TD - Feuille 2

X A
X→Spec(A)A→Γ(X, OX)
X X
f:Z→X
Y⊂X f Z →Y→X
Y0
//X
Z
OO
f
==
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Y0X Y →Y0X
Y f
Z Y f
n A n ≥1
µn,A n
A S n Γ(S, OS)
A S →µn,A
n A
m:µn,A ×Spec(A)µn,A →µn,A
µp,Qµp,Fp0p06=p µp,Fp
Y ν3:P1
k→P3 k
ν3(x:y)=(x3:x2y:xy2:y3)ν3P1
kY
Y
Y⊂Pn
kr Y
I(Y)n−r Y
Y n −r
Y
Y k[x, y, z]
Y
Y22Y33Y=Y2∩Y3
Y2∩Y3Y
P3
kX Y ⊂X
IY X X
In+1
PnPn
k

M A
M{p∈Spec(A), Mp6= 0}I={a∈A, aM = 0}M
M V (I) = {p∈Spec A, I ⊂p}
A=k[t]M A M
k M t ∈End(M)M
P P t
A K
A6=K A v :K→Z∪ {∞}
{0,∞}
v−1(∞) = {0}
x, y K v(xy) = v(x) + v(y)
x, y K v(x+y)≥min(v(x), v(y))
A={x∈K, v(x)≥0}
A
A
A
π∈A x ∈A x =uπn
u∈A×n∈N
⇒ ⇒ ⇒ ⇒ ⇒ I=∩n≥1(πn)
A I ⊂A M A
M=IM a ∈A a ≡1 mod I aM = 0
A
A m M =mM
M= 0
A1k[X1, . . . , Xn]
f∈k[X1, . . . , Xn]
An
kA2=k[[x]]
A3Zp
A1A2A3