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arXiv · 2510.23959

Grothendieck topologies with logarithmic modifications

Xianyu Hu·Maximilian Schimpf
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Abstract

Many concepts in log geometry are invariant under log blow-ups. To formalize this invariance, we introduce the m-open, m-\'etale, m-smooth, m-fppf, and m-fpqc topologies for fs log schemes. These refine the standard topologies from scheme theory by treating abstract log modifications as covers. For example, the m-\'etale topology is a subtopology of full log \'etale topology, characterized by a stronger lifting property than for log \'etale maps. Along the way, we identify and correct an error in the definition of the full log \'etale topology. We also prove a global integralization theorem by logarithmic blow-ups and use it to describe the m-open topos as a limit over log blow-ups. Finally, we characterize the sheaves on the m-type sites and connect the m-open site to Kato's valuative space.

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Xianyu Hu, Maximilian Schimpf. 2025-10-28. Grothendieck topologies with logarithmic modifications. https://arxiv.org/abs/2510.23959

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