Perturbation of Hilbert-Samuel function relative to an Artinian module
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Let $(R, \mathfrak{m})$ be a Noetherian local ring and $M$ a co-Cohen-Macaulay $R$-module. Suppose $f_1, \ldots, f_r$ form an $M$-coregular sequence and let $I = (f_1, \ldots, f_r)$. Consider an ideal $J$ of $R$ such that $\lambda(0:_M(I,J)) < \infty$. In this paper, we will prove that the Hilbert-Samuel function of $J$ relative to $\overline{M} = 0:_MI$ is preserved under $J$-adic perturbations of $I$.
Abstract
Let $(R, \mathfrak{m})$ be a Noetherian local ring and $M$ a co-Cohen-Macaulay $R$-module. Suppose $f_1, \ldots, f_r$ form an $M$-coregular sequence and let $I = (f_1, \ldots, f_r)$. Consider an ideal $J$ of $R$ such that $\lambda(0:_M(I,J)) < \infty$. In this paper, we will prove that the Hilbert-Samuel function of $J$ relative to $\overline{M} = 0:_MI$ is preserved under $J$-adic perturbations of $I$.