CMA Research Report

MRR03-001

Triviality of the functor Coker(K₁(F) -> K₁(D)) for division algebras

Abstract:  Let D be a division algebra with centre F. 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, assuming D is a product of cyclic algebras, the group CK₁(D) is trivial if and only if D is an ordinary quaternion division algebra over a real Pythagorean field F. Likewise if the index of D equals four then CK₁(D) is non-trivial. If the index of a cyclic division algebra 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.

AMS Classification:  16K20, 16E20, 19B99
Date:  19 February 2003

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