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R. HAZRAT AND A. R. WADSWORTH

C o r . 3 . 6 ( i i i ) w i t h a l ( E 0 / T 0 ) b i c y c l i c ( a n d M , P , a n d I a s a b o v e ) ,

S K 1 ( E ) = B r ( E 0 / T 0 )



B r ( M / T 0 ) + B r ( P / T 0 ) + h I 0 i .

(3.10)

Details will appear in our paper [HW2]. There is an analogous formula in the Henselian valued setting of

Cor. 4.10(iii) with Gal(D/F ) bicyclic.

For a graded division algebra E with center T , define C K 1 ( E ) = E ( T E 0 ) .

(3.11)

This is the graded analogue to CK1(D) for a division algebra D, which is defined as CK1(D) = D(F D0), w h e r e F = Z ( D ) . T h a t i s , C K 1 ( D ) i s t h e c o k e r n e l o f t h e c a n o n i c a l m a p K 1 ( F ) K 1 ( D ) . S e e [ H 1 ] f background on CK1(D). Notably, it is known that CK1(D) is torsion of bounded exponent n = ind(D), and CK1 has functorial properties similar to SK1. The CK1 functor was used in [HW1] in showing that for “nearly all” division algebras D, the multiplicative group Dhas a maximal proper subgroup. It is conjectured (see [HW1] and its references) that if CK1(D) is trivial, then D is a quaternion division algebra (necessarily over a real Pythagorean field). o r

Now, for the graded division algebra E with center T , the degree map (2.1) induces a surjective map E E / T w h i c h h a s k e r n e l T E 0 . O n e c a n t h e n o b s e r v e t h a t t h e r e i s a n e x a c t s e q u e n c e EE 0 T CK1(E) 0 1 1. T E 0 s 0 u n r a m i fi e d , C K 1 ( E ) E 0 if E i / ( T 0 E 0 ) Th d E T E 0 . I t t h e n f o l l o w s t ng us h a t E 0 E 0 an , y i e l d i = = =

CK1(E)

=

C K 1 ( E 0 ) . A

t t h e o t h e r e x t r e m e , w h e n

Ei

s t o t a l l y r a m i fi e d t h e n

E 0

( T 0 E 0 )

=

1

, s o t h e e x a c t

s e q u e n c e a b o v e y i e l d s C K 1 ( E )

=

E/

T.

4. SK1 of a valued division algebra and its associated graded division algebra

The aim of this section is to study the relation between the reduced Whitehead group (and other related functors) of a valued division algebra with that of its corresponding graded division algebra. We will prove that SK1 of a tame valued division algebra over a henselian field coincides with SK1 of its associated graded division algebra. We start by recalling the concept of λ-polynomials introduced in [MW]. We keep the

notations introduced in §2. Let F be a field with valuation v, let gr(F ) be the associated graded field, and F alg

the algebraic closure

of F . For a F , let ea gr(F )v(a)

e be the image of a in gr(F ), let 0 = 0gr(F )

, and for f =

P

aixi F [x],

e let f = P eaixi gr(F )[x].

De nition 4.1. Take any λ in the divisible hull of

F

and let f = anxn +

. . . + aixi + . . . + a0 F [x] with

a

n a 0 6 =

0.

Take

any

extension

of

v

to

F alg

. We say that f is a λ-polynomial if it satisfies the following

equivalent conditions: (a) Every root of f in F alg

has value λ;

(b) v(ai) (n

i)λ + v(an) for all i and v(a0) = + v(an);

(c)

Take h ( 0 ) =6 any 0.

c F alg

with v(c) = λ and let h =

1 anc

n f(cx) F alg

[ x ] ; t h e n h i s m o n i c i n V F a l g [ x ] a n d

If f is a λ-polynomial, let

f (λ)

=

n

P

a 0 i x i g r ( F ) [ x ] ,

(4.1)

i=0

a 0 where i = 0

a if 0 i

v(ai) > (n is the image of ai in gr(F ) i)λ + v(an) ). Note that (n i)λ+v(an) f ( λ )

(so a

0 0

= e a 0 a n d a

is

a

homogenizable

0 n

= f a n , b u t f o r 1 i n

polynomial

in

gr(F )[x],

i.e.,

f (λ)

1, is

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