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