SK_{1 }

OF GRADED DIVISION ALGEBRAS

7

# Hence,

Nrd_{E}(a) = ^{}( 1) = N Z ( E 0 ) / T m t_{0 } 0 N n/m L / T = 0 ( a )

δ ( 1)^{m}t_{0 } 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 Trd_{E}(a) is proved analogously.

In the rest of this section we study the reduced Whitehead group SK_{1 }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 SK_{1 }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 [P_{1}] of division algebras with nontrivial SK_{1 }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 SK_{1 }of unramified, semiramified, and totally ramified graded division algebras. This will be applied in §4, where it will be shown that SK_{1 }of a tame division algebra over a henselian field coincides with SK_{1 }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 N^{n }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 ^{1}b where a, b ∈ E^{∗ }= E^{h }\ {0}. We will use the notation [a, b] = aba ^{1}b ^{1 }for a, b ∈ E^{h}. 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 N^{n }⊆ Nrd_{E}(N)[E^{∗}, N].(ii)

SK

_{1}(E) is n-torsion.

# Proof. Let a ∈ N and let h_{a }∈ q(T )[x] be the minimal polynomial of a over q(T ), and let m = deg(h_{a}). 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 ⊆ E^{h}. 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 ∈ E^{∗ }g N r o u p where each o Note f E a ∗ It follows that E^{∗}.

Nrd_{E}(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

∈ Nrd_{E}(N)[E^{∗}, N], yielding (i). (ii) is immediate from (i) by taking N = E^{(1)}.

# The fact that SK_{1}(E) is n-torsion is also deducible from the injectivity of the map SK_{1}(E) → SK_{1}(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 SK_{1}(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}.