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SK1

OF GRADED DIVISION ALGEBRAS

9

Corollary 3.6. Let E be a graded division ring with center T .

( i ) I f E i SK1(E) s u n r a m i e d , t h e n S K 1 ( E 0 ) n = SK1(E) . ( i i ) I f E i s t o t a l l y r a m i e d , t h e n = µ ( T 0 ) / µ e ( T 0 )

w h e r e n = i n d ( E )

a n d e i s t h e e x p o n e n t o f

E/

T.

( i i i ) I f E i s s e m i r a m i e d , t h e n f o r

G

=a

l ( E 0 / T 0 )

=

E/

T

t h e r e i s a n e x a c t s e q u e n c e

G G b H 1 ( G , E 0 ) S K 1 ( E ) 1 .

(3.8)

(iv) If E has maximal graded sub elds L and K which are respectively unrami ed and totally rami ed

over

T , t h

en

E i s s e m i r a m i e d a n

d SK1(E)

=

b H

1(

a

l ( E 0 / T 0 ) , E 0 )

.

Proof. See §2 for the definitions of unramified, totally ramified, and semiramified graded division algebras. ( i ) S i n c e E i s u n r a m i fi e d o v e r T , w e h a v e E 0 i s a c e n t r a l T 0 - d i v i s i o n a l g e b r a , i n d ( E 0 ) = i n d ( E ) , a n d E b = E 0 T . I t f o l l o w s t h a t G = a l ( Z ( E 0 ) / T 0 ) i s t r i v i a l , a n d t h u s H 1 ( G , N r d E 0 ( E 0 ) ) i s t r i v i a l ; a l s o ,

δ = 1 , a n d f r o m ( 3 . 3 ) , N r d E 0 ( a ) = N r d E ( a ) f o r a l l a E 0 . F u r t h e r m o r e , [ E 0 , E ] = [ E 0 , E 0 T as T is central. Plugging this information into the exact top row of diagram (3.5) and noting that the ] = [ E 0 , E 0 ] e x a c t s e q u e n c e e x t e n d s t o t h e l e f t b y 1 [ E 0 , E ] / [ E 0 , E 0 ] S K 1 ( E 0 ) , p a r t ( i ) f o l l o w s .

e ( i i ) W h e n E i s t o t a l l y r a m i fi e d , E 0 = T 0 , δ = n , N i s t h e i d e n t i t y m a p o n T 0 , a n d [ E , E 0 ] = [ E , T 0 ] = 1 .

B

P

y p l u g g i n g a l l t h i s i n t o t h e e x a c t c o l u m n o ( 3 . 5 ) , i t f o l l o w s t h a t E ( f di 1 ) = µ r o p . 2 . 1 , E 0 = µ e ( T 0 ) w h e r e e i s t h e agram e x p o n e n t o f t h e t o r s i o n a b e l i a n g r o u p E / n T .

( T 0 ) . A l

s o b y [ H w W 2 ]

P

art

(ii)

n o w f o l l o w s .

(iii) As recalled at the beginning of the proof of Th. 3.4, for any graded division algebra E with cen-

T , w e h a v e Z ( E 0 ) i s G a l o i s o v e r T 0 , a n d t h e r e i s a n e p i m o r p h i s m θ : E a l ( Z ( E 0 ) / T 0 E 0 E / a l ( Z ( E ). E Clearly, is semi- ter 0 ) / T 0 and T lie in ker(θ), so θ induces an epimorphism θ0 : ) . W h e T n r a m i fi e d , b y d e fi n i t i o n [ E 0 : T 0 ] = | T | = i n d ( E ) a n d E 0 i s a fi e l d . L e t G = a l ( E 0 / T 0 ) . B e c : E a u s e

| G | = [ E 0 : T 0 ] = | T | , t h e m a p θ 0 m u s t b e a n i s o m o r p h i s m . I n d i a g r a m ( 3 . 5 ) , b s i n c e S K 1 ( E 0 : E ) = 1 a n d c l e a r l y δ , t h e e x a c t t o p r o w a n d c o l u H 1 m n y i e l d E ( 1 ) [ E 0 , E ] 1 ( G , E 0 ) . T h e r e f o r e , t h e e x = = a c t

row

( 3 . 8 ) f o l l o w s f r o m t h e e x a c t s e c o n d r o w o

f di

agram

( 3 . 5 )

a n d t h e i s o m o r p h i s m

E/

T

=

G g i v e n b y θ 0

.

(iv) ind(E)

Since = [K

L and K : T] = |

K are :

maximal subfields

T|

|

E

:

T |.

of E, we It follows

have from

ind(E) = [L : T ] (2.2) that these

= [ L 0 : T 0 ] [ inequalities are E 0 : T 0 ] a n d equalities,

s o E 0 = L 0 a n d

E

=

K . Hence, E is semiramified, and (iii) applies. Take any η, ν

E/

T , and

a n y i n v e r s e i m a g e s a , b o f η , ν i n E . T h e l e f t m a p i n ( 3 . 8 ) s e n d s η ν t o a b a 1 b 1 m o d I G ( E 0 ) . S i n = K , these a and b can be chosen in K, so they commute. Thus, the left map of (3.8) is trivial here, yielding the isomorphism of (iv). c e E

Remark 3.7. In the setting of Cor. 3.6(iii), there is a further interesting and new formula for SK1(E) w h e n a l ( E 0 / T 0 ) i s b i c y c l i c , w h i c h w e d e s c r i b e h e r e w i t h o u t p r o o f . S u p p o s e G = a l ( E 0 / T 0 ) = h σ i h τ i . L e t M a n d P b e t h e fi x e d fi e l d s , M = E σ 0 a n d P = E τ 0 . S o , M a n d P a r e c y c l i c G a l o i s o v e r T 0 a n d

E 0

= M T 0

P . Then, there is an isomorphism

b H

1 ( G , E 0 )

=

B r ( E 0 / T 0 )



B r ( M / T 0 ) + B r ( P / T 0 ) ,

(3.9)

w h e r e B r ( T 0 ) i s t h e B r a u e r g r o u p o f T 0

a n d B r ( E 0 / T 0 ) = k e r

B r ( T 0 ) B r ( E 0 ) . A l l t h e e x p l i c i t c a l -

culations of SK1 of division algebras in [P1] and [P2] reduce to calculations of relative Brauer groups, using the formula for SK1 obtained by replacing the relative Brauer group term for the H 1 term in the valued version of Cor. 3.6(iv). Now, as our graded division algebra E is semiramified, it is known that E is graded Brauer equivalent to I T N, where I and N are graded division rings with center T , such that I i s i n e r t i a l , i . e . , I = I 0 T 0 T , a n d N i s n i c e l y s e m i r a m i fi e d , i . e . , i t h a s a m a x i m a l g r a d e d s u b fi e l which is unramified over T and another which is totally ramified over T . Furthermore I and N are split d

b y t h e u n r a m i fi e l d g r a d e d fi e l d e x t e n s i o n E 0 T o f T , s o I 0

i s s p l i t b y E 0 . O n e s h o w t h a t i n t h e s e t t i n g o f

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