Log syntomic cohomology of truncated polynomials and coordinate axes

TL;DR

This paper advances the understanding of logarithmic syntomic cohomology for log schemes, proving descent properties, representability, and computing specific cases like coordinate axes, with applications to topological cyclic homology.

Contribution

It establishes descent and representability results for logarithmic syntomic cohomology and computes explicit examples for coordinate axes and truncated polynomials.

Findings

Proves saturated descent for logarithmic prismatic and syntomic cohomology under free log structures.

Shows certain presheaves are representable and invariant in the logarithmic motivic stable homotopy category.

Computes syntomic cohomology for projective log coordinate axes and determines logarithmic topological cyclic homology for specific examples.

Abstract

We study the logarithmic syntomic cohomology of fine and saturated log schemes and its realization in the logarithmic motivic stable homotopy category $\mathrm{logSH}(\mathrm{pt}_\mathbb{N})$ of a log point. We prove that logarithmic prismatic and syntomic cohomology satisfy saturated descent under the sole assumption that the log structure is free, and that the presheaves $\mathrm{logTHH}$, $\mathrm{logTC}$, $\widehat{\mathbf{\Delta}}$, and $\mathbb{Z}_p^\mathrm{syn}(i)$ are representable and $\square$-invariant in $\mathrm{logSH}_{\mathrm{k\acute{e}t}}^{\mathrm{eff}}(\mathrm{pt}_\mathrm{N})$. As an application, we compute $\mathbb{Z}_p^\mathrm{syn}(i)$ for the projective log coordinate axes $D$ in $\mathbb{P}^2$, obtaining \[ \mathbb{Z}_p^\mathrm{syn}(i)(D) \simeq \mathbb{Z}_p^\mathrm{syn}(i)(k,\mathbb{N})\oplus \mathbb{Z}_p^\mathrm{syn}(i-1)(k,\mathbb{N})[-2] \] Moreover, we determine logarithmic topological cyclic homology for truncated polynomial and semistable examples, directly from the syntomic calculations.