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Non-vanishing of the structure-sheaf cohomology of affine henselian schemes and henselian affinoids

  • Yoshinori Gongyo
  • Fumiharu Kato
  • Takayuki Uchiba
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Abstract

Let (A,I) be a henselian pair such that A is a domain of characteristic zero and I≠0. We prove that any flat affine henselian scheme of finite presentation over (A,I) has nonzero first structure-sheaf cohomology whenever one of its fibers over a point in V(I) has positive dimension. This in particular gives a negative answer to the question by Devadas in the mixed-characteristic case. As an application, we also prove that the first structure-sheaf cohomology does not vanish for henselian affinoids of positive dimensions.

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Cite

BibTeXRIS
BibTeX
@misc{gongyo2026nonvanishing,
  title = {{Non-vanishing of the structure-sheaf cohomology of affine henselian schemes and henselian affinoids}},
  author = {Yoshinori Gongyo and Fumiharu Kato and Takayuki Uchiba},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30851},
  eprint = {2609.30851},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30851},
}
RIS
TY  - UNPB
TI  - Non-vanishing of the structure-sheaf cohomology of affine henselian schemes and henselian affinoids
AU  - Yoshinori Gongyo
AU  - Fumiharu Kato
AU  - Takayuki Uchiba
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.30851
UR  - https://arxiv.org/abs/2609.30851
AB  - Let (A,I) be a henselian pair such that A is a domain of characteristic zero and I≠0. We prove that any flat affine henselian scheme of finite presentation over (A,I) has nonzero first structure-sheaf cohomology whenever one of its fibers over a point in V(I) has positive dimension. This in particular gives a negative answer to the question by Devadas in the mixed-characteristic case. As an application, we also prove that the first structure-sheaf cohomology does not vanish for henselian affinoids of positive dimensions.
ER  -