Comparison of Hyodo-Kato and de Rham Fargues-Fontaine Cohomology Theories
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916959028772864 |
|---|---|
| author | Cao, Kaixing |
| author_facet | Cao, Kaixing |
| contents | We prove that, for adic étale motives over $\mathbb{C}_p$, the vector bundles on the Fargues-Fontaine curve arising from their Hyodo-Kato cohomology coincide with their de Rham-Fargues-Fontaine cohomologies, where the latter provides an overconvergent refinement of crystalline vector bundles, albeit constructed on the generic fiber. This equivalence is established in the setting of symmetric monoidal $\infty$-categories and respects the full motivic structure. Furthermore, we enrich both realizations with Galois actions, yielding $G_{\breve{\mathbb{Q}}_{p}}$-equivariant solid quasi-coherent sheaves on the Fargues-Fontaine curve; in this equivariant context, the comparison isomorphism becomes canonical. As an application, we show that the de Rham-Fargues-Fontaine cohomology of any smooth quasi-compact rigid analytic variety over $\mathbb{C}_p$ admits a finite slope-increasing filtration. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_16726 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Comparison of Hyodo-Kato and de Rham Fargues-Fontaine Cohomology Theories Cao, Kaixing Algebraic Geometry Number Theory We prove that, for adic étale motives over $\mathbb{C}_p$, the vector bundles on the Fargues-Fontaine curve arising from their Hyodo-Kato cohomology coincide with their de Rham-Fargues-Fontaine cohomologies, where the latter provides an overconvergent refinement of crystalline vector bundles, albeit constructed on the generic fiber. This equivalence is established in the setting of symmetric monoidal $\infty$-categories and respects the full motivic structure. Furthermore, we enrich both realizations with Galois actions, yielding $G_{\breve{\mathbb{Q}}_{p}}$-equivariant solid quasi-coherent sheaves on the Fargues-Fontaine curve; in this equivariant context, the comparison isomorphism becomes canonical. As an application, we show that the de Rham-Fargues-Fontaine cohomology of any smooth quasi-compact rigid analytic variety over $\mathbb{C}_p$ admits a finite slope-increasing filtration. |
| title | Comparison of Hyodo-Kato and de Rham Fargues-Fontaine Cohomology Theories |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2509.16726 |