Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniform arithmetic in local rings via ultraproducts

Clay Adams, Francesca Cantor, Anese Gashi, Semir Mujevic, Sejin Park, Austyn Simpson, Jenna Zomback

Abstract

We reinterpret various properties of Noetherian local rings via the existence of some $n$-ary numerical function satisfying certain uniform bounds. We provide such characterizations for seminormality, weak normality, generalized Cohen-Macaulayness, and $F$-purity, among others. Our proofs that such numerical functions exist are nonconstructive and rely on the transference of the property in question from a local ring to its ultrapower or catapower.

Uniform arithmetic in local rings via ultraproducts

Abstract

We reinterpret various properties of Noetherian local rings via the existence of some -ary numerical function satisfying certain uniform bounds. We provide such characterizations for seminormality, weak normality, generalized Cohen-Macaulayness, and -purity, among others. Our proofs that such numerical functions exist are nonconstructive and rely on the transference of the property in question from a local ring to its ultrapower or catapower.
Paper Structure (10 sections, 19 theorems, 42 equations)

This paper contains 10 sections, 19 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Sketch of proof technique
  3. Notational conventions
  4. Preliminaries
  5. Background on ultraproducts
  6. Invariants of a (Quasi-)Local Ring
  7. Variants of Normality
  8. F-singularities
  9. Variants of the Cohen--Macaulay property
  10. Questions

Key Result

Theorem 1.1

Ree61 Let $(R, \mathfrak{m})$ be a Noetherian local ring. Then its $\mathfrak{m}$-adic completion $\widehat{R}$ is a domain if and only if there exists a binary numerical function $\varphi_R$ such that for all $x, y\in R$ the following inequality holds:

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Ł oś's Theorem
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • ...and 30 more