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arXiv · 2609.30851

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 $I\ne0$. 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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Yoshinori Gongyo, Fumiharu Kato, Takayuki Uchiba. 2026-09-25. Non-vanishing of the structure-sheaf cohomology of affine henselian schemes and henselian affinoids. https://arxiv.org/abs/2609.30851

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