On the first generalized Hilbert coefficient and depth of associated graded rings
Bull. Korean Math. Soc. 2020 Vol. 57, No. 2, 407-417
Published online October 24, 2019
Printed March 31, 2020
Amir Mafi, Dler Naderi
University Of Kurdistan; University of Kurdistan
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 $\mathfrak{m}$ is not an associated prime of $R/I$. 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 $I$ has almost minimal $j$-multiplicity, $G(I)$ is Cohen-Macaulay and $r_J(I)$ is at most 2, where $J=(x_1,\ldots,x_d)$ is a general minimal reduction of $I$ and $J_i=(x_1,\ldots,x_i)$. In addition, the last theorem is in the spirit of a result of Sally who has studied the depth of associated graded rings and minimal reductions for $\mathfrak{m}$-primary ideals.
Keywords : Generalized Hilbert coefficient, minimal reduction, associated graded ring
MSC numbers : 13A30, 13D40, 13H15, 13C14
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