14

R. HAZRAT AND A. R. WADSWORTH

( i ) I f D i SK_{1}(D) SK_{1}(D) s u n ∼ r a m i e d t h e n = SK_{1}(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

SK_{1}(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 [M_{1}], 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,[HwW_{2}], 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

SK_{1}(D)

∼

=

SK_{1}(

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 d_{τ }∈ D^{∗ }analogously. Then, b unique, though its image in F ^{1}(G, D^{∗}) v([d_{σ}, d_{τ }]) = 0 and N_{D/F }([d_{σ}, d_{τ }]) = 1 since Nrd_{D}([d_{σ}, d_{τ }]) = 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 ∈ D^{∗ }with N_{D/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 ) → SK_{1}(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 Nrd_{D}(ac

^{1}) = 1. The map

image

of

ac

1

in

SK_{1}(D).

Recall that the reduced norm residue group of D is defined as SH^{0}(D) = F ^{∗}/Nrd_{D}(D^{∗}). It is known that SH^{0}(D) coincides with the first Galois cohomology group H^{1}(F, D^{(1)}) (see [KMRT], §29). We now show that for a tame division algebra D over a henselian field, SH^{0}(D) coincides with SH^{0 }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

SH^{0}(D)

∼

=

SH^{0}(

gr

(D))

.

Proof. Consider the diagram with exact rows,

1

//

1 + M D

//

D^{∗ }

ρ

//

gr(D)^{∗ }

//

1

Nrd_{D }

Nrd_{gr(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 SK_{1}, if D is tame and unramified, then

SH^{0}(D)

∼ =

SH^{0}(

gr

(D))

∼ =

SH^{0}(

gr

(D)_{0})

∼ =

SH^{0}(D)

.