Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algebraic $K$-theory of coherent spaces

Georg Lehner

TL;DR

This work develops a universal, motive-based framework for computing finitary localizing invariants, notably algebraic K-theory, of ∞-categories of sheaves on locally coherent spaces. By leveraging Stone duality with lower bounded distributive lattices and introducing the spectrum of motives $\mathcal{M}(D)$, the authors reduce complex K-theory computations to finite combinatorial data via $M(D)$ and valuations, and they extend known results from locally compact spaces to the broader class of locally coherent spaces. A key outcome is the main formula $F^{cont}(\mathrm{Sh}(X,\mathcal{C})) \simeq \mathcal{M}(\mathcal{K}^o(X)) \otimes F^{cont}(\mathcal{C})$, which specializes to $\pi_n F^{cont}(\mathrm{Sh}(X,\mathcal{C})) \cong M(\mathcal{K}^o(X)) \otimes_{\mathbb{Z}} \pi_n(F^{cont}(\mathcal{C}))$ when $F$ takes values in spectra. The paper then applies this framework to scissors-congruence $K$-theory and to K-theory of measure spaces, revealing deep links between THH of dualizable ∞-categories and classical invariants in geometry and measure theory. Overall, the results provide a cohesive, computable bridge between abstract ∞-category theory and concrete invariants in geometry, topology, and analysis. This ushers in a pathway to understand K-theory of a wide class of spaces through motive spectra and valuation-theoretic data.

Abstract

We give a description of the value of a finitary localizing invariant, such as algebraic $K$-theory, on the category of sheaves on a locally coherent space $X$. This in particular includes all spaces that arise as spectra of commutative rings. As applications we discuss the connection between scissors congruence $K$-theory and Topological Hochschild Homology of certain locally coherent spaces, as well as the algebraic $K$-theory of a measure space.

Algebraic $K$-theory of coherent spaces

TL;DR

This work develops a universal, motive-based framework for computing finitary localizing invariants, notably algebraic K-theory, of ∞-categories of sheaves on locally coherent spaces. By leveraging Stone duality with lower bounded distributive lattices and introducing the spectrum of motives , the authors reduce complex K-theory computations to finite combinatorial data via and valuations, and they extend known results from locally compact spaces to the broader class of locally coherent spaces. A key outcome is the main formula , which specializes to when takes values in spectra. The paper then applies this framework to scissors-congruence -theory and to K-theory of measure spaces, revealing deep links between THH of dualizable ∞-categories and classical invariants in geometry and measure theory. Overall, the results provide a cohesive, computable bridge between abstract ∞-category theory and concrete invariants in geometry, topology, and analysis. This ushers in a pathway to understand K-theory of a wide class of spaces through motive spectra and valuation-theoretic data.

Abstract

We give a description of the value of a finitary localizing invariant, such as algebraic -theory, on the category of sheaves on a locally coherent space . This in particular includes all spaces that arise as spectra of commutative rings. As applications we discuss the connection between scissors congruence -theory and Topological Hochschild Homology of certain locally coherent spaces, as well as the algebraic -theory of a measure space.

Paper Structure

This paper contains 18 sections, 60 theorems, 119 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Acknowledgements
  4. Locales and Grothendieck topologies on posets
  5. Coherent spaces via finitary Grothendieck topologies
  6. Locally coherent spaces
  7. Boolean algebras and profiniteness
  8. Generalities on sheaves and higher topoi
  9. Alexandroff spaces and Birkhoff's theorem
  10. Valuations and motives for distributive lattices
  11. Presentable, compactly generated and dualizable $\infty$-categories
  12. Dualizable $\infty$-categories
  13. Localizing invariants and semiorthogonal decompositions
  14. The ring spectrum of motives
  15. Algebraic $K$-theory of (locally) coherent spaces
  16. ...and 3 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 91: Spumellaria, Public domain, via Wikimedia Commons

Theorems & Definitions (126)

  • 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{['sheavesfiltereddualizablelowerbounded']}
  • Corollary 1.4
  • Theorem 1.5: See Theorem \ref{['ktheorycoherent']} and Corollary \ref{['ktheorycoherentvaluesspectra']}
  • Definition 1.6
  • Theorem 1.7: See Theorem \ref{['spaceofidempotents']}
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 116 more