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Algebraic $K$-theory of stably compact spaces

Georg Lehner

Abstract

We compute the value of finitary localizing invariants, including algebraic $K$-theory, on categories of sheaves over stably locally compact spaces $X$. Our formula simultaneously generalizes the cases of locally compact Hausdorff and coherent (spectral) spaces and recovers several smaller $K$-theory calculations as special instances.

Algebraic $K$-theory of stably compact spaces

Abstract

We compute the value of finitary localizing invariants, including algebraic -theory, on categories of sheaves over stably locally compact spaces . Our formula simultaneously generalizes the cases of locally compact Hausdorff and coherent (spectral) spaces and recovers several smaller -theory calculations as special instances.
Paper Structure (22 sections, 85 theorems, 82 equations, 1 figure, 2 tables)

This paper contains 22 sections, 85 theorems, 82 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. The proof strategy
  4. Acknowledgements
  5. Preliminaries
  6. Compactly assembled $\infty$-categories
  7. Dualizable stable $\infty$-categories and localizing invariants
  8. Topoi
  9. Locales
  10. Sublocales
  11. Stable local compactness
  12. Stably locally compact topoi
  13. Stably locally compact spaces
  14. Sheaves on stably locally compact spaces
  15. de Groot and Verdier duality for stably compact spaces
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $X$ be a locally compact Hausdorff space. Then where $\mathrm{Sp}$ is the $\infty$-category of spectra and the right-hand side refers to compactly supported sheaf cohomology of $X$ with respect to the local system given by the $K$-theory of the sphere spectrum.

Figures (1)

  • Figure : Ernst Haeckel, Kunstformen der Natur, 1904, plate 71: Stephoidea, Public domain, via Wikimedia Commons

Theorems & Definitions (204)

  • Theorem 1.1: efimov2025ktheorylocalizinginvariantslarge Theorem 6.11, see also krause_nikolaus_puetzstueck Theorem 3.6.1
  • Definition 1.2
  • Theorem 1.3: See Theorem \ref{['maintheoremmain2']}
  • Corollary 1.4: See Corollary \ref{['sheavesequalcosheaves']}
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7: aoki2025sheavesspectrumadjunction
  • Remark 1.8
  • Theorem 1.9: See Theorem \ref{['properandcofilteredexcision']}
  • Remark 1.10
  • ...and 194 more