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

Abstract

Abstract. Let D be a cyclic division algebra over its centre F of index n. Consider the group CK1(D) = D * /F * D where D * is the group of invertible elements of D and D is its commutator subgroup. In this note we shall show that the group CK1(D) is trivial if and only if D is an ordinary quaternion division algebra over a real Pythagorean field F . This in particular shows that if the index of D is an odd prime p, then the exponent of CK1 is p. We show that the converse does not hold by exhibiting a division algebra D and a division subalgebra A ⊂ D such that CK1(A) ∼ = CK1(D). Using valuation theory, the group CK1(D) is computed for some valued division algebras.

Reduced K-theory of Azumaya algebras

“…The conjecture that CK 1 of a division algebra D is not trivial if D is not a quaternion algebra still remains open. This has connections with the study of normal and maximal subgroups of D * (see [9] and references there). Proof.…”

“…It has been conjectured that if CK 1 (D) is trivial then D is a quaternion algebra. In [9], it is proved that if D is a tensor product of cyclic algebras then CK 1 (D) is trivial if and only if D is a quaternion division algebra over a real Pythagorean field F . The group CK 1 has been computed in [8] for certain division algebras and its connection with SK 1 studied.…”

Nontriviality of certain quotients of K1 groups of division algebras

“…Since every division algebra over a global field is a cyclic algebra, the result quoted from [5] above shows that every global field is NKNT. Likewise, every nonreal local field is NKNT.…”

“…It was shown in [5] that if C is a noncommutative cyclic algebra with NK 1 (C) = 1, then C ∼ = ( −1,−1 F ) with F a real Pythagorean field. We will show here that NK 1 (D) is nontrivial for a great many other noncommutative division algebras D. Of course, whenever NK 1 (D) = 1, we also have CK 1 (D) = 1 (so D * has a maximal subgroup).…”

“…The preceding observation reduces the question to the case when CK 1 (D) is trivial. In [5,Theorem 2.12] it was proved that if D is a tensor product of cyclic algebras then CK 1 (D) is trivial if and only if D is a quaternion division algebra ( −1,−1 F ) where F is a real Pythagorean field (see also [9]). It has been conjectured in [7] that if CK 1 (D) is trivial then D is a quaternion division algebra.…”