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SK1

OF GRADED DIVISION ALGEBRAS

19

  • -

    module.

also acts on R = Z(T ) by ring automorphisms, and on Div(R), and all the maps in the

commutative diagram below (see Prop. 5.4) are

  • -

    module homomorphisms.

Q i

δT

//

Div(T )

NrdQ

i

Nrd

(5.7)

Z(Qi)

δ R

//

Div(R)

Since inner automorphisms of Qi act trivially on Div(T ) (see Remark 5.1(iii)), and on Z(Qi) and Div(R), these -modules are actually G-modules. Let

= Nrd(Div(T )) Div(R).

Because Nrd : Div(T ) Div(R) is injective (see Prop. 5.4),

is a G-module isomorphic to Div(T ), so

is a permutation G-module.

Within

we have two distinguished

  • -

    submodules,

IG(

0

=

)

=

k e r ( N G

) , w h e r e N G

:

{β

σ(β) | β

, σ G}

i s t h e n o r m , g i v e n b y N G

0.

(b) =

PσG

σ(b); and

By

definition,

b H

1(G,

)

=

0 / I G (

).

But,

because

b is a permutation G-module, H

1(G,

) = 0.

(This is well known, and is an easy calculation, as

is a direct sum of G-modules of the form Z[G/H] for

subgroups H of G.) That is,

  • 0

    = IG(

Take any generator β

σ(β) of IG(

). ),

where

σ

G

and

β

, say β = Nrd(η), where η Div(T ).

T a k e a n y b Q i w i t h δ T ( b ) = η , a n d c h o o s e u E w h i c h i s s o m e p r o d u c t o f t h e ϕ j ( i + 1 j n ) , s u c h Z(Qi) t h a t i n t ( u ) | Z ( Q i ) = σ| . T h e n , δ R ( N r d i ( b ) ) = N r d ( δ T ( b ) ) = β ( s e e ( 5 . 7 ) ) . A l s o , b e c a u s e i n t ( u is Qi ) | Q a Q Q 1 n 1) = u Nrd 1)u 1 1u 1 i (ub i (b a u t o m o r p h i s m o f Q i , w e h a v e N r d [ Q i , Q ] Q i a n d Thus, bub u .

NrdQ

i (bub

1u

1) = NrdQ = NrdQ

i (b) NrdQi (ub i(b)σ(Nrd Q u i (b)). 1

1) = NrdQi (b) u Nrd

Q

i (b

1)u

1

Hence, in Div(R),

Since such β

δR NrdQi(bub σ(β) generate IG(

1u ),

1

it

)

  • = δR NrdQi (b)/σNrdQi (b)

follows that for any γ IG(

=β ), there

σ(β). is c

[ Q i

, Q] Qi,

with

γ = δR(NrdQi (c)) = Nrd(δT (c)) (see(5.7)). To prove (5.6), we need a formula for NrdQ

for an element of Qi.

For this, note that

1 i+1 1] which E = S i [ Qi[xi+1, x x Now, let C = = Zεi+1 . . . Zεn. can be considered a graded S i i s a g r a d e d d i v i s i o n r i n g w i t ring over = Qi and ,x , . . . , xn, xn i+1 1 i+1 h C . 1] Q. This C , . . . , xn, xn C 0 S i n c e E C Q = q ( E ) , w e h a v e q ( C ) = Q . F o r t h e g r a d e d fi e l d Z ( C ) w e h a v e Z ( C ) 0 c o n s i s t s o f t h o s e e l - e m e n t s o f Z ( C 0 ) = Z ( Q i ) c e n t r a l i z e d b y x i + 1 , . . . , x n , i . e . , Z ( C ) 0 i s t h e fi x e d fi e l d Z ( Q i ) = Z ( Q i ) G . S i n c e ,

a s n o t e d e a r l i e r ,

G i n j e c t s i n t o

A

ut

(Z(Qi)

, w e h a v e

G

=

a

l ( Z ( Q i ) / Z ( C ) 0 ) . T h

u s , f o r a n y q

Qi

=

C 0 ,

by Prop. 3.2(i) and (iv),

NrdQ(q) = Nrdq(C) ( q ) = N r d C ( q ) = N Z ( C 0 ) / Z ( C 0 ) = N Z ( Q i ) / Z ( Q i ) G ( N r d Q i ( q ) ) m ,

G (Nrd

C 0 ( q ) )

m

w h e r e m = i n d ( Q ) / i n d ( Q i ) [ Z ( Q i ) : Z ( Q i ) G ] . T o v e r i f y ( 5 . 6 ) , t a k e a n y a Q i Q ( 1 ) . T h u s ,

1 = N r d Q ( a ) = N Z ( Q i ) / Z ( Q i )

G (Nrd

Qi

(a))m.

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