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4

R. HAZRAT AND A. R. WADSWORTH

definitions are motivated by analogous definitions for valued division algebras ([W2]). Indeed, if a valued division algebra is unramified, semiramified, or totally ramfied, then so is its associated graded division algebra (see §4).

A main theme of this paper is to study the correspondence between SK1 of a valued division algebra and that of its associated graded division algebra. We now recall how to associate a graded division algebra to a valued division algebra.

Let D be a division algebra finite dimensional over its center F , with a valuation v : Da totally ordered abelian group, and v satisifies the conditions that for all a, b D,

. So

is

  • (1)

    v(ab) = v(a) + v(b);

  • (2)

    v(a + b) min{v(a), v(b)}

( b =6

a).

Let

V D M D

= {a D: v(a) 0} ∪ {0}, the valuation ring of v; = { a D : v ( a ) > 0 } { 0 } , t h e u n i q u e m a x i m a l l e f t ( a n d r i g h t ) i d e a l o f V D ;

D = V D / M D , t h e r e s i d u e d i v i s i o n r i n g o f v o n D ; a = im(v), the value group of the valuation. n d D

For background on valued division algebras, see [JW] or the survey paper [W2]. One associates to D a graded division algebra as follows: For each γ D, let

Dγ D gr(D) > γ γ

= {d D: v(d) γ} ∪ {0}, an additive subgroup of D; = {d D: v(d) > γ} ∪ {0}, a subgroup of Dγ; and = DγD.

Then define

gr(D)

=

L

gr(D)γ.

γ

D

Because DDδ + DγDD>(γ+δ) for all γ, δ D, the multiplication on gr(D) induced by multi- plication on D is well-defined, giving that gr(D) is a graded ring, called the associated graded ring of D.

The multiplicative property (1) of the valuation v implies that gr(D) is a graded division ring. Clearly, we D . F o r d D , w e w r i t e e d f o r t h e i m a g e d + D > v ( d ) of d in gr(D)v(d) have gr(D)0 = D and = e gr(D) .

T h u s , t h e m a p g i v e n b y d 7 d i s a g r o u p e p i m o r p h i s m D g r ( D ) w i t h k e r n e l 1 + M D .

The restriction v|F of the valuation on D to its center F is a valuation on F , which induces a corre- sponding graded field gr(F ). Then it is clear that gr(D) is a graded gr(F )-algebra, and by (2.2) and the

Fundamental Inequality for valued division algebras, [gr(D) : gr(F )] = [D : F ] |

D

:

F | [D : F ] < .

Let F be a field with a henselian valuation v. Recall that a field extension L of F of degree n < is

said to field L

be tamely rami ed or tame over is a separable field extension of

F F

if, with respect to the

and

char(F )

n [ L

:

unique extension of v to L, the residue F ]. Such an L is necessarily defectless

over F , i.e., [L : F ] = [L : F ] | L : with center F (so, by convention

F | = [gr(L) : gr(F )]. Along the same lines, let D be a division algebra , [D : F ] < ); then v on F extends uniquely to a valuation on D.

With respect to this valuation, D is said to be tamely rami ed or tame if Z(D) is separable over F i and char(F ) ind(D) Z(D)/F is separable, n d ( D ) [ Z ( D ) : F ] . it is abelian Galois. Recall from [JW], Prop. 1.7 that whenever the It is known (cf. Prop. 4.3 in [HwW2]) that D field extension is tame if and

only if [gr(D) : gr(F )] = [D : F ] and Z(gr(D)) = gr(F ), if and only if ramified extension of F , if and only if char(F ) = 0 or char(F ) = p =6 D is split by the maximal tamely 0 and the p-primary component

of D

is inertially split, i.e., split by the maximal unramified extension of F .

We say D

is strongly

tame

if

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