Bounds for the first Hilbert coefficients of ${\mathfrak m}$-primary ideals

TL;DR

This paper characterizes Noetherian local rings where the first Hilbert coefficients of m-primary ideals take only finitely many values, providing insights into their structure and examples.

Contribution

It introduces a characterization of rings with finitely many first Hilbert coefficients for m-primary ideals, advancing understanding of their algebraic properties.

Findings

Identification of conditions for finitely many first Hilbert coefficients

Examples illustrating the characterization

Insights into the structure of Noetherian local rings

Abstract

This paper purposes to characterize Noetherian local rings $(A, {\mathfrak m})$ of positive dimension such that the first Hilbert coefficients of ${\mathfrak m}$-primary ideals in $A$ range among only finitely many values. Examples are explored.