Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the $p$-adic deformation problem for the $K$-theory of semistable schemes

Federico Binda, Tommy Lundemo, Alberto Merici, Doosung Park

TL;DR

The paper addresses the p-adic deformation problem for the algebraic K-theory of semistable schemes by establishing a semistable Beilinson–Bloch–Esnault–Kerz fiber square that relates K-theory to logarithmic TC. It shows that obstructions to lifting K-theory classes are governed by the Hyodo–Kato Chern character, yielding a precise lifting criterion in terms of Hyodo–Kato–filtered de Rham cohomology, and extends the framework to higher K-groups via a logarithmic trace approach. A key thematic achievement is a purely K-theoretic proof of Yamashita’s semistable p-adic Lefschetz (1,1) theorem, together with a characteristic-zero analogue using relative logarithmic de Rham data. The methods combine log geometry, HKR-type filtrations, and trace techniques to connect K-theory with Hyodo–Kato cohomology, providing new insight into semistable degeneration and p-adic deformation phenomena with potential for further logarithmic trace developments.

Abstract

We establish a semistable generalization of the Beilinson-Bloch-Esnault-Kerz fiber square, relating the algebraic K-theory of a semistable scheme to its logarithmic topological cyclic homology. We prove that the obstruction to lifting K-theory classes is governed by the Hyodo-Kato Chern character. This answers the $p$-adic deformation problem for continuous K-theory in the semistable case, extending the work of Antieau-Mathew-Morrow-Nikolaus. As an application, we provide a purely K-theoretic proof of Yamashita's semistable $p$-adic Lefschetz $(1,1)$-theorem.

On the $p$-adic deformation problem for the $K$-theory of semistable schemes

TL;DR

The paper addresses the p-adic deformation problem for the algebraic K-theory of semistable schemes by establishing a semistable Beilinson–Bloch–Esnault–Kerz fiber square that relates K-theory to logarithmic TC. It shows that obstructions to lifting K-theory classes are governed by the Hyodo–Kato Chern character, yielding a precise lifting criterion in terms of Hyodo–Kato–filtered de Rham cohomology, and extends the framework to higher K-groups via a logarithmic trace approach. A key thematic achievement is a purely K-theoretic proof of Yamashita’s semistable p-adic Lefschetz (1,1) theorem, together with a characteristic-zero analogue using relative logarithmic de Rham data. The methods combine log geometry, HKR-type filtrations, and trace techniques to connect K-theory with Hyodo–Kato cohomology, providing new insight into semistable degeneration and p-adic deformation phenomena with potential for further logarithmic trace developments.

Abstract

We establish a semistable generalization of the Beilinson-Bloch-Esnault-Kerz fiber square, relating the algebraic K-theory of a semistable scheme to its logarithmic topological cyclic homology. We prove that the obstruction to lifting K-theory classes is governed by the Hyodo-Kato Chern character. This answers the -adic deformation problem for continuous K-theory in the semistable case, extending the work of Antieau-Mathew-Morrow-Nikolaus. As an application, we provide a purely K-theoretic proof of Yamashita's semistable -adic Lefschetz -theorem.
Paper Structure (19 sections, 28 theorems, 71 equations)

This paper contains 19 sections, 28 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. The fiber square for log rings
  3. $p$-adic deformation of K-theory classes and Hyodo-Kato cohomology
  4. A semistable $p$-adic Lefschetz (1,1)-Theorem
  5. A characteristic zero story
  6. Notation
  7. Acknowledgments
  8. A logarithmic Bloch--Esnault--Kerz square
  9. Vertical log structures
  10. Removing vertical log structures
  11. From absolute to relative
  12. Involving K-theory
  13. The Hyodo-Kato Chern class
  14. Hyodo--Kato cohomology
  15. Chern orientations
  16. ...and 4 more sections

Key Result

Theorem A

Let $(R, P)$ be an ${\mathcal{O}}_K^\sharp$-algebra which is vertical, regular, and log regular. If $R$ is Henselian along $(p)$, then there exists a natural cartesian square \begin{tikzcd} K(R;\Q_p)\ar[d]\ar[r]& K(R/p;\Q_p)\ar[d] \\ \HC^-((R,P)/\cO_K^\sharp;\Q_p)\ar[r]& \HP((R,P)/\cO_K^\sharp;\Q_p)

Theorems & Definitions (64)

  • Theorem A: See Theorem \ref{['thm:beilfibernologktheory']}
  • Theorem B: See Theorem \ref{['thm:hkchern2']}
  • Theorem C: See Theorem \ref{['prop:old-thm-A']}
  • Definition 2.2
  • Example 2.3
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • Proposition 2.8
  • ...and 54 more