TR with logarithmic poles and the de Rham-Witt complex

TL;DR

This paper explores the relationship between log topological restriction homology and the log de Rham--Witt complex for smooth schemes over p-adic rings, focusing on p-completely smooth formal schemes over p-adic complex fields.

Contribution

It extends conjectures relating TR and de Rham--Witt complexes to the setting of p-completely smooth formal schemes over p-adic complex fields, and studies the motivic filtration and homotopy fixed points.

Findings

Confirmed conjectural relations in the p-adic formal scheme setting.

Analyzed the motivic filtration of log TR and its S^1-homotopy fixed points.

Provided new insights into the structure of log TR in p-adic geometry.

Abstract

In the article of Hesselholt [Hes05], a set of conjectures is laid out. Given a smooth scheme $X$ over the ring of integers $\mathcal{O}_K$ of a $p$-adic field $K$, these conjectures concern the expected relation between log topological restriction homology $\mathrm{TR}^r (X,M_X)$ and the absolute log de Rham--Witt complex $W_r\Omega_{(X,M_X)}$. In this note, which is companion to [And24a], we discuss the case of a $p$-completely smooth $p$-adic formal scheme $X$ over $\mathrm{spf} \mathcal{O}_C$, where $C$ is the field of $p$-adic complex numbers. Along the way, we study the motivic filtration of log $\mathrm{TR}^r$ and its $S^1$-homotopy fixed points, following ideas of [Bin+23].