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 e1(I) = e0(I) − ℓA(A/I) + 4, where e0(I) and e1(I) denote the multiplicity and the first Hilbert coefficient, respectively.