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

Roozbeh Hazrat, Uzi Vishne

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


Let D be a division algebra with center F. 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, assuming a division algebra D is a product of cyclic algebras, the group CK1(D) is trivial if and only if D is an ordinary quaternion algebra over a real Pythagorean field F. We also characterize the cyclic central simple algebras with trivial CK1 and show that CK 1 is not trivial for division algebras of index 4. Using valuation theory, the group CK1(D) is computed for some valued division algebras.

Original languageEnglish
Pages (from-to)1427-1435
Number of pages9
JournalCommunications in Algebra
Issue number5
StatePublished - 2005


  • Division algebras
  • Reduced K-theory


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