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SK1

OF GRADED DIVISION ALGEBRAS

21

of twisted polynomial rings. (See [PY], Lemma 8.) By way of proof of [PY], Lemma 8, the authors say nothing more than that it follows by induction from the rank 1 case. It is not clear whether the proof given here coincides with their unstated proof, since the transition from rank 1 to finite rank is not transparent.

(ii) So far the functor CK1 has manifested properties similar to SK1. However, the similarity does not hold here, since the functor CK1 is not (homotopy) stable. In fact, for a division algebra D over its center F of index n, one has the following split exact sequence,

1 CK1(D) CK1(D(x)) L Z/(n/np)Z 1

p

where p runs over irreducible monic polynomials of F [x] and np is the index of central simple algebra D F F [x]/(p)(see Th. 2.10 in [H1]). This is provable by mapping the exact sequence (5.1) with

T = F [x] to the sequence for T = D[x] and taking cokernels. Example 5.8. Let E be a semiramified graded division ring with

E

=

Zn

, an

dl

et

T

=

is a torsion group, there are a base {γ1, s u c h t h a t { r 1 γ 1 , . . . , r n γ n } i s a b a s e o f C h o o s e a n y n o n z e r o z . . . , γn} of the free abelian group and E i E γ i a n d x i L e t F = T 0 a n d M = E 0 , a al(M/F ). Because E is semiramified, T. n d l e t G =

Z ( E ) . S i E / s o m e r 1 , . . . , r n T r i γ i , 1 M is Galois over nce i

T

N n. F

with [M : F ] = |

w i t h z ri i

= b i x i .

E

:

T|

= i n d ( E )

L e t u i j = z i z j z i

= r 1 . 1 . . r n , a n T . M. S dG = E/ i n c e z i E ri r Let σi G be the automorphism of M determined by i γ i = E 0 x i , t h e r e i s b i M 1 z j

E/ c o n j G u g dG b a t hi i h o n y z i . F r o m t h e i s o m o r p T , e a c h σ i a s o r sm d e r r i i n G = h σ 1 i an × . . . × h σ n i . C = l e a r l y , T = F [ x 1 , x 1 1 , . . . x n , x 1 n ] , a n i t e r a t e d L a u r e n t p o l y n o m i a l r i n g , a n d E = M [ z 1 , z 1 1 , . . . , z n , z 1 n ] , a n i t e r a t e d t w i s t e d L a u r e n t p o l y n o m i a l r i n g w h o s e m u l t i p l i c a t i o n i s c o m p l e t e l y d e t e r m i n e d b y t h e b i the uij M, and the action of the σi on M. M ,

Let D = q(E), which is a division ring with center q(T ) = F (x1, . . . , xn), a rational function field over F . Then, D is the generic abelian crossed product determined by M/F , the base {σ1, . . . , σn} of G, the bi and the uij, as defined in [AS]. As was pointed out in [BM], all generic abelian crossed products arise this way as rings of quotients of semiramified graded division algebras. Generic abelian crossed products were used in [AS] to give the first examples of noncyclic p-algebras, and in [S1] to prove the existence of noncrossed product p-algebras. It is known by [T], Prop. 2.1 that D is determined up to F -isomorphism by M and

the uij. By Cor. 3.6(iii) and Th. 5.7, there is an exact sequence

b G G H 1(G, M) SK1(D) 1,

(5.8)

w h e r e t h e l e f t m a p i s d e t e r m i n e d b y s e n d i n g σ i σ j t o u i j m o d I G ( M ) . A n i m p o r t a n t c o n d i t i o n i n - t r o d u c e d b y A m i t s u r a n d S a l t m a n i n [ A S ] w a s n o n d e g e n e r a c y o f { u i j } . T h i s c o n d i t i o n w a s e s s e n t i a the noncyclicity results in [AS], and is also key to the results on noncyclicity and indecomposability of generic abelian crossed products in recent work of McKinnie in [Mc1], [Mc2] and Mounirh [M2]. The orig- inal definition of nondegeneracy in [AS] was somewhat mysterious. A cogent characterization was given recently in [Mc3], Lemma 5.1: A family {uij} in M(meeting the conditions to appear in a generic abelian l f o r

b crossed product) is nondegenerate iff for every rank 2 subgroup H of G, the map H H H 1(H, M)

a p p e a r i n g i n t h e c o m p l e x ( 5 . 8 ) f o r t h e g e n e r i c a b e l i a n c r o s s e d p r o d u c t C D ( M H ) i s n o n z e r o . I n t h e fi r s t ij n }i G o n t r i v i a l c a s Zp Zp e , w h e r e w i t h p a p r i m e n u m b e r , w e h a v e { u s n o n d e g e n e r a t e i ff t h e m a p × =

bb G G H 1(G, M) is nonzero, iff the epimorphism H 1(G, M) SK1(D) is not injective. Thus, the

nondegeneracy is encoded in SK1(D), and it occurs just when SK1(D) is not “as large as possible.”

Appendix A. The Wedderburn factorization theorem

Let D be a noncommutative division ring with center F , and let a D with minimal polynomial f in F [x]. Any conjugate of a is also a root of this polynomial. Since the number of conjugates of a is infinite ([L], 13.26), this suggests that f might split completely in D[x]. In fact, this is the case, and it is called

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