Log syntomic cohomology of truncated polynomials and coordinate axes
Doosung Park, Paul Arne Østvær
TL;DR
This work develops logarithmic versions of prismatic and syntomic cohomology within the logarithmic motivic framework. It introduces log animated rings and the notion of log Cartier smoothness, proves saturated descent for free log structures, and establishes representability of key log cohomology theories in logSH(pt_N). It also provides explicit calculations for the log syntomic cohomology of truncated polynomial algebras and the projective log coordinate axes, and analyzes logTC via syntomic inputs. The results unify divided powers, connections, and derived de Rham cohomology with motivic realizations, extending classical p-adic cohomology to semistable and log-smooth settings with concrete computational tools.
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 $\mathrm{logTHH}$, $\mathrm{logTC}$, $\widehat{\mathbfΔ}$, 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.