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SK1

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 [HwW2]. 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 ] = qk [D : F ] |

D

:

F|

(2.4)

where q = char(D) and k Z with k 0 (and qk = 1 if char(D) = 0). If qk = 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 cγ =6

0.

by

λ(

P

cγ) = cδ,

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 Qas 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: QEis

well-defined and is a group homomorphism. Since the composition EQEis the identity, λ is a splitting map for the injection EQ. (In Lemma 5.5 below, we will observe that this map induces a monomorphism from SK1(E) to SK1(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 n2 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 charA(x, a) = det(x

(a 1)) = xn

TrdA(a)xn 1

  • +

    . . . + ( 1)nNrdA(a).

Using descent theory, one shows that charA(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 NrdA(a) is invertible in R (see Knus [K], III.1.2, and Saltman [S2], 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

SK1(A)

A/A

0

N r d R

SH0(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[HwW2], Cor. 1.2), its reduced Whitehead group SK1(E) is defined.

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