R. Hazrat: SK_1 like functors for division algebras


Submission: 2000, Mar 20

We introduce a Dieudonne/reduced Dieudonne functor for central simple algebras A, which embraces different groups which are associated to A in different contexts, including the reduced Whitehead group SK_1(A). We show that a reduced Dieudonne functor has most important functorial properties of the reduced Whitehead group SK_1(A). Specializing a Dieudonne functor to the group G(A)=A^*/F^*A', where F is the center of A and A' its commutator subgroup, we establish a fundamental connection between this group, its residue version and relative valued group when A is a henselian division algebra. The structure of G(A) turns out to carry significant information about the arithmetic of A. Along these lines, we employ G(A) to compute the group SK_1(A). As an application, we obtain theorems of reduced K-theory which requires heavy machinery, as simple examples of our method.

1991 Mathematics Subject Classification: 12E15

Keywords and Phrases: Division algebra. Valuation theory. Dieudonne functor.

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