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SK1

OF GRADED DIVISION ALGEBRAS

27

Proof of Claim. Take T 1 + J. The idea is that the process of bringing T to upper triangular f o r m b y r o w o p e r a t i o n s i s c a r r i e d o u t e n t i r e l y w i t h i n 1 + J . W r i t e T = I ` + Z w i t h Z = ( z i j ) J . S o ,

w ( z i i ) > γ i

γ i = 0 f o r a l l i , i . e . , z i i M C . T h u s , f o r a l l i , j , w e h a v e

tii

= 1 + zii

1 + M C

and

tij

= zij,

so w(tij) > γi

γ j

when

i =6

j.

Fix k with 1 k `

1. Since tkk

1 + M C , w ( t k k ) = 0 , s o t k k

=6 0.

Let Y

= ( y i j ) M `

(C) be the

matrix for the row operations to bring 0’s to all entries in the k-th column of T

below the main diagonal,

i.e., the i-th row of Y T

is:

(the i-th row of T )

rows unchanged). So, yii = 1 for all i; yik otherwise. For i > k,we have

=

t i k t (t ik 1 k k

t 1 k k

· the k-th row of T ) for k < i ` (with the first k

for our fixed k and all i with k < i `; and yij = 0

w(yik) = w(tik)

w ( t k k ) > γ i

γk.

Hence, Y 1 + J and Y is a unipotent lower triangular matrix. Since 1 + J is closed under multiplication, we have Y T 1 + J. To bring T to upper triangular form we apply the row operations successively

for columns 1 to `

1.

W e e n d u p w i t h a n u p p e r t r i a n g u l a r m a t r i x T 0

= Y ` 1 Y ` 2

. . . Y 2 Y 1 T

1 + J,

w h e r e e a c h Y k 1 + J i s t h e m a t r i x f o r z e r o i n g m a t r i x Y k 1 . . . Y 1 T 1 + J ( n o t t o T ) . S a y T 0

the k-th = ( t 0 i j ) .

E column as described above, but applied to the a c h Y k i s u n i p o t e n t a n d l o w e r t r i a n g u l a r , s o

d d e t ( Y k ) = 1 C / C 0 , S o , d d e t ( T 0 ) = d d e t ( Y k 1

) . . . d d e t ( Y 1 ) d d e t ( T ) = d d e t ( T ) i n C / C 0 . S i n c e T 0 i s

u p p e r t r i a n g u l a r w i t h e a c h t 0 i i 1 + M C , w e h a v e d d e t ( T ) = d d e t ( T 0 ) = t 0 1 1 . . . t 0 ` `

1 + M C ( e q u a l i t y m o d u l o C 0 ) ,

proving the Claim.

References

[AS] [B] [BM]

S. A. Amitsur, D. J. Saltman, Generic Abelian crossed products and p-algebras, J. Algebra, 51 (1978), 76–87. 21 M. Boulagouaz, Le gradue´ d’une alge`bre a` division value´e, Comm. Algebra, 23 (1995), 4275–4300. 1, 5

M. Boulagouaz, K. Mounirh, Generic abelian crossed products and graded division algebras, pp. 33–47 in Algebra and Number Theory, eds. M. Boulagouaz and J.-P. Tignol, Lecture Notes in Pure and Appl. Math., Vol. 208, Dekker, New York, 2000. 21

[D]

P. Draxl, Skew Fields, London Math. Soc. Lecture Note Series, Vol. 81, Cambridge Univ. Press, Cambridge, 1983. 23, 24

[EP] [E]

A. J. Engler, A. Prestel, Valued Fields, Springer-Verlag, Berlin, 2005. 11, 24

Yu. Ershov, Henselian valuations of division rings and the group S 1, Mat. Sb. (N.S.), 117 (1982), 60–68 (in Russian); English trans., Math. USSR-Sb., 45 (1983), 63–71. 2, 7, 12, 13, 14, 23

[G]

P. Gille, Le proble´me de Kneser-Tits, expose´ Bourbaki, No. 983, to appear in Aste´risque; preprint available at: http://www.dma.ens.fr/ gille/ . 1

[H1] [H2]

R. Hazrat, SK1-like functors for division algebras, J. Algebra, 239 (2001), 573–588. 10, 21

R. Hazrat, Wedderburn’s factorization theorem, application to reduced 311–314. 23

  • -

    theory, Proc. Amer. Math. Soc., 130 (2002),

[H3] [H4]

  • R.

    Hazrat, On central series of the multiplicative group of division rings, Algebra Colloq., 9 (2002), 99–106. 15

  • R.

    Hazrat, SK1 of Azumaya algebras over Hensel pairs, to appear in Math. Z.; preprint available (No. 282) at:

http://www.math.uni bielefeld.de/LAG/ . 1

[HW1] R. Hazrat, A. R. Wadsworth, On maximal subgroups of the multiplicative group of a division algebra, to appear in J. Algebra; preprint available (No. 260) at: http://www.math.uni bielefeld.de/LAG/ . 10

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