Perfectoid towers and their tilts : with an application to the étale cohomology groups of local log-regular rings

Ishiro, Shinnosuke; Nakazato, Kei; Shimomoto, Kazuma

doi:10.2140/ant.2025.19.2307

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Perfectoid towers and their tilts: with an application to the étale cohomology groups of local log-regular rings

Shinnosuke Ishiro, Kei Nakazato and Kazuma Shimomoto

Vol. 19 (2025), No. 12, 2307–2358

DOI: 10.2140/ant.2025.19.2307

Abstract

To initiate a systematic study on the applications of perfectoid methods to Noetherian rings, we introduce the notions of perfectoid towers and their tilts. We mainly show that the tilting operation preserves several homological invariants and finiteness properties. Using this, we also provide a comparison result on étale cohomology groups under the tilting. As an application, we prove finiteness of the prime-to- $p$ -torsion subgroup of the divisor class group of a local log-regular ring that appears in logarithmic geometry in the mixed characteristic case.

Keywords

perfectoid tower, perfectoid ring, local log-regular ring, Frobenius map, tilting, small tilts, étale cohomology, log-regularity

Mathematical Subject Classification

Primary: 14G45

Secondary: 13B02, 13B40, 13F35, 14A21

Milestones

Received: 9 March 2023

Revised: 8 August 2024

Accepted: 23 December 2024

Published: 20 October 2025

Authors

Shinnosuke Ishiro
National Institute of Technology Gunma College Gunma 371-8530 Japan

Kei Nakazato
Proxima Technology Inc. Tokyo 111-0005 Japan

Kazuma Shimomoto
Department of Mathematics Institute of Science Tokyo Tokyo 152-8551 Japan

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Vol. 19, No. 12, 2025