SK1-like Functors for Division Algebras

Abstract

We investigate the group valued functor G D = D * /F * D where D is a division algebra with center F and D the commutator subgroup of D * . We show that G has the most important functorial properties of the reduced Whitehead group SK 1 . We then establish a fundamental connection between this group, its residue version, and relative value group when D is a Henselian division algebra. The structure of G D turns out to carry significant information about the arithmetic of D. Along these lines, we employ G D to compute the group SK 1 D . As an application, we obtain theorems of reduced K-theory which require heavy machinery, as simple examples of our method.

Triviality of the Functor Coker( K 1 ( F ) → K 1 ( D )) for Division Algebras

“…what would be imposed on the algebraic structure of D. We mention that if D is a totally ramified division algebra then exp(CK 1 (D)) = i(D) if and only if D is cyclic [5]. In fact from the above theorem it follows that if D is a cyclic division algebra of index p, an odd prime, then the exponent of CK 1 (D) is exactly p. The converse is not true, as the following example shows.…”

“…If R is semilocal, since the stable rank of R and A are one, CK 1 (A) = A * /R * A where A * and R * are the group of invertible elements of A and R respectively and A is the derived subgroup of A * . A study of this group in the case of central simple algebras is initiated in [6] and further in [5]. It has been established that despite of a "different nature" of this group from the reduced Whitehead group SK 1 , these two groups have the same functorial properties.…”

“…In [6] this functor is determined for totally ramified division algebras and in particular for any finite abelian group G, a division algebra D such that CK 1 (D) = G × G is constructed. Further in [5], this functor is studied in more cases and examples of cyclic CK 1 (even over non local fields) constructed. It was conjectured in [6] that if the functor CK 1 is trivial then the index of the division algebra is 2.…”

Wedderburn’s factorization theorem application to reduced $K$-theory

“…Let C be the field of complex numbers and r be a nonnegative integer. Let D 1 = C((x 1 )) and define σ 1 : In [4], as another application of Lemma 1, we obtain theorems of reduced Ktheory which previously required heavy machinery, as a simple consequence of this approach.…”

“…is a torsion group of bounded exponent n. Some algebraic properties of this group are studied in [4].…”