gl—sses ré—lis—˜les d9extensions non —˜éliennes

k OkΓ
N/k
ΓONNM
Okk[Γ] Ok[Γ] Cl(M)
M
N/k
ONM ⊗Ok[Γ] ON
[M ⊗Ok[Γ] ON]Cl(M)R(M)
c∈Cl(M)
N/k
Γ [M ⊗Ok[Γ] ON] = c
R(M)
Cl(M) Γ = VoρC V F2
r≥2C2r−1
ρ C V
A4
k
k

k OkΓ
N/k
ΓONNMOk
k[Γ] Ok[Γ] Cl(M)
N/k
ON
M ⊗Ok[Γ] ON[M ⊗Ok[Γ] ON]Cl(M)
R(M)c∈Cl(M)
N/k
Γ [M ⊗Ok[Γ] ON] = c
R(M)Cl(M) Γ = VoρC V F2
r≥2C2r−1
ρ C V
A4
k
k
K OK
Cl(K)
k k k Gal(k/k)
Γ
π Gal(k/k) Γ N k
Ker(π)N/k
Gal(N/k)Gal(k/k)/Ker(π)
π Gal(N/k) Γ π
ONOk[Γ]
x∈ONγ∈Γγx =π−1(γ)(x)M
Okk[Γ] Ok[Γ]
N/k ON[ON]
Cl(Ok[Γ]) Ok[Γ]
Ok[Γ]
M ⊗Ok[Γ] ON[M ⊗Ok[Γ] ON]
Cl(M)M M
R(Ok[Γ]) R(M)c
Cl(Ok[Γ]) Cl(M)N/k
Γ [ON] = c[M ⊗Ok[Γ]
ON] = c c N/k R(Ok[Γ])
R(M)

R(Ok[Γ]) ⊂Cl◦(Ok[Γ]) R(M)⊂Cl◦(M)
Cl◦(Ok[Γ]) Cl◦(M)
Cl(Ok[Γ]) →Cl(k)Cl(M)→Cl(k)
Ok[Γ] →OkM → Ok
R(Ok[Γ]) Cl◦(Ok[Γ])
R(M)Cl◦(M)
1 2
Ok[Γ] MEx :Cl(Ok[Γ]) →Cl(M)
Ex(R(Ok[Γ])) = R(M)
k=QΓ
Q
Γ
ΓR(Z[Γ])
Cl(Z[Γ])
R(M)
kΓ
Γ
lq l q k/Q
lq Q
R(M)
ΓD4k
Q(i)Qi2=−1
kR(M) = Cl◦(M)
Γ
4l l

R(M)Cl◦(M) 2
l k
A4
k k/Q
3Q
R(Ok[A4]) = Cl◦(Ok[A4])
kR(Ok[D4]) = Cl◦(Ok[D4])
k4Ok
Γ
F2Z/2Z
VF2r≥2C
2r−1
ρ:C→AutF2(V)
C V Γ = VoρC
ρ
k[C]
k[C]'
n
Y
i=0
k(χi),
n+ 1 k
C
i∈ {0,1, ..., n}χi
χ0k(χi)k
k χi
M(C)Okk[C]C
Cl(M(C)) '
n
Y
i=0
Cl(k(χi)),
Cl◦(M(C)) '
n
Y
i=1
Cl(k(χi)).
R(M(C))
k
CR(M(C)) Cl◦(M(C))
Qn
i=1 Cl(k(χi)).
ρ
k

Γk[Γ]
k[Γ] '
n
Y
i=0
k(χi)×M2r−1(k),
M2r−1(k) 2r−1
k
MOkk[Γ] Ok[Γ]
Cl(M)'
n
Y
i=0
Cl(k(χi)) ×Cl(k),
Cl◦(M)'
n
Y
i=1
Cl(k(χi)) ×Cl(k).
Cl◦(M)Qn
i=1 Cl(k(χi)) ×Cl(k)
(1.1)
K/k
NK/k K/k G
m Gm
m G
kΓ = VoρC
ρ Cl◦(M)Qn
i=1 Cl(k(χi))×
Cl(k)R(M)Cl◦(M)A
A={(c1, c2, ..., cn, x
n
Y
i=1
Nk(χi)/k(ci))|
(c1, c2, ..., cn)∈ R(M(C)), x ∈Cl(k)2r−2}.
R(M)
R(M(C)) ×Cl(k)2r−2.
A4
Γ
r= 2
k
Γ = A4R(M)Cl◦(M)
j3R(M)'
Cl(k)×Cl(k)×Cl(k)j∈kR(M)'Cl(k(j)) ×Cl(k)