SK_{1 }

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 CK_{1 }has manifested properties similar to SK_{1}. However, the similarity does not hold here, since the functor CK_{1 }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 → CK_{1}(D) → CK_{1}(D(x)) → ^{L }Z/(n/n_{p})Z → 1

p

where p runs over irreducible monic polynomials of F [x] and n_{p }is the index of central simple algebra D ⊗_{F }F [x]/(p)^{ }(see Th. 2.10 in [H_{1}]). 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

∼

=

Z^{n }

, an

dl

et

T

=

is a torsion group, there are a base {γ_{1}, 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 . . . , γ_{n}} 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 r_{i }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 r_{i }r Let σ_{i }∈ 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 u_{ij }∈ M, and the action of the σ_{i }on M. M ,

Let D = q(E), which is a division ring with center q(T ) = F (x_{1}, . . . , x_{n}), a rational function field over F . Then, D is the generic abelian crossed product determined by M/F , the base {σ_{1}, . . . , σ_{n}} of G, the b_{i }and the u_{ij}, 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 [S_{1}] 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 u_{ij}. By Cor. 3.6(iii) and Th. 5.7, there is an exact sequence

b G ∧ G → H ^{1}(G, M^{∗}) → SK_{1}(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 [Mc_{1}], [Mc_{2}] and Mounirh [M_{2}]. The orig- inal definition of nondegeneracy in [AS] was somewhat mysterious. A cogent characterization was given recently in [Mc_{3}], Lemma 5.1: A family {u_{ij}} 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 Z_{p }Z_{p }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^{∗}) → SK_{1}(D) is not injective. Thus, the

nondegeneracy is encoded in SK_{1}(D), and it occurs just when SK_{1}(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