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2

R. HAZRAT AND A. R. WADSWORTH

group of certain valued division algebras, by passing to the graded setting; this shows the utility of the

graded approach (see Cor. 4.10). LγIn the other direction, if E =

E

E γ i s a g r a d e d d i v i s i o n a l g e b r a w h o s e g r a d e g r o u p

E is torsion-free

abelian, then E has a quotient division algebra q(E) which has the same index as E. The same question on comparing the reduced Whitehead groups of these objects can also be raised here. It is known that when the grade group is Z, then E has the simple form of a skew Laurent polynomial ring D[x, x 1, ϕ], where D is a

division algebra and ϕ is an automorphism of D. In this setting the quotient division algebra of D[x, x

1, ϕ]

is D(x, ϕ). In [PY], Platonov and Yanchevski˘ı compared SK1(D(x, ϕ)) with SK1(D). In particular, they

s h o w e D[ x, x 1 d t h a t n n e r if ϕ i a u t o m o r p h s an i i s m t h e n , ϕ ] i s a n u n r a m i fi e d g r a d e d d i v i s i o n a l g e b r a a n SK1(D( x, ϕ d )) = w e p r o v e t h a SK1(D) t

. I n f a c t SK1(D[ , x, x if

1

ϕ i s i n n e r , t h e n , ϕ ] ) = S K 1 ( D )

(Cor. 3.6(i)). By combining these, one concludes that the reduced Whitehead group of the graded division algebra D[x, x 1, ϕ], where ϕ is inner, coincides with SK1 of its quotient division algebra. In Section 5, we show that this is a very special case of stability of SK1 for graded division algebras; namely, for any graded division algebra with torsion-free grade group, the reduced Whitehead group coincides with the reduced Whitehead group of its quotient division algebra. This allows us to give a formula for SK1 for

generic abelian crossed product algebras.

The paper is organized as follows: In Section 2, we gather relevant background on the theory of graded division algebras indexed by a totally ordered abelian group and establish several homomorphisms needed in the paper. Section 3 studies the reduced Whitehead group SK1 of a graded division algebra. We establish a n a l o g u e s t o E r s h o v s l i n k e d e x a c t s e q u e n c e s [ E ] i n t h e g r a d e d s e t t i n g , e a s i l y d e d u c i n g f o r m u l a s f o r S K 1 o f u n r a m i fi e d , t o t a l l y r a m i fi e d , a n d s e m i r a m i fi e d g r a d e d d i v i s i o n a l g e b r a s . I n S e c t i o n 4 , w e p r o v e t h a t S K 1 a tame division algebra over a henselian field coincides with SK1 of its associated graded division algebra. Section 5 is devoted to proving that SK1 of a graded division algebra is isomorphic to SK1 of its quotient division algebra. We conclude the paper with two appendices. Appendix A establishes the Wedderburn factorization theorem in the setting of graded division rings, namely that the minimal polynomial of a homogenous element of a graded division ring E splits completely over E (Th. A.1). Appendix B provides a complete proof of the Congruence Theorem for all tame division algebras over henselian valued fields. This theorem was originally proved by Platonov for the case of complete discrete valuations of rank 1, and it was a key tool in his calculations of SK1 for certain valued division algebras. o f

2. Graded division algebras

In this section we establish notation and recall some fundamental facts about graded division algebras indexed by a totally ordered abelian group, and about their connections with valued division algebras. In addition, we establish some important homomorphisms relating the group structure of a valued division algebra to the group structure of its associated graded division algebra.

Let R = LγRγ be a graded ring, i.e., Rγ is a subgroup of (R, +) and Rγ · Rδ Rγ+δ

is an abelian group, and R is a unital ring such that each for all γ, δ . Set

R

=

h R

=

{γ Sγ

R

| R Rγ , γ

=6 0},

the

set

the

of

grade

set

of

homogeneous

R;

elements

of

R.

For a homogeneous element of R of degree γ, i.e., an r Rγ \ 0, we write deg(r) = γ. Recall that R0 is a

, the group Rγ is a left and right R0-module. A subring S of R is subring of R and that for each γ R R Lγ( S R γ ) . F o r e x a m p l e , t h e c e n t e r o f R , d e n o t e d Z ( R ) , i s a g r a d e d s u b r i n g a graded subring if S =

of R. If T =

Lγ

T γ i s a n o t h e r g r a d e d r i n g , a g r a d e d r i n g h o m o m o r p h i s m i s a r i n g h o m o m o r p h i s m

f : R T w i t h f ( R γ ) T γ

for all γ

. If f is also bijective, it is called a graded ring isomorphism; we

t h e n w r i t e

R

=gr

T.

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