$K$-theory of rings of continuous functions

Ko Aoki

$K$-theory of rings of continuous functions

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arxiv 2402.05257 v1 pith:VOJCHC7J submitted 2024-02-07 math.KT math.ACmath.AGmath.GN

Ko Aoki This is my paper · ORCID

classification math.KTmath.ACmath.AGmath.GN

keywords theoryalgebraiccasecontinuousfunctionslocalringrosenberg

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abstract

We study the algebraic $K$-theory of the ring of continuous functions on a compact Hausdorff space with values in a local division ring, e.g., a local field: We compute its negative $K$-theory and show its $K$-regularity. The complex case reproves the results of Rosenberg, Friedlander--Walker, and Corti\~nas--Thom. Our consideration in the real case proves two previously unconfirmed claims made by Rosenberg in 1990. The algebraic nature of our methods enables us to deal with the nonarchimedean and noncommutative cases analogously.

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