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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, Kazuma Shimomoto

TL;DR

The paper develops a tower-theoretic framework to import perfectoid methods into Noetherian ring theory by defining perfectoid towers and their tilts. It proves that tilting preserves several finiteness and homological invariants and enables a comparison of étale cohomology between mixed and equal characteristic settings, particularly via a tilt-compatible base-change formalism. A key application shows finiteness of the prime-to-$p$ torsion in the divisor class group for local log-regular rings in mixed characteristic, built from a tilt-based analysis and strong $F$-regularity arguments. Overall, the work bridges Noetherian, log-regular, and perfectoid techniques, providing a Noetherian analogue of the tilting correspondence and new finiteness results for invariants of singularities with log structures.

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.

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

TL;DR

The paper develops a tower-theoretic framework to import perfectoid methods into Noetherian ring theory by defining perfectoid towers and their tilts. It proves that tilting preserves several finiteness and homological invariants and enables a comparison of étale cohomology between mixed and equal characteristic settings, particularly via a tilt-compatible base-change formalism. A key application shows finiteness of the prime-to- torsion in the divisor class group for local log-regular rings in mixed characteristic, built from a tilt-based analysis and strong -regularity arguments. Overall, the work bridges Noetherian, log-regular, and perfectoid techniques, providing a Noetherian analogue of the tilting correspondence and new finiteness results for invariants of singularities with log structures.

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--torsion subgroup of the divisor class group of a local log-regular ring that appears in logarithmic geometry in the mixed characteristic case.
Paper Structure (26 sections, 62 theorems, 83 equations)

This paper contains 26 sections, 62 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Log-regularity
  3. Preliminaries on monoids
  4. Basic terms
  5. $c$-times maps on integral monoids
  6. Local log-regular rings
  7. Definition of local log-regular rings
  8. Log-regularity and strong $F$-regularity
  9. Log-regularity and splinters
  10. Perfectoid towers and small tilts
  11. Perfect towers
  12. Purely inseparable towers and inverse perfection
  13. Axioms for perfectoid towers
  14. Remarks on torsion
  15. Perfectoid towers and pillars
  16. ...and 11 more sections

Key Result

Proposition 2.8

Let $\mathcal{Q}$ be an integral monoid, and let $\mathcal{Q}'\subseteq \mathcal{Q}$ be a submonoid. Let $\theta: \mathcal{Q}'\hookrightarrow \mathcal{Q}$ be the inclusion map, and let $\mathbb{Z}[\theta]: \mathbb{Z}[\mathcal{Q}']\to \mathbb{Z}[\mathcal{Q}]$ be the induced ring map. Set $G:=\mathcal

Theorems & Definitions (167)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: $\mathcal{Q}$-module
  • Definition 2.4: Monoid algebras
  • Definition 2.5
  • Definition 2.6: Exact homomorphisms
  • Definition 2.7
  • Proposition 2.8: cf. Ogus18
  • proof
  • Remark 2.9
  • ...and 157 more