Log syntomic cohomology of truncated polynomials and coordinate axes
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866908687940976640 |
|---|---|
| author | Park, Doosung Østvær, Paul Arne |
| author_facet | Park, Doosung Østvær, Paul Arne |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02886 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Log syntomic cohomology of truncated polynomials and coordinate axes Park, Doosung Østvær, Paul Arne Algebraic Geometry K-Theory and Homology 19D55, 14A21, 14F42 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. |
| title | Log syntomic cohomology of truncated polynomials and coordinate axes |
| topic | Algebraic Geometry K-Theory and Homology 19D55, 14A21, 14F42 |
| url | https://arxiv.org/abs/2512.02886 |