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arXiv · 2512.02886

Log syntomic cohomology of truncated polynomials and coordinate axes

Doosung Park·Paul Arne Østvær
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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.

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Doosung Park, Paul Arne Østvær. 2025-12-02. Log syntomic cohomology of truncated polynomials and coordinate axes. https://arxiv.org/abs/2512.02886

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