Three-Standardness of the Maximal Ideal

TL;DR

This paper investigates the concept of n-standardness of the maximal ideal in Cohen-Macaulay local rings, extending previous definitions and results, and explores conditions for three-standardness with applications to invariance of certain lengths.

Contribution

It extends the notion of n-standardness for ideals, particularly analyzing three-standardness in Cohen-Macaulay rings across different characteristics, and applies this to invariance properties of minimal reductions.

Findings

Established conditions for three-standardness in prime characteristic.

Extended results to equicharacteristic zero via reduction methods.

Proved length invariance for minimal reductions of the maximal ideal.

Abstract

We study a notion called $n$-standardness (defined by M. E. Rossi and extended in this paper) of ideals primary to the maximal ideal in a Cohen-Macaulay local ring and some of its consequences. We further study conditions under which the maximal ideal is three-standard, first proving results when the residue field has prime characteristic and then using the method of reduction to prime characteristic to extend the results to the equicharacteristic zero case. As an application, we extend a result due to T. Puthenpurakal and show that a certain length associated to a minimal reduction of the maximal ideal does not depend on the minimal reduction chosen.