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An application of Fontaine's monoidal maps to perfectoid towers

Kazuki Hayashi, Shinnosuke Ishiro, Kazuma Shimomoto

Abstract

To connect arithmetic and ring-theoretic properties of rings of mixed characteristic with those of positive characteristic, we introduce monoidal maps for perfectoid towers. Using these maps, we discuss the almost integrality of perfectoid towers and of their tilts. We also show that the towers constructed by F. Andreatta via ramification theory become perfectoid towers, and we apply the monoidal maps to deduce the normality of their small tilts.

An application of Fontaine's monoidal maps to perfectoid towers

Abstract

To connect arithmetic and ring-theoretic properties of rings of mixed characteristic with those of positive characteristic, we introduce monoidal maps for perfectoid towers. Using these maps, we discuss the almost integrality of perfectoid towers and of their tilts. We also show that the towers constructed by F. Andreatta via ramification theory become perfectoid towers, and we apply the monoidal maps to deduce the normality of their small tilts.
Paper Structure (10 sections, 36 theorems, 43 equations)

This paper contains 10 sections, 36 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Almost integrality in cartesian diagrams
  3. Monoidal maps for perfectoid towers
  4. Review on Fontaine's monoidal maps
  5. Perfectoid towers
  6. Monoidal maps for perfectoid towers
  7. Perfectoid towers arising from ramification theory
  8. Construction
  9. Proof of the surjectivity of Frobenius projections
  10. Flatness of completions

Key Result

Theorem 1.1

Let $\{R_i,t_i\}_{i\geq 0}$ be a perfectoid tower arising from some pair $(R,f_0)$ with $f_0\in R$. If $R$ is $f_0$-torsion free, then the following conditions are equivalent. Moreover, if $\{R_i,t_i\}_{i\geq 0}$ satisfies these conditions, then so does the tilt $\{R_i^{s.\flat},t_i^{s.\flat}\}_{i\geq 0}$.

Theorems & Definitions (86)

  • Theorem 1.1: \ref{['lem:citower', 'tiltingintegral']}
  • Theorem 1.2: \ref{['thm:Sn perfectoid', 'cor:B']}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • ...and 76 more