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SK1

OF GRADED DIVISION ALGEBRAS

7

Hence,

NrdE(a) = ( 1) = N Z ( E 0 ) / T m t0 0 N n/m L / T = 0 ( a )

δ ( 1)mt0 δ [ L : T 0 ] / m = N Z ( E 0 ) / T 0 N r d

= N L / T 0 ( a ) E 0 ( a ) δ .

δ

The formula for TrdE(a) is proved analogously.

In the rest of this section we study the reduced Whitehead group SK1 of a graded division algebra. As we mentioned in the introduction, the motif is to show that working in the graded setting is much easier than in the non-graded setting.

The most successful approach to computing SK1 for division algebras over henselian fields is due to Ershov in [E], where three linked exact sequences were constructed involving a division algebra D, its residue d i v i s i o n a l g e b r a D , a n d i t s g r o u p o f u n i t s U D ( s e e a l s o [ W 2 ] , p . 4 2 5 ) . F r o m t h e s e e x a c t s e q u e n c e s , E r s h o recovered Platonov’s examples [P1] of division algebras with nontrivial SK1 and many more examples as well. In this section we will easily prove the graded version of Ershov’s exact sequences (see diagram (3.5)), yielding formulas for SK1 of unramified, semiramified, and totally ramified graded division algebras. This will be applied in §4, where it will be shown that SK1 of a tame division algebra over a henselian field coincides with SK1 of its associated graded division algebra. We can then readily deduce from the graded results many established formulas in the literature for the reduced Whitehead groups of valued division algebras (see Cor. 4.10). This demonstrates the merit of the graded approach. v

If N is a group, we denote by Nn the subgroup of N generated by all n-th powers of elements of N. A homogeneous multiplicative commutator of E, where E is a graded division ring, has the form aba 1b where a, b E= Eh \ {0}. We will use the notation [a, b] = aba 1b 1 for a, b Eh. Since a and b are h o m o g e n e o u s , n o t e t h a t [ a , b ] E 0 . I f H a n d K a r e s u b s e t s o f E , t h e n [ H , K ] d e n o t e s t h e s u b g r o u p o f E g e n e r a t e d b y { [ h , k ] : h H , k K } . T h e g r o u p [ E , E ] w i l l b e d e n o t e d b y E 1 0 .

Proposition 3.3. Let E = Then,

L

E be a graded division algebra with graded center T , with ind(E) = n.

  • (i)

    If N is a normal subgroup of E, then Nn NrdE(N)[E, N].

  • (ii)

    SK1(E) is n-torsion.

Proof. Let a N and let ha q(T )[x] be the minimal polynomial of a over q(T ), and let m = deg(ha). As

n o t e d i n t h e p r o o f o f P r o p . 3 . 2 , h a T [ x ] a n d N r d E ( a ) = [ ( 1 ) m h a ( 0 ) ] n / . By the graded Wedderburn m d 1 a d 1 1 ) . . . is normal in Eh. 1 ( = (x , since ) Factorization Theorem A.1, we have h x d m a d m d i t h a t [ E , N ] i s a n o r m a l s u b Eg N r o u p where each o Note f E a It follows that E.

NrdE(a)

=

d 1 a d 1

1

. . . d m a d 1 m

  • n/m

=

[ d 1 , a ] a [ d 2 , a ] a . . . a [ d m , a ] a

  • n/m

= a n d a

where

d a

[E, N].

T h e r e f o r e , a n = N r d E ( a ) d a

1

NrdE(N)[E, N], yielding (i). (ii) is immediate from (i) by taking N = E(1).

The fact that SK1(E) is n-torsion is also deducible from the injectivity of the map SK1(E) SK1(q(E))

shown in Lemma 5.5 below.

b We recall the definition of the group H

1(G, A), which will appear in our description of SK1(E). For

a n y fi n i t e g r o u p G a n d a n y G - m o d u l e A , d e fi n e t h e n o r m m a p N G : A A a s f o l l o w s : f o r

N G ( a ) = P g G g a . C o n s i d e r t h e G - m o d u l e I G ( A ) g e n e r a t e d a s a n a b e l i a n g r o u p b y { a C l e a r l y , I G ( A ) k e r ( N G ) . T h e n ,

ga

:

any a A, let a A and g

b H

1 ( G , A ) = k e r ( N G ) I G ( A ) .

(3.4)

G}.

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