Variation of the first Hilbert coefficients of parameters with a common integral closure

TL;DR

This paper investigates how the first Hilbert coefficients of parameter ideals sharing the same integral closure vary within a local ring, providing both positive results and counterexamples.

Contribution

It offers new insights into the variation of Hilbert coefficients for parameter ideals with a common integral closure, addressing a problem posed by Vasconcelos.

Findings

Affirmative results on the variation of Hilbert coefficients

Counterexamples illustrating limitations

Enhanced understanding of parameter ideals with shared integral closure

Abstract

A problem posed by Wolmer V. Vasconcelos on the variation of the first Hilbert coefficients of parameter ideals with a common integral closure in a local ring is studied. Affirmative answers are given and counterexamples are explored as well.