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

2017/10/11 by Mafi, Amir, Naderi, Dler
#13A30 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1710.03926

Abstract

Let (R,\mathfrakm) be a d-dimensional Cohen-Macaulay local ring with infinite residue field. Let I be an ideal of R that has analytic spread ℓ(I)=d, satisfies the G_d condition, the weak Artin-Nagata property AN_d-2^- and depth(R/I)≥min\lbrace 1,dim R/I \rbrace. In this paper, we show that if j₁(I) = λ(I/J) +λ[R/(J_d-1 :_R I+(J_d-2 :_RI+I) :_R, \mathfrakm^∞)]+1, then depth(G(I))≥ d -1 and r_J(I)≤ 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 ,\mathfrakm-primary ideals.

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