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The perfection can be a non-coherent GCD domain

Austyn Simpson

Abstract

We show that there exists a complete local Noetherian normal domain of prime characteristic whose perfection is a non-coherent GCD domain, answering a question of Patankar in the negative concerning characterizations of $F$-coherent rings. This recovers and extends a result of Glaz using tight closure methods.

The perfection can be a non-coherent GCD domain

Abstract

We show that there exists a complete local Noetherian normal domain of prime characteristic whose perfection is a non-coherent GCD domain, answering a question of Patankar in the negative concerning characterizations of -coherent rings. This recovers and extends a result of Glaz using tight closure methods.
Paper Structure (2 sections, 5 theorems, 6 equations)

This paper contains 2 sections, 5 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Results

Key Result

Theorem 1.1

(= cor:gcd-fpure + ex:fpure-ufd-non-CMex:fpure-ufd-non-fregular) There exists a complete local Noetherian normal domain $R$ of prime characteristic $p>0$ such that $R_{\operatorname{perf}}$ is a non-coherent GCD domain.

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • Corollary 2.4
  • proof
  • Example 2.5
  • Example 2.6
  • Corollary 2.7
  • ...and 2 more