Hilbert coefficients and the associated graded rings

Abstract

Let ( R , m ) (R, \mathfrak {m}) be a d d -dimensional Cohen-Macaulay local ring with infinite residue field. Let I I be an m \mathfrak {m} -primary ideal of R R . In this paper, we prove that if ∑ n = 1 ∞ λ ( I n / I n − 1 J ) − e 1 ( I ) = 1 \sum _{n=1}^{\infty } \lambda (I^n/I^{n-1}J)-e_1(I)=1 for some minimal reduction J J of I I , then depth G ( I ) ≥ d − 2 G(I)\geq d-2 .