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Construction of logarithmic cohomology theories I

Doosung Park

TL;DR

This work develops a coherent framework to construct and study cohomology theories for logarithmic schemes with boundary, by building logarithmic motivic homotopy theory over the field with one element $\mathbb{F}_1$ and introducing a logarithm functor that extends presheaves from schemes to fs log schemes. It shows that fundamental invariants such as algebraic K-theory, topological Hochschild homology, and topological cyclic homology admit natural logarithmic extensions and, in many cases, are representable in the logarithmic motivic category, notably via a log version of the Snaith theorem and a log cyclotomic trace. The paper also constructs a robust theory of log motives over $\mathbb{F}_1$, establishes equivalences between log and non-log frameworks through dividing covers and admissible blow-ups, and proves several descent and purity results that enable practical computations, including connections to Nygaard prisms, syntomic cohomology, and log motivic filtrations. Overall, the results provide a unified, representability-driven approach to logarithmic cohomology theories with potential applications to K-theory computations for non-regular schemes and to log-geometry in a motivic setting, tying together Suslin functors, logarithmic THH/TC, and cyclotomic traces within a coherent $\mathbb{F}_1$-based framework.

Abstract

We propose a method for constructing cohomology theories of logarithmic schemes with strict normal crossing boundaries by employing techniques from logarithmic motivic homotopy theory over $\mathbb{F}_1$. This method recovers the K-theory of the open complement of a strict normal crossing divisor from the K-theory of schemes as well as logarithmic topological Hochschild homology from the topological Hochschild homology of schemes. In our applications, we establish that the K-theory of non-regular schemes is representable in the logarithmic motivic homotopy category, and we introduce the logarithmic cyclotomic trace for the regular log regular case.

Construction of logarithmic cohomology theories I

TL;DR

This work develops a coherent framework to construct and study cohomology theories for logarithmic schemes with boundary, by building logarithmic motivic homotopy theory over the field with one element and introducing a logarithm functor that extends presheaves from schemes to fs log schemes. It shows that fundamental invariants such as algebraic K-theory, topological Hochschild homology, and topological cyclic homology admit natural logarithmic extensions and, in many cases, are representable in the logarithmic motivic category, notably via a log version of the Snaith theorem and a log cyclotomic trace. The paper also constructs a robust theory of log motives over , establishes equivalences between log and non-log frameworks through dividing covers and admissible blow-ups, and proves several descent and purity results that enable practical computations, including connections to Nygaard prisms, syntomic cohomology, and log motivic filtrations. Overall, the results provide a unified, representability-driven approach to logarithmic cohomology theories with potential applications to K-theory computations for non-regular schemes and to log-geometry in a motivic setting, tying together Suslin functors, logarithmic THH/TC, and cyclotomic traces within a coherent -based framework.

Abstract

We propose a method for constructing cohomology theories of logarithmic schemes with strict normal crossing boundaries by employing techniques from logarithmic motivic homotopy theory over . This method recovers the K-theory of the open complement of a strict normal crossing divisor from the K-theory of schemes as well as logarithmic topological Hochschild homology from the topological Hochschild homology of schemes. In our applications, we establish that the K-theory of non-regular schemes is representable in the logarithmic motivic homotopy category, and we introduce the logarithmic cyclotomic trace for the regular log regular case.

Paper Structure

This paper contains 20 sections, 121 theorems, 318 equations.

Table of Contents

  1. Introduction
  2. Logarithmic extensions of non-A1-invariant cohomology theories
  3. Logarithmic extensions of A1-invariant cohomology theories
  4. Suslin functors in non-A1-invariant motivic homotopy theories
  5. Logarithmic motivic homotopy theory over F1
  6. Logarithm functor
  7. Motivic representability of K-theory
  8. Log cyclotomic trace
  9. Logarithmic motivic homotopy theory over F1
  10. Suslin functor
  11. Logarithm functor: Case of monoid schemes
  12. Logarithm functor: Case of schemes
  13. Logarithm of A1-invariant sheaves
  14. Logarithmic sheaves of spectra
  15. Motivic representability of logarithmic K-theory
  16. ...and 5 more sections

Key Result

Theorem A

Let $S\in \mathrm{Sch}$ and $q\in \mathbb{N}$. Then the Nisnevich sheaves of spectra on $\mathrm{Sm}/S$ are logarithmic. Moreover, if $S=\mathop{\mathrm{Spec}}\nolimits(k)$ with a perfect field $k$, then the Nisnevich sheaf of weight $q$ motivic cohomology complexes $R\Gamma_\mathrm{mot}(-,\mathbb{Z}(q))$ on $\mathrm{Sm}/k$ is also logarithmic.

Theorems & Definitions (342)

  • Theorem A: See Theorems \ref{['boundarify.31']}, \ref{['boundarify.6']}, \ref{['boundarify.21']}, and \ref{['boundarify.23']}
  • Theorem B: See Theorem \ref{['boundarify.20']}
  • Theorem C: See Theorem \ref{['boundarify.19']}
  • Theorem D: See Theorem \ref{['boundarify.34']}
  • Theorem E: See Theorem \ref{['boundarify.29']}
  • Theorem F: See Theorem \ref{['Trace.2']}
  • Theorem G: See Theorem \ref{['trace.8']}
  • Definition 2.1: cf. Arndt
  • Remark 2.2
  • Remark 2.3
  • ...and 332 more