10

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 [HW_{2}]. 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 CK_{1}(D) for a division algebra D, which is defined as CK_{1}(D) = D^{∗}(F ^{∗}D^{0}), 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 CK_{1}(D). Notably, it is known that CK_{1}(D) is torsion of bounded exponent n = ind(D), and CK_{1 }has functorial properties similar to SK_{1}. The CK_{1 }functor was used in [HW_{1}] in showing that for “nearly all” division algebras D, the multiplicative group D^{∗ }has a maximal proper subgroup. It is conjectured (see [HW_{1}] and its references) that if CK_{1}(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 ∗ _{E} → E 0 T → CK_{1}(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 = = =

CK_{1}(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. SK_{1 }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 SK_{1 }of a tame valued division algebra over a henselian field coincides with SK_{1 }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 = 0_{gr(F }_{) }

, and for f =

P

a_{i}x^{i }∈ F [x],

e let f = ^{P }ea_{i}x^{i }∈ gr(F )[x].

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

### F

and let f = a_{n}x^{n }+

. . . + a_{i}x^{i }+ . . . + a_{0 }∈ 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(a_{i}) ≥ (n

i)λ + v(a_{n}) for all i and v(a_{0}) = nλ + v(a_{n});

(c)

Take h ( 0 ) =6 any 0.

c ∈ F ^{alg }

with v(c) = λ and let h =

1 a_{n}c

_{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(a_{i}) > (n is the image of a_{i }in gr(F ) i)λ + v(a_{n}) ). Note that (n i)λ+v(a_{n}) 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