SK_{1 }

OF GRADED DIVISION ALGEBRAS

15

We conclude this section by establishing a similar result for the CK_{1 }functor of (3.11) above. Note that here, unlike the situation with SK_{1 }(Th. 4.8) or with SH^{0 }(Th. 4.12), we need to assume strong tameness here.

# Theorem 4.13. Let F be a eld with henselian valuation v and let D be a strongly tame F -central division

a l g e b r a .

# Th

en

CK_{1}(D)

∼

=

CK_{1}(

gr

(D))

.

→ gr(D)^{∗ }g i v e n b y a 7 → e a , w i t h k e r n e l 1 + M D Proof. Consider the canonical epimorphism ρ: D ∗ . S _{∗} i n c e ρ m a p s D 0 o n t o g r ( D ) 0 a n d F ∗ o n t o g r ( F ) ∗ , i t i n d u c e s a n i s o m o r p h i s m D F ∗ D 0 ( 1 + M D ∼ = )

gr(D)^{∗ }

g r ( F ) ∗ g r ( D ) 0 . W e h a v e g r ( F ) = Z ( g r ( D ) ) a n d b y L e m m a 2 . 1 i n [ H 3 ] , a s D i s s t r o n g l y t a m e ,

1 + M D

=

(1

+

M F ) [ D ∗ , 1 +

M D ]

⊆

F ∗ D 0

.

Th

us,

CK_{1}(D)

∼

=

CK_{1}(

gr

(D))

.

# 5. Stability of the reduced Whitehead group

# The goal of this section is to prove that if E is a graded division ring (with

E

a torsion-free abelian

g r o u p ) , t h e n S K 1 ( E ) ∼ = S K 1 ( q ( E ) ) , w h e r e q ( E ) i s t h e q u o t i e n t d i v i s i o n r i n g o f E . W h e n

E

∼ = Z , t h i s w a s

essentially proved by Platonov and Yanchevski˘ı in [PY], Th. 1 (see the Introduction). Their argument was based on properties of twisted polynomial rings, and our argument is based on their approach. So, we will first look at twisted polynomial rings. For these, an excellent reference is Ch. 1 in [J].

Let D be a division ring finite dimensional over its center Z(D). Let σ be an automorphism of D whose restriction to Z(D) has finite order, say `. Let T = D[x, σ] be the twisted polynomial ring, with multiplication given by xd = σ(d)x, for all d ∈ D. By Skolem-Noether, there is w ∈ D^{∗ }with σ^{` }= int(w ^{1}) (= conjugation by w ^{1}); moreover, w can be chosen so that σ(w) = w (by a Hilbert 90 argument, see [J], Th. 1.1.22(iii) or [PY], Lemma 1). Then Z(T ) = K[y] (a commutative polynomial ring), where K = Z(D)^{σ}, the fixed field of Z(D) under the action of σ, and y = wx^{`}. Let Q = q(T ) = D(x, σ), the division ring of

quotients of T . Since T is a finitely-generated Z(T )-module, Q is the central localization T ⊗_{Z(T }_{) }

q(Z (T ))

o f T . N o t e t h a t Z ( Q ) = q ( Z ( T ) ) = K ( y ) , a n d i n d ( Q ) = ` i n d ( D ) . O b s e r v e t h a t w i t h i n Q w e h a v e t h e

twisted Laurent polynomial ring T [x

^{1}] = D[x, x

^{1}, σ] which is a graded division ring, graded by degree

in x, and T ⊆ T [x ^{1}] ⊆ q(T ), so that q(T [x ^{1}]) = Q. Recall that, since we have left and right division algorithms for T , T is a principal left (and right) ideal domain.

Let S denote the set of isomorphism classes [S] of simple left T -modules S, and set

Div(T )

=

# L

Z[S],

[S]∈S

the free abelian group with base S. For any T -module M satisfying both ACC and DCC, the Jordan-H¨older Theorem yields a well-defined element jh(M) ∈ Div(T ), given by

jh(M)

=

# P

n_{[S] }

(M)[S],

[S]∈S

where n_{[S]}(M) is the number of appearances of simple factor modules isomorphic to S in any composition series of M. Note that for any f ∈ T \ {0}, the division algorithm shows that dim_{D}(T/T f) = deg(f) < ∞. Hence, T/T f has ACC and DCC as a T -module. Therefore, we can define a divisor function

δ : T \ {0} → Div(T ), Remark 5.1. Note the following properties of δ:

given by

δ(f) = jh(T/T f).

( i ) F o r a n y

f , g ∈

T \ { 0 } , δ ( f g ) =

δ(f)

+

δ ( g ) . T h i s f o l l o w s f r o m t h e i s o m o r p h i s m

T g / T f g ∼ =

T /T f

(as T has no zero divisors). (ii) We can extend δ to a map δ : Q^{∗ }→ Div(T ), where Q = q(T ), by δ(fh

^{1}) = δ(f)

δ(h) for any

f ∈ T \ {0}, h ∈ Z(T ) \ {0}. It follows from (i) that δ is well-defined and is a group homomorphism on Q^{∗}. Clearly, δ is surjective, as every simple T -module is cyclic.