Some properties of ideals in Cohen-Macaulay local rings

TL;DR

This paper investigates properties of ideals in Cohen-Macaulay local rings, focusing on how certain invariants behave under algebraic operations and characterizing special types of ideals like Elias, Burch, and Ulrich.

Contribution

It introduces new conditions for ideals in one-dimensional Cohen-Macaulay rings to be Elias or Burch and explores their relationships with Ulrich ideals.

Findings

Relations between index, Gorenstein length, and multiplicity are preserved under regular sequences and localization.

Conditions for ideals to be Elias or Burch are established in one-dimensional Cohen-Macaulay rings.

The study clarifies the connections among Elias, Burch, and Ulrich ideals.

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.