The first Hilbert coefficient of stretched ideals

TL;DR

This paper investigates the algebraic properties of stretched $rak{m}$-primary ideals with small first Hilbert coefficient in Cohen-Macaulay local rings, focusing on their associated graded rings and specific Hilbert coefficient relations.

Contribution

It characterizes the structure of stretched $rak{m}$-primary ideals satisfying a particular Hilbert coefficient equality, advancing understanding of their algebraic properties.

Findings

Connected the first Hilbert coefficient to the almost Cohen-Macaulayness of associated graded rings.

Provided structural descriptions for ideals with $ ext{e}_1(I)= ext{e}_0(I)- ext{length}_A(A/I)+4$.

Enhanced the classification of stretched ideals with small first Hilbert coefficients.

Abstract

In this paper we explore the almost Cohen-Macaulayness of the associated graded ring of stretched $\mathfrak{m}$-primary ideals with small first Hilbert coefficient in a Cohen-Macaulay local ring $(A,\mathfrak{m})$. In particular, we explore the structure of stretched $\mathfrak{m}$-primary ideals satisfying the equality $\mathrm{e}_1(I)=\mathrm{e}_0(I)-\ell_A(A/I)+4$ where $\mathrm{e}_0(I)$ and $\mathrm{e}_1(I)$ denote the multiplicity and the first Hilbert coefficient respectively.