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$K$-theory of rings of continuous functions

Ko Aoki

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 Cortiñas--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.

$K$-theory of rings of continuous functions

Abstract

We study the algebraic -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 -theory and show its -regularity. The complex case reproves the results of Rosenberg, Friedlander--Walker, and Cortiñas--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.
Paper Structure (26 sections, 68 theorems, 27 equations, 4 figures)

This paper contains 26 sections, 68 theorems, 27 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Methods
  3. Application: examples in K-theory
  4. Future plans
  5. Compacta
  6. Weight
  7. The category of compacta
  8. (Strong) shape and cohomology
  9. The cd topology on compacta
  10. The cd topology
  11. Cd sheaves
  12. Variant: profinite sets
  13. Cdh covers underlie cd covers
  14. Digression: compacta with Ind-smooth rings of continuous functions
  15. F-spaces
  16. ...and 11 more sections

Key Result

Theorem A

For a compactum $X$, we have canonical isomorphisms for ${*}\leq0$. Here the right-hand sides are the sheaf cohomology groups.

Figures (4)

  • Figure 1: A graph of the section dependency
  • Figure 2: A typical cd cover that is not a cover in the closed topology
  • Figure 3: The cover $[-1,0]\amalg[0,1]\to[-1,1]$ inside the real points of $\operatorname{Spec}\mathbf{R}[x]\amalg\operatorname{Spec}\mathbf{R}[y]\to\operatorname{Spec}\mathbf{R}[x,y]/\langle xy\rangle$
  • Figure 4: Part of The construction in \ref{['dcd3146c40']}

Theorems & Definitions (172)

  • Theorem A
  • Theorem B
  • Theorem C
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem D
  • Corollary 1.6
  • ...and 162 more