The first Hilbert coefficient of stretched ideals

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.

17 pages. arXiv admin note: substantial text overlap with arXiv:2109.12918