x2= 0
X= Spec(A)Y= Spec(B)y q ⊂B k(y) = Bq/qBq
f−1(y) = Spec(A⊗Bk(y)) X={y=x2}Y={x= 0}
0∈Y f :X→Y f(x, y) = y
X= Spec(k[x, y]/(y−x2)) Y= Spec(k[y]) f
k[y]→k[x, y]/(y−x2)y7→ y0∈YSpec(k)→Y
k[y]→k y 7→ 0I= (y)A=k[x, y]/(y−x2)
A⊗k[y]k=A/yA =k[x]/x2f−1(0) = Spec(k[x]/x2)
F⊂Y f(F)Xr´ed →X
X Y F f(F)f(F)∩Ui=f(F∩Vi)
Uii X Y X = Spec(A)Y= Spec(B)
F= Spec(B/J)X f
f A →B
f(F) = {q∩A , q ⊃J}=V(J∩A)
f U =A1
k\ {0}

f|U:U→f(U)f p = (0,0) ∈X
{0} ∈ A1
k
A=k[x, y]/(y2−x3)f A →k[t]
x7→ t2y7→ t3p A/(x, y) = k
k[t]⊗Ak=k[t]/(x, y)k[t] = k[t]/(t2, t3)k[t] = k[t]/t2
f f
f f A →k[t]k[t2, t3]k[t]
A1t f
f
f
p= (0,0) Spec(k[t]/t2)A1
kX
p f
A→k[t]t
S⊂R
{q∈Spec(R), q ∩S=∅} → Spec(S−1R)q7→ S−1q q07→ q0∩R
S=R\p p q q ∩S=∅
p
p0Ap00
0
f∈p0Ap0
p0Ap0f
fa/s = 0 a∈A s 6∈ p0t6∈ p0tfa = 0 A ta = 0 f
a/s = 0 Ap0p0
p0p0(p0(p1(· · · (pmm+ 1 ≤dim(A)
rt −s2k[r, s, t, u]
A=k[r, s, t, u]/(rt −s2) dim(A) =
4−1=3 ru −st A 0
k[r, s, t, u]/(rt −s2, ru −st) 2
su −t20
k[r, s, t, u]/(rt −s2, ru −st)
dim(A) = dim(Ar´ed)
k[r, s, t, u]/(rt −s2, ru −st, su −t2)k[r, s, t, u]/(rt −s2, ru −st) 2