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SK1

OF GRADED DIVISION ALGEBRAS

13

We can now prove the main result of this section: Theorem 4.8. Let F be a eld with henselian valuation v and let D be a tame F -central division algebra.

Th

en

SK1(D)

=

SK1(

gr

(D))

.

Proof. ker(ρ)

Consider = 1 + M D the . If

canonical surjective

a D(1)

V D

then

group homomorphism ρ: Dea gr(D)0 and by Cor. 4.4,

gr(D)

given

by

a

7→

ea.

Clearly,

^ N r d g r ( D ) ( e a ) = N r d D ( a ) = 1 .

This shows that ρ(D(1)

) gr(D)

(1). Now consider the diagram

//

D (1 + M ) D0

//

D (1 + M ) D(1)

1

1

//

D0

//

D

(1)

ρ

//

gr(D)0

//

gr(D)

(1)

//

1

//

1

(4.3)

The top row of the above diagram is clearly exact. The Congruence Theorem (see Th. B.1 in Appendix B), implies that the left vertical map in the diagram is an isomorphism. Once we prove that ρ(D(1)) = gr(D)(1), we will have the exactness of the second row of diagram (4.3), and the theorem follows by the exact sequence

for cokernels.

To prove the needed surjectivity, take any b gr(D)with Nrdgr(D)

(b) = 1. Thus b gr(D)0 by Th. 3.3.

C h o o s e a V D

such that ea = b.

Then we have,

^

NrdD(a) = NrdD(a) = Nrdgr(D)(b) = 1.

Thus NrdD(a) 1 NrdD(c) = Nrd(a)

1 + M F . B y C o r . 4 . 7 , s i n c e N r d D ( 1 + M . Then, ac D(1) and ρ(ac) = ρ(a) = b. D

)

=

1

+

M F

,

there

is

c

1

+

M D

such

that

Recall from §2 that starting from any graded division algebra E with center T and any choice of total ordering on the torsion-free abelian group E, there is an induced valuation v on q(E), see (2.6). Let h(T ) be the henselization of T with respect to v, and let h(E) = q(E) q(T ) h(T ). Then, h(E) is a division ring by Morandi’s henselization theorem ([Mor], Th. 2 or see [W2], Th. 2.3), and with respect to the unique extension of the henselian valuation on h(T ) to h(E), h(E) is an immediate extension of q(E), i.e.,

gr

(h(E))

=

gr

g r ( q ( E ) ) . F u r t h e r m o r e , a s

[h(E) : h(T )] = [q(E) : q(T )] = [E : T ] = [gr(q(E)) : gr(q(T ))] = [gr(h(E) : gr(h(T ))]

and

Z(

gr

(h(E)))

=

gr

Z(

g r ( q ( E ) ) )

=

gr

T

=gr

gr(h(T )) = gr(Z(h(E))),

h(E) is tame (see the characterizations of tameness in §2).

C

o r o l l a r y

4 . 9 . L

et

E b e a g r a d e d d i v i s i o n a l g e b r a .

Th

en

SK1(h(E))

=

SK1(E)

.

P r o o f . S i

nce

h(E) i

s a t a m e v a l u e

d d i v i s i o n a l g e b r a ,

b

y

T h . 4 . 8 , S K 1 ( h ( E ) )

=

SK1(

gr

(h(E)))

.

But

gr

(h(E))

=

gr

g r ( q ( E ) )

=

gr

E , s o t h e c o r o l l a r y f o l l o w s .

Having now established that the reduced Whitehead group of a division algebra coincides with that of its associated graded division algebra, we can easily deduce stability of SK1 for unramified valued division algebra, due originally to Platonov (Cor. 3.13 in [P1]), and also a formula for SK1 for a totally ramified division algebra ([LT], p. 363, see also [E], p. 70), and also a formula for SK1 in the nicely semiramified case ([E], p. 69), as natural consequences of Th. 4.8:

Corollary center F .

4.10.

Let

F

be

a

eld

with

Henselian

valuation,

and

let

D

be

a

tame

division

algebra

with

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