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14

R. HAZRAT AND A. R. WADSWORTH

( i ) I f D i SK1(D) SK1(D) s u n r a m i e d t h e n = SK1(D) ( i i ) I f D i s t o t a l l y r a m i e d t h e n = µ

n ( F ) / µ e ( F )

w h e r e n = i n d ( D )

a n d e i s t h e e x p o n e n t

o f D / F .

( i i i ) I f D i l(D/F ) s G s e m F . Then, there is an exact sequence i r a m i e d , l e t D =a / G = G b H 1 ( G , D ) S K 1 ( D ) 1 .

(4.4)

( i v ) I f D i s n i c e l y s e m i r a m i e d , t h e n

SK1(D)

=

b H

1(

a

l ( D / F ) , D )

.

See Remark 4.11 below for a description of the maps in (4.4)

Proof. Because D is tame, Z(gr(D)) = gr(F ) and ind(gr(D)) = ind(D). Therefore, for D in each case (i)–(iv) here, gr(D) is in the corresponding case of Cor. 3.6. (In case (iii), that D is semiramified means [D : F ] = | D : F | = ind(D) and D is a field. Hence gr(D) is semiramified. In case (iv), since D is nicely semiramified, by definition (see [JW], p. 149) it contains maximal subfields K and L, with K unramified over F and L totally ramified over F . (In fact, by [M1], Th. 2.4, D is nicely semiramified if and only if it has such maximal subfields.) Then, gr(K) and gr(L) are maximal graded subfields of gr(D) by dimension count and the graded double centralizer theorem,[HwW2], Prop. 1.5(b), with gr(K) unramified over gr(F ) and gr(L) totally ramified over gr(F ). So, gr(D) is then in case (iv) of Cor. 3.6.) Thus, in each case

C

or.

4 . 1 0 f o r

D f o l l o w s f r o m

C

or.

3 . 6 f o r g r

(D)

t o g e t h e r w i t h t h e i s o m o r p

hi

sm

SK1(D)

=

SK1(

gr

(D))

given by Th. 4.8.

Remark 4.11. By tracing through the isomorphisms used in their construction, one can see that the maps in (4.4) can be described as follows: Let v be the valuation on D. For each σ G = al(D/F ) there is

b y [ J W ] , P r o p . 1 . 7 o r [ E ] , P r o p . 1 s o m e d σ D s u c h t h a t d 1 σ a d σ = σ ( a ) f o r a l l a V D . T h i s d σ i s n o t

D/ is uniquely determined. For τ G, choose dτ Danalogously. Then, b unique, though its image in F 1(G, D) v([dσ, dτ ]) = 0 and ND/F ([dσ, dτ ]) = 1 since NrdD([dσ, dτ ]) = 1, by (4.2). The map G G H

b s e n d s σ τ t o t h e i m a g e o f [ d σ , d τ ] i n 1(G, D). Now, take any b Dwith ND/F (b) = 1, and let H a b e a n y i n v e r s e i m a g e o f b i n V D . B y ( 4 . 2 ) ( s i n c e h e r e δ = 1 a n d Z ( D ) = D ) , N r d D ( a ) 1 + M F , s o b y

H 1 (G, D C o r . 4 . 7 t h e r e i s c 1 + M b ) SK1(D) sends D

the image of b in w i t h N r d D ( c ) = N r d D ( a ) . T h e n a b H c 1 (G, D) to the

1

= b and NrdD(ac

1) = 1. The map

image

of

ac

1

in

SK1(D).

Recall that the reduced norm residue group of D is defined as SH0(D) = F /NrdD(D). It is known that SH0(D) coincides with the first Galois cohomology group H1(F, D(1)) (see [KMRT], §29). We now show that for a tame division algebra D over a henselian field, SH0(D) coincides with SH0 of its associated graded division algebra.

Theorem 4.12. Let F be a

eld with a henselian valuation v and let D be a tame F -central division

a l g e b r a .

Th

en

SH0(D)

=

SH0(

gr

(D))

.

Proof. Consider the diagram with exact rows,

1

//

1 + M D

//

D

ρ

//

gr(D)

//

1

NrdD

Nrdgr(D)

(4.5)

1

//

1 + M F

//

F

//

gr(F )

//

1

where Cor. 4.4 guarantees that the diagram is commutative. By Cor. 4.7, the left vertical map is an epimorphism. The theorem follows by the exact sequence for cokernels.

Remark. As with SK1, if D is tame and unramified, then

SH0(D)

=

SH0(

gr

(D))

=

SH0(

gr

(D)0)

=

SH0(D)

.

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