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

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 $A{N}_{d-2}^{-}$ and depth$(R/I)\ge \min \{1,\dim R/I\}$. 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$(G(I))\ge d-1$ and $r_J(I)\le 2$, 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.

9 pages, comments welcome. arXiv admin note: text overlap with arXiv:1312.0651, arXiv:1401.1571 by other authors

References in corpus (1)

• Integral closure of ideals