What is a system of parameters?

TL;DR

This paper explores generalizations of a theorem characterizing systems of parameters in Noetherian local rings, extending it beyond Cohen-Macaulay cases to broader classes of rings.

Contribution

It removes the Cohen-Macaulay assumption from existing theorems, providing new criteria for systems of parameters in more general Noetherian rings.

Findings

Extended the theorem to non-Cohen-Macaulay rings

Provided new criteria for systems of parameters

Broadened understanding of ring dimension and parameters

Abstract

In this paper we discuss various refinements and generalizations of a theorem of Sankar Dutta and Paul Roberts. Their theorem gives a criterion for $d$ elements in a $d$-dimensional Noetherian Cohen-Macaulay local ring to be a system of parameters, i.e., to have height $d$. We chiefly remove the assumption that the ring be Cohen-Macaulay and discuss similar theorems.