On the first generalized Hilbert coefficient and depth of associated graded rings

TL;DR

This paper investigates the relationship between the first generalized Hilbert coefficient and the depth of associated graded rings in Cohen-Macaulay local rings, providing conditions under which the depth is at least d-1 and the reduction number is at most 2.

Contribution

It establishes a new criterion linking the first generalized Hilbert coefficient to the depth of the associated graded ring and the reduction number in a broad class of ideals.

Findings

Depth of G(I) is at least d-1 under certain conditions.

Reduction number r_J(I) is at most 2 when the first generalized Hilbert coefficient satisfies a specific formula.

Extends Sally's results to a wider class of ideals beyond m-primary cases.

Abstract

Let $(R,\mathfrak{m})$ be a $d$-dimensional Cohen-Macaulay local ring with infinite residue field. Let $I$ be an ideal of $R$ that has analytic spread $\ell(I)=d$, satisfies the $G_d$ condition, the weak Artin-Nagata property $AN_{d-2}^-$ and depth$(R/I)\geq\min\lbrace 1,\dim R/I \rbrace$. In this paper, we show that if $j_1(I) = \lambda (I/J) +\lambda [R/(J_{d-1} :_{R} I+(J_{d-2} :_{R}I+I) :_R, \mathfrak{m}^\infty)]+1$, then depth$(G(I))\geq d -1$ and $r_J(I)\leq 2$, where $J$ is a general minimal reduction of $I$. In addition, we extend the result by Sally who has studied the depth of associated graded rings and minimal reductions for an $,\mathfrak{m}$-primary ideals.