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Noncatenary splinters in prime characteristic

S. Loepp, Austyn Simpson

Abstract

We construct a local Noetherian splinter (in fact, a weakly $F$-regular domain) in prime characteristic which is not catenary, which we view as an analogue of a theorem of Ogoma in equal characteristic zero. Moreover, we construct a weakly $F$-regular local UFD which is not Cohen-Macaulay. Both of these examples are obtained via finding sufficient conditions ensuring that a complete local ring of prime characteristic is the completion of some weakly $F$-regular local domain, which we expect to be of independent interest.

Noncatenary splinters in prime characteristic

Abstract

We construct a local Noetherian splinter (in fact, a weakly -regular domain) in prime characteristic which is not catenary, which we view as an analogue of a theorem of Ogoma in equal characteristic zero. Moreover, we construct a weakly -regular local UFD which is not Cohen-Macaulay. Both of these examples are obtained via finding sufficient conditions ensuring that a complete local ring of prime characteristic is the completion of some weakly -regular local domain, which we expect to be of independent interest.
Paper Structure (7 sections, 41 theorems, 25 equations)

This paper contains 7 sections, 41 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Weakly F-regular precompletions
  3. A weakly F-regular UFD which is not Cohen--Macaulay
  4. F-regular UFD precompletions of a complete local Gorenstein ring
  5. A demonstration of Theorem D
  6. Noncatenary prime characteristic splinters
  7. Questions

Key Result

Theorem A

(= theorem:noncatenary) There exists a local splinter of prime characteristic which is not catenary.

Theorems & Definitions (75)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • ...and 65 more