20

R. HAZRAT AND A. R. WADSWORTH

# Hence, for

= δ_{T }(a) ∈ Div(T ), using the identification of G with al(Z(Q_{i})/Z(C)_{0}) and the commutative

diagram (5.7), 0 = δ R ( N r d Q ( a ) ) = δ R N Z ( Q i ) / Z ( Q i )

## G (Nrd

### Q

_{i }(a))

_{m}

=

P

σ∈G

σ δ_{R}(Nrd_{Q }

i ( a ) m )

=

N G

δ_{R}(Nrd

Q

_{i }(a))

_{m}

= m N G ( N r d ( δ T ( a ) ) ) = m N G ( N r d ( ) )

.

as we saw above, S i n c e D i v ( R ) i s t o r s i o n - f r Nrd(δ_{T }(a^{0})) = Nrd(δ_{T }(a)) e e , w e h a v e N G ( N r d ( ) ) = 0 , i . e . , N r d ( ) ∈ k e r ( N G ) = t h e r e i s c ∈ [ Q ∗ i , Q ∗ ] ∩ Q ∗ i w i t Nrd(δ_{T }(c)) = h N r d ( ) = N r d ( δ T ( c ) ) . L e t a Nrd( ) Nrd( 0 =

0

= I

_{G}( ). Therefore,

a / c ∈ Q ) = 0. ∗ i

.

Then,

B e c a u s e N r d : D i v ( T ) → D i v ( R ) i s i n j e c t i v e ( s e e P r o p . 5 . 4 ) , i t f o l l o w s t h a t δ T ( a 0 ) = 0 i n D i v ( T ) . T h e r e - f o r e , a s T = Q i 1 [ x , ϕ i ] a n d q ( T ) = Q i , b y P r o p . 5 . 3 t h e r e i s a 0 0 ∈ Q i 1 w i t h a 0 0 ≡ a 0 ( m o d Q 0 i ) . S o , a 0 0 ≡ a

( m o d [ Q ∗ i , Q ∗ ] ) , a n d h e n c e N r d Q ( a 0 0 ) = N r d Q ( a ) = 1 , i . e . , a 0 0 ∈ Q proving (5.6). ∗ i 1

∩Q^{(1) }

. T h u s , a ∈ ( Q ∗ i 1

∩ Q^{(1) }

) [ Q ∗ i , Q ∗ ] ,

# The inclusion (5.6) shows that for any i, 1 ≤ i ≤ n and any a ∈ Q^{(1) }

with b ≡ a (mod Q^{0}).

Hence, by downward induction on i, for any q

∩ Q i t h e r e i s b ∈ Q ( 1 ) ∈ Q ( 1 ) = Q^{(1) }∩ Q_{n }there is ∩ Q_{i }_{1 }

d ∈ Q 0 ∩ Q ( 1 ) = E 0 ∩ Q ( 1 ) w i t h d ≡ q m o d Q 0 ) . S o , Q ( 1 ) completing the proof of Case I.

⊆ (Q^{(1) }

∩ E 0 ) Q 0 . T h e r e v e r s e i n c l u s i o n i s c l e a r ,

# Case

II.

# Suppose

### E

is

not

a

nitely

generated

abelian

group.

The basic point is that E is a direct limit of sub-graded division algebras with finitely generated grade group, so we can reduce to Case I. But we need to be careful about the choice of the sub-division algebras to assure that they have the same index as E, so that the reduced norms are compatible.

# Let F = Z(E). Since | _{E}/ _{F }| < ∞, there is a finite subset, say {γ_{1}, . . . , γ_{k}} of

### E

whose images in

_{E}/ generate this group. Let ∆_{0 }be any finitely generated subgroup of _{E}, and let ∆ be the subgroup F E generated by ∆_{0 }and γ_{1}, . . . , γ_{k}. Then, ∆ is also a finitely generated subgroup of of _{E}, but with the

added property that ∆ +

F

=

_{E}. Let

E_{∆ }

=

L E δ ,

δ∈∆

which is a graded sub-division ring of E, with E_{∆,0 }

= E 0 a n d

E

= ∆. Since ∆ +

### F

=

_{E}, we have

a E_{∆}F = E. (For, take n d c ∈ F η . T h e n , E that γ any γ ∈ = d c E 0

E and write γ = δ + η with δ ∈ ∆ and η ∈ ⊆ E ∆ F . ) B e c a u s e E ∆ F = E , w e h a v e Z ( E F ∆ , and any nonzero d ∈ E_{∆,δ }) = F ∩ E_{∆ }= F_{∆∩ F }. Note

[E_{∆ }: Z(E_{∆})] = [E_{∆,0 }: F_{∆∩ }= [ E 0 : F 0 ] | E

: _{F ,0}] | ∆ :( _{∆}∩ _{F }| = [E : F ].

F ) | = [ E 0 : F 0 ] | (

∆

+

_{F}) :

_{F}|

## The graded homomorphism E_{∆ }⊗_{Z(E ) }

# F → E is onto as E_{∆}F = E, and is then also injective by di-

m e n s i o n c o u n t ( o r b y t h e g r a d e d s i m p l i c i t y o

f E_{∆ }

⊗

### Z(E )

# F)

.

# Th

us,

E_{∆ }

⊗

### Z(E )

# F

∼

=

E . I t f o l l o w s t h a t

q(E_{∆}) ⊗_{q(Z(E }_{)) }

q

# (F )

∼ = q

( E ) . S

pec

i fi c a l l

y,

q(E_{∆}) ⊗_{q(Z(E }_{)) }

q

(F )

∼

=

( E ∆

⊗_{Z(E ) }

q(Z(E_{∆}))) ⊗_{q(Z(E }_{)) }

q

# (F )

∼

=

E ∆

⊗_{Z(E ) }

q(F )

Therefore, for any a ∈ q(E_{∆}), Nrd_{q(E ) }∼ = (E_{∆ }Z(E ) ⊗ (a) = Nrd_{q(E) }F) ⊗ F (a). q

(F )

∼ =

E

⊗ F q

(F )

∼ (E) =q .

# Now, if we take any a ∈ Q^{(1) }

where Q = q(E), there is a subgroup ∆ ⊆

E

with ∆ finitely generated

and ∆ +

F

a ∈ q(E_{∆})

(1)

(a) = Nrd_{Q}(a) = 1, we have, by Case I applied to E_{∆}, and a ∈ E_{∆}. Since Nrd_{q(E ) }= E ∩ E 0 q ( E ∆ ) 0 ⊆ ( Q ( 1 ) ∩ E 0 ) Q 0 , c o m p l e t i n g t h e p r o o f f o r C a s e I I .

# Remark. (i) Prop. 5.6 for those E with

E

∼

=

Z

was prove

di

n

[PY]

, a n d o u r p r o o f o f t h i s i s e s s e n t i a l l y

the same as theirs, expressed in a somewhat different language. Platonov and Yanchevski˘ı also in effect assert Prop. 5.6 for E with _{E }finitely generated, expressed as a result for iterated quotient division rings