Roozbeh Hazrat: On triviality of the functor $\text{Coker}(K_1(F) \rightarrow K_1(D))$ for division algebras

Submission: 2003, Jan 25

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 $F$. We construct a division algebra $D$ and a division subalgebra $A \subset D$ such that $CK_1(A)\cong CK_1(D)$. Using valuation theory, the group $CK_1(D)$ is computed for some valued division algebras.

2000 Mathematics Subject Classification:

Keywords and Phrases:

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