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