On triviality of the functor Coker(K_1(F) --> K_1(D)) for division algebras, by Roozbeh Hazrat

Let D be a cyclic division algebra over its centre F of index n. Consider the group CK_1(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 CK_1(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 CK_1 is p. We show that the converse does not hold by exhibiting a division algebra D and a division subalgebra A in D such that CK_1(A) is isomorphic to CK_1(D). Using valuation theory, the group CK_1(D) is computed for some valued division algebras.

Roozbeh Hazrat <hazrat@maths.anu.edu.au>