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Some properties of ideals in Cohen-Macaulay local rings

Richard F. Bartels

Abstract

For a Cohen-Macaulay local ring $(R,\mathfrak{m})$ with canonical module, we study how relations between $\text{index}(R)$ and $\text{g}\ell\ell(R)$ and between $\text{index}(R)$ and $e(R)$ are preserved when factoring out regular sequences and localizing at prime ideals. We then give conditions for when ideals in a one-dimensional Cohen-Macaulay local ring are Elias and Burch, and use these conditions to study the relationship between Elias, Burch, and Ulrich ideals.

Some properties of ideals in Cohen-Macaulay local rings

Abstract

For a Cohen-Macaulay local ring with canonical module, we study how relations between and and between and are preserved when factoring out regular sequences and localizing at prime ideals. We then give conditions for when ideals in a one-dimensional Cohen-Macaulay local ring are Elias and Burch, and use these conditions to study the relationship between Elias, Burch, and Ulrich ideals.

Paper Structure

This paper contains 3 sections, 27 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Invariants of Cohen-Macaulay local rings
  3. Criteria for Elias and Burch ideals

Key Result

Lemma 2.2

Let $R$ and $S$ be rings. Let $\varphi:R \longrightarrow S$ be a ring map and $M$ an $R$-module. Then

Theorems & Definitions (69)

  • Definition 2.1
  • Lemma 2.2: 13, Lemma 1.5
  • Proposition 2.3: 9, Propositions 2.4
  • Corollary 2.4
  • proof
  • Definition 2.5: 17, page 114
  • Definition 2.6: 10, page 1
  • Corollary 2.7
  • proof
  • Lemma 2.8: 9, Lemmas 1.7 and 2.2
  • ...and 59 more