TD14 : Lissité, anneaux de valuation discrète, anneaux de Dedekind

k
V:x2−xz −y2−1 = yz −x2−z2= 0
A3
k
K c ∈K×C y2=x3−c
Cdim C= 1
C K
K
C
k C
x2+y2=z2A3
k(0,0,0)
O(C)
k A k l
k A ⊗kl
l/k A ⊗kl
A B A M B

A d :B→M A B M
d(b1b2) = b1db2+b2db1b1, b2∈B da = 0 a∈A
A(B, M)A B M
BΩ1
B/A A d :B→
Ω1
B/A B M B(Ω1 B/A, M)→
A(B, M), φ 7→ φ◦d(Ω1
B/A, d)
B A
(Ω1
B/A, d) (I/I2, δ)I
B⊗AB→B, b ⊗b07→ bb0δ:B→I/I2, b 7→ b⊗1−1⊗b
(Ω1
B/A, d)
B=A[T1, ..., Tn]n≥0
B A
B A
A A0B0=B⊗AA0
B0Ω1
B0/A0∼
=Ω1
B/A ⊗BB0
B→C A
Ω1
B/A ⊗BC→Ω1 C/A →Ω1 C/B →0.
S B S−1Ω1
B/A ∼
=Ω1
S−1B/A
C=B/J J
I/I2→Ω1
B/A ⊗BC→Ω1 C/A →0.
B A
Ω1
B/A B(Ω1 B/A, d)
B=A[X1, ..., Xn]/(f)f∈A[X1, ..., Xn]
A=k B k
m∈B B/m∼
=km/m2
Ω1
B/k ⊗BB/m
A=k B
k X =B
XΩ1
B/k B
dim B B M
npB f ∈B\p
M[f−1]B[f−1]n

k A k
X=A X
Q
Fq(t)
K v1, ..., vn
K x1, ..., xnK m ∈N
x∈K vi(x−xi)> m i
L/K v K
L v
C
C
K v
ApA
f, g, h ∈A[X]f f =gh ∈A/p[X]g
h G, H ∈A[X]
f=GH deg G= deg g G =g H =h
f=a0+a1X+... +anXn∈K[X]
a0an6= 0 min{v(ai)|0≤i≤n}= min{v(a0), v(an)}
an= 1 a0∈A f ∈A[X]
Z[√5],Z[1 + √5
2],Z[ζ81],R[X, Y ]/(X2+Y2−1),
R[X, Y ]/(Y2−X3−X2),R[X, Y, Z]/(X2+Y2+Z).

I A
K a ∈K×aI A I
A
A
A I J A
A f :I⊗AJ→IJ, i ⊗j7→ ij
A