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Perfectoid pure thresholds of lifts of rational double points

Teppei Takamatsu, Shou Yoshikawa

Abstract

We study the perfectoid pure threshold with respect to $p$, an invariant of singularities in mixed characteristic $(0,p)$ arising from perfectoid purity. In this paper, we compute perfectoid pure thresholds for lifts of rational double points. We show that the set of such thresholds is contained in $\mathbb{Q}$ and satisfies the ascending chain condition. In characteristic $2$, all reciprocals of positive integers occur, and $0$ is the unique accumulation point.

Perfectoid pure thresholds of lifts of rational double points

Abstract

We study the perfectoid pure threshold with respect to , an invariant of singularities in mixed characteristic arising from perfectoid purity. In this paper, we compute perfectoid pure thresholds for lifts of rational double points. We show that the set of such thresholds is contained in and satisfies the ascending chain condition. In characteristic , all reciprocals of positive integers occur, and is the unique accumulation point.

Paper Structure

This paper contains 12 sections, 20 theorems, 210 equations, 1 table.

Table of Contents

  1. Introduction
  2. Multi-height and perfectoid pure threshold of p
  3. Definition of multi-height and Fedder-type criterion
  4. Variants of Fedder-type criterion
  5. Naive multi-height
  6. Pre-periodic
  7. List of perfectoid pure thresholds of rational double points
  8. On the types D(2n)r and D(2n+1)r
  9. Proof.
  10. Lifts of D(2n)r and D(2n+1)r without the monomial xz
  11. Lifts of D(2n)r and D(2n+1)r with the monomial xz
  12. Rationality and the Ascending Chain Condition

Key Result

Theorem 1.1

Let $d$ be a non-negative integer. Then the following two sets coincide:

Theorems & Definitions (50)

  • Theorem 1.1: Yoshikawa25-cri*Theorem C
  • Theorem A: \ref{['ACC']}
  • Definition 1
  • Definition 2
  • Remark 1
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Proposition 1
  • ...and 40 more