SK_{1 }

OF GRADED DIVISION ALGEBRAS

5

char(F ) ind(D). Note that strong tameness implies tameness. This is clear from the last characterization of tameness, or from (2.4) below. For a detailed study of the associated graded algebra of a valued division algebra refer to §4 in [HwW_{2}]. Recall also from [Mor], Th. 3, that for a valued division algebra D finite dimensional over its center F (here not necessarily henselian), we have the “Ostrowski theorem”

[D : F ] = q^{k }[D : F ] |

D

:

_{F}|

(2.4)

where q = char(D) and k ∈ Z with k ≥ 0 (and q^{k }= 1 if char(D) = 0). If q^{k }= 1 in equation (2.4), then

D is said to be defectless over F . Let E be a graded division algebra with, as we always assume,

## E

a torsion-free abelian group. After

fixing some total ordering on

_{E}, define a function

where

δ

is

minimal

among

the

γ

λ: ∈

# E \ {0} → E^{∗ }

## E

with c_{γ }=6

0.

by

λ(

P

c_{γ}) = c_{δ},

# Note that λ(a) = a for a ∈ E^{∗}, and

λ(ab) = λ(a)λ(b) for all a, b ∈ E \ {0}.

(2.5)

Let Q = q(E). We can extend λ to a map defined on all of Q^{∗ }as follows: for q ∈ Q^{∗}, write q = ac

1

with a ∈ E \ {0}, c ∈ Z(E) \ {0}, and set λ(q) = λ(a)λ(c)

^{1}. It follows from (2.5) that λ: Q_{∗ }→ E_{∗ }is

well-defined and is a group homomorphism. Since the composition E^{∗ }→ Q^{∗ }→ E^{∗ }is the identity, λ is a splitting map for the injection E^{∗ }→ Q^{∗}. (In Lemma 5.5 below, we will observe that this map induces a monomorphism from SK_{1}(E) to SK_{1}(Q).)

Now, by composing λ with the degree map of (2.1) we get a map v

Q ∗

λ

// v

E ∗

deg

(2.6)

!!

E

This v is in fact a valuation on Q: for a, b ∈ Q^{∗}, v(ab) = v(a) + v(b) as v is the composition of two group homomorphisms, and it is straightforward to check that v(a + b) ≥ min(v(a), v(b)) (check this first for a, b ∈ E \ {0}). It is easy to see that for the associated graded ring for this valuation on q(E), we have

g r ( q ( E ) )

∼

=

gr

E ; t h i s i s a s t r o n g i n d i c a t i o n o f t h e c l o s e c o n n e c t i o n b e t w e e n g r a d e d a n d v a l u e d s t r u c t u r e s .

# 3. Reduced norm and reduced Whitehead group of a graded division algebra

Let A be an Azumaya algebra of constant rank n^{2 }over a commutative ring R.

# Then there is a

c o m m u t a t i v e r i n g

S f a i t h f u l l y fl

at over

R

w

h i c h

s p l i t s

A , i . e . ,

A

⊗

R

S

∼

=

M n ( S ) . F o r a ∈

A

, cons

i d e r i n g

a ⊗ 1 a s a n e l e m e n t o f M n ( S ) , o n e t h e n d e fi n e s t h e r e d u c e d c h a r a c t e r i s t i c p o l y n o m i a l , t h e r e d u c e d t r a c e ,

and the reduced norm of a by char_{A}(x, a) = det(x

(a ⊗ 1)) = x^{n }

Trd_{A}(a)x^{n 1 }

+

. . . + ( 1)

^{n}Nrd_{A}(a).

# Using descent theory, one shows that char_{A}(x, a) is independent of S and of the choice of isomorphism

# A

⊗

## R

# S

∼

=

M n ( S )

, a n d t h a t c h a r

_{A}(

x, a

) li

e s i n

R [ x ] ; f u r t h e r m o r e , t h e e l e m e n t a i s i n v e r t

i b l e i n A i f

an

d

only if Nrd_{A}(a) is invertible in R (see Knus [K], III.1.2, and Saltman [S_{2}], Th. 4.3). Let A^{(1) }

denote the

set of elements of A with the reduced norm 1. One then defines the reduced Whitehead group of A to be S K 1 ( A ) = A ( 1 ) / A 0 , w h e r e A 0 d e n o t e s t h e c o m m u t a t o r s u b g r o u p o f t h e g r o u p A ∗ o f i n v e r t i b l e e l e m e n t s o f A . T h e r e d u c e d n o r m r e s i d u e g r o u p o f A i s d e fi n e d t o b e S H 0 ( A ) = R ∗ / N r d A ( A ∗ ) . T h e s e g r o u p s a r e r e l a t e d

by the exact sequence:

1

→ SK_{1}(A)

→ A^{∗}/A

0

N r d → R ∗

→ SH^{0}(A)

→1

(3.1)

Now let E be a graded division algebra with center T . Since E is an Azumaya algebra over T ([B], Prop. 5.1 or[HwW_{2}], Cor. 1.2), its reduced Whitehead group SK_{1}(E) is defined.