The reduction number of stretched ideals

TL;DR

This paper investigates the homological properties of associated graded rings of stretched m-primary ideals in Cohen-Macaulay local rings, focusing on cases with minimal or near-minimal reduction numbers, and provides detailed descriptions for these cases.

Contribution

It characterizes the almost Cohen-Macaulayness of associated graded rings of stretched ideals with small reduction numbers, extending understanding of their structure in Cohen-Macaulay rings.

Findings

Associated graded rings are almost Cohen-Macaulay under certain conditions.

Complete descriptions of these rings are provided for small reduction numbers.

Results apply to stretched m-primary ideals in Cohen-Macaulay local rings.

Abstract

The homological property of the associated graded ring of an ideal is an important problem in commutative algebra and algebraic geometry. In this paper we explore the almost Cohen-Macaulayness of the associated graded ring of stretched $\mathfrak{m}$-primary ideals in the case where the reduction number attains almost minimal value in a Cohen-Macaulay local ring $(A,\mathfrak{m})$. As an application, we present complete descriptions of the associated graded ring of stretched $\mathfrak{m}$-primary ideals with small reduction number.