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26

R. HAZRAT AND A. R. WADSWORTH

The following proposition will complete the proof of the Congruence Theorem. Proposition B.3. Let F be a henselian valued eld, and let D be an F -central division algebra which is

d e f e c t l e s s o v e r

F . L

et

L b e a e l d , F

L

D , a n

d l e t C

=

CD(L)

, so

L

FD

=

M ` ( C )

w h

ere

`

=

[ L : F ]

.

T a k e a n y a

1

+

M D

.

Th

en,

f o r

1

a

L

FD

=

M ` ( C )

,

where

ddet

denotes

the

d d e t ( 1 a ) 1 + M Dieudonne´determinant. C

(mod

C 0 ) ,

Proof. D is an L-D bimodule via multiplication in D. Hence (as L is commutative) D is a right L F D-

P P module, with module action given by a( ` i d i ) = ` i a d i . I n f a c t , D i s s i m p l e a s a r i g D-module, F h t L s (D), where i d n c e i t i s a l r e a y a s i H L D m p l e r i F g h t D - m o d u l e . b y W e d d e r b u r ence, n s T h E n eorem, d E = (D) ( n d L F D a c t i n g o n Si o n t h e l D D D D e f t ) . ( f o r a c t i n g o n o n t h e r i g h t ) E n nce d D ( D = = )

( e l e m e n t s o f D a c t i n g o n D b y l e f t m u l t i p l i c a t i o n ) E n d L F D

(D) consists of left multiplication by elements

  • o

    fD

w h i c h c o m m u t e w i t h t h e l e f t a c t i o n o

fL

on

D , i . e . ,

=

CD(L)

=

C . S

o,

L

FD

=

E n d ( D )

=

E n d C ( D )

=

M ` ( C )

,

where ` = [D : C] = [L : F ]. The last isomorphism is obtained by choosing a base {b1, . . . , b`} of D as a left C-vector space (D = Cb1 . . . Cb`) and writing the matrix for an element of L F D acting C-linearly on D (on the right) relative to this base, with matrix entries in C.

Because D is defectless over F , D is also defectless over C, i.e., [D : C] = [gr(D) : gr(C)]; thus, the valuation w on D extending v on F is a w|C -norm by [RTW], Cor. 2.3. This means that we can choose our base {b1, . . . , b`} to be a splitting base for w over w|C , i.e., satisfying, for all c1, . . . , c` C,

`

w

P

i=1

c i b i

= m i n 1 i ` w ( c i ) + w ( b i ) .

(B.2)

Let γi = w(bi) for 1 i `. Let

R = { A = ( a i j ) M ` ( C ) : w ( a i j ) γ i J = { A = ( a i j ) M ` ( C ) : w ( a i j ) > γ i

γ j f o r a l l i , j } ; γ j f o r a l l i , j } ;

1 + J = { I ` + A : A J } , w h e r e I ` M ` ( c ) i s t h e i d e n t i t y m a t r i x .

B e c a u s e w i s a v a l u a t i o n , i t i s e a s y t o c h e c k t h a t R i s a s u b r i n g o f M ` ( C ) a n d J i s a n i d e a l o f R . T h e r e f o r e , 1 + J i s c l o s e d u n d e r m u l t i p l i c a t i o n . T a k e a n y f E n d C ( D ) ( w h i c h a c t s o n D o n t h e r i g h t ) , a n d l e t A ` j=1 P = ( a i j ) b e t h e m a t r i x o f f r e l a t i v e t o t h e C - b a s e { b 1 , . . . b ` } o f D , i . e . , b i f = a i j b j f o r a l l i . S o ,

`

w(bif) = w

P

j=1

a i j b j

= min

1j`

w ( a i j ) + γ j .

Thus, w(bif) w(bi) = γi iff w(aij) γi

γ j f o r 1 j ` . F r o m t h i s

it

is

clear

that

A = (aij) R

iff

w(bif) w(bi) for all i.

Analogously, A J

iff w(bif) > w(bi) for all i.

N o w , t a k e a n y u 1 + M D , s a y u = 1 + m w i t h m M D . T h e n , 1 m L F D c o r r e s p o n d s t o ρ m E n d C ( D ) , w h e r e d ρ m = d m f o r a l l d D . L e t S M ` ( C ) b e t h e m a t r i x f o r ρ m . S i n c e w ( m ) > 0 have , w e

w ( b i ρ m ) = w ( b i m ) = w ( b i ) + w ( m ) > w ( b i )

for all i.

Hence, S J, by the preceding paragraph. C l a i m . F o r a n y m a t r i x T 1 + J , w e h a v e d d e t ( T ) 1 + M C ( m o d C 0 ) . T h e P r o p o s i t i o n f o l l o w s a t o n c e f r o m t h i s c l a i m , s i n c e t h e m a t r i x f o r 1 ( 1 + m ) i s I ` + S 1 + J .

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