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$F$-pure and $F$-injective singularities in equal characteristic zero

Tatsuki Yamaguchi

TL;DR

The paper develops ultra-$F$-purity and ultra-$F$-injectivity as equal-characteristic-zero analogues obtained via ultraproducts to study descent of dense $F$-pure type and dense $F$-injective type under pure morphisms. It introduces the ultra-perfect closure and $p$-standard sequences to bridge characteristic $p$ methods to characteristic zero, enabling a descent framework for log canonicity and Du Bois-type phenomena. The main results show that, under a pure (resp. strongly pure) local morphism $R\to S$ between reduced local rings essentially of finite type over $\mathbb{C}$ with $R$ $\mathbb{Q}$-Gorenstein, density of $F$-purity (resp. $F$-injectivity) in $S$ transfers to $R$ for corresponding pairs $(R,\mathfrak{a}^t)$; in the $F$-injective setting, additional purity conditions are required. This provides a nonstandard analytic bridge linking positive-characteristic singularities to zero-characteristic analogues, with implications for descent of canonical singularities and connections to Du Bois theory.

Abstract

Inspired by Schoutens' results, we introduce a variant of sharp $F$-purity and sharp $F$-injectivity in equal characteristic zero via ultraproducts. As an application, we show that if $R\to S$ is pure and $S$ is of dense $F$-pure type, then $R$ is of dense $F$-pure type.

$F$-pure and $F$-injective singularities in equal characteristic zero

TL;DR

The paper develops ultra--purity and ultra--injectivity as equal-characteristic-zero analogues obtained via ultraproducts to study descent of dense -pure type and dense -injective type under pure morphisms. It introduces the ultra-perfect closure and -standard sequences to bridge characteristic methods to characteristic zero, enabling a descent framework for log canonicity and Du Bois-type phenomena. The main results show that, under a pure (resp. strongly pure) local morphism between reduced local rings essentially of finite type over with -Gorenstein, density of -purity (resp. -injectivity) in transfers to for corresponding pairs ; in the -injective setting, additional purity conditions are required. This provides a nonstandard analytic bridge linking positive-characteristic singularities to zero-characteristic analogues, with implications for descent of canonical singularities and connections to Du Bois theory.

Abstract

Inspired by Schoutens' results, we introduce a variant of sharp -purity and sharp -injectivity in equal characteristic zero via ultraproducts. As an application, we show that if is pure and is of dense -pure type, then is of dense -pure type.
Paper Structure (7 sections, 26 theorems, 47 equations)

This paper contains 7 sections, 26 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $F$-pure and $F$-injective singularities
  4. Ultraproducts
  5. $\operatorname{p}$-standard sequences and ultraproducts
  6. Ultra-$F$-purity
  7. Ultra-$F$-injectivity

Key Result

Theorem 1.1

Let $R\to S$ be a pure local $\mathbb{C}$-algebra homomorphism between reduced local rings essentially of finite type over $\mathbb{C}$, $\mathfrak{a}$ be a nonzero ideal of $R$ and $t$ be a positive real number. Suppose that $R$ is $\mathbb{Q}$-Gorenstein normal and $(S,(\mathfrak{a} S)^t)$ is of d

Theorems & Definitions (82)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3: HR76, Schw08, Tak04
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Proposition 2.7
  • proof
  • ...and 72 more