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A procdh topology

Shane Kelly, Shuji Saito

TL;DR

This work introduces the procdh topology on Sch_S to capture procdh excision and develops its topos-theoretic consequences: hypercompleteness, finite homotopy dimension, and conservative fibre functors, together with a detailed local-ring description. It then applies this framework to algebraic K-theory and motivic cohomology, proving that non-connective K-theory is the procdh sheafification of connective K-theory and that Elmanto–Morrow’s non-$A^1$ motivic cohomology coincides with procdh sheafifications of Voevodsky’s theory. The paper further develops Nisnevich–Riemann–Zariski spaces, descent theory, and conservativity results essential for hyperdescent arguments, and constructs a procdh motivic complex Z(n)^{procdh} with a spectral sequence converging to K-theory via procdh cohomology. Overall, the procdh approach provides a robust tool for organizing excision phenomena and yields new structural insights for K-theory and motivic cohomology in finite-dimensional Noetherian settings.

Abstract

In this article we propose a definition of a procdh topos. We show that it encodes procdh excision, has bounded homotopy dimension and therefore is hypercomplete and admits a conservative family of fibre functors. We also describe the local rings. As an application, we show that nonconnective $K$-theory is the procdh sheafification of connective $K$-theory, and that the motivic cohomology recently proposed by Elmanto and Morrow is the procdh sheafification of Voevodsky's motivic cohomology.

A procdh topology

TL;DR

This work introduces the procdh topology on Sch_S to capture procdh excision and develops its topos-theoretic consequences: hypercompleteness, finite homotopy dimension, and conservative fibre functors, together with a detailed local-ring description. It then applies this framework to algebraic K-theory and motivic cohomology, proving that non-connective K-theory is the procdh sheafification of connective K-theory and that Elmanto–Morrow’s non- motivic cohomology coincides with procdh sheafifications of Voevodsky’s theory. The paper further develops Nisnevich–Riemann–Zariski spaces, descent theory, and conservativity results essential for hyperdescent arguments, and constructs a procdh motivic complex Z(n)^{procdh} with a spectral sequence converging to K-theory via procdh cohomology. Overall, the procdh approach provides a robust tool for organizing excision phenomena and yields new structural insights for K-theory and motivic cohomology in finite-dimensional Noetherian settings.

Abstract

In this article we propose a definition of a procdh topos. We show that it encodes procdh excision, has bounded homotopy dimension and therefore is hypercomplete and admits a conservative family of fibre functors. We also describe the local rings. As an application, we show that nonconnective -theory is the procdh sheafification of connective -theory, and that the motivic cohomology recently proposed by Elmanto and Morrow is the procdh sheafification of Voevodsky's motivic cohomology.
Paper Structure (20 sections, 50 theorems, 89 equations)

This paper contains 20 sections, 50 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Future work
  3. Notation and conventions
  4. Acknowledgements
  5. Definition of a procdh topology
  6. Local rings
  7. Nisnevich-Riemann-Zariski spaces
  8. Conservativity of the fibre functors
  9. Excision
  10. Homotopy dimension
  11. Valuative dimension
  12. Homotopy dimension
  13. Hypercompleteness
  14. Points of ∞-topoi
  15. Counterexample
  16. ...and 5 more sections

Key Result

Theorem 1.2

Let $S$ be a scheme and consider the following conditions on a presheaf of spaces $F \in \mathrm{PSh}(\mathrm{Sch}_S, \mathcal{S}$).As usual, $\mathcal{S} := N\mathcal{K} an$ is the quasi-category associated to the simplicial category of Kan complexes, and $\mathrm{PSh}(\mathrm{Sch}_S, \mathcal{S})$ If $S$ is qcqs then (Excision) $\Leftrightarrow$ (Čech descent). If $S$ is qcqs, has finite valuati

Theorems & Definitions (139)

  • Definition 1.1: Definition \ref{['defi:procdh']}
  • Theorem 1.2: Theorem \ref{['theo:descentConditions']}, Corollary \ref{['coro:procdhHypercomplete']}, Proposition \ref{['prop:hyperExciPoint']}
  • Theorem 1.3: Theorem \ref{['theo:bounded']}, Example \ref{['exam:counterCohDim']}
  • Corollary 1.4: Corollary \ref{['coro:procdhHypercomplete']}
  • Theorem 1.5: Theorem \ref{['theo:procdhEnoughPoints']}, Corollary \ref{['coro:enoughInfinityPoints']}
  • Proposition 1.6: Proposition \ref{['prop:enoughPoints']}
  • Proposition 1.7: Characterisation of procdh local rings
  • Theorem 1.8: Theorem \ref{['thm;apcdhKconn=K']}
  • Definition 1.9: Definition \ref{['def;Znpcdh']}
  • Theorem 1.10: Theorem \ref{['thm;AH-Kpcdh']}
  • ...and 129 more