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On deformation of perfectoid purity in Gorenstein domains

Benjamin Baily, Karina Dovgodko, Austyn Simpson, Jack Westbrook

Abstract

If $(R,\mathfrak{m})$ is a complete local ring of mixed characteristic $(0,p)$ and $R/pR$ is an $F$-pure Gorenstein domain, we find a sufficient condition for $R$ to be perfectoid pure. This condition is related to the Cohen-Macaulayness of the absolute integral closures of Gorenstein local domains of mixed characteristic which are not necessarily excellent. Along the way, we show that the problem of lifting $F$-purity of $R/pR$ to perfectoid purity of $R$ is equivalent to a similar deformation problem for the splinter property.

On deformation of perfectoid purity in Gorenstein domains

Abstract

If is a complete local ring of mixed characteristic and is an -pure Gorenstein domain, we find a sufficient condition for to be perfectoid pure. This condition is related to the Cohen-Macaulayness of the absolute integral closures of Gorenstein local domains of mixed characteristic which are not necessarily excellent. Along the way, we show that the problem of lifting -purity of to perfectoid purity of is equivalent to a similar deformation problem for the splinter property.

Paper Structure

This paper contains 5 sections, 14 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Deformation of Gorenstein splinters, modulo Conjecture 1.2
  4. A Heitmann-type completion lemma
  5. Proof of Theorems A and B

Key Result

Theorem A

(= thm:perfd-pure-deformation) conj:CM implies conj:F-pure.

Theorems & Definitions (29)

  • Conjecture 1.1
  • Conjecture 1.2
  • Theorem A
  • Conjecture 1.3
  • Theorem B
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 4.1
  • ...and 19 more