Degrees of multiplicity functions for equimultiple ideals

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• 作者：Cătălin Ciupercă ^{catalin.ciuperca@ndsu.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：13D40 ; 13H15 ; 13B22
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 April 2016
• 年：2016
• 卷：452
• 期：Complete
• 页码：106-117
• 全文大小：334 K

文摘

Let J be an equimultiple ideal of height a   in a formally equidimensional local ring (R,m)$(R,\mathfrak{m})$. If I is a proper ideal that contains J  , we show that the degree of the multiplicity function f_J,I(n)=e(Iⁿ/Jⁿ)$f_{J,I}(n)=\mathrm{e}(I^n/J^n)$ is at most a with equality if and only if J is not a reduction of I  . As a consequence, we are able to define a unique filtration $J\subseteq J_{[a]}\subseteq \dots \subseteq J_{[1]}\subseteq \bar{J}$ between the ideal J   and its integral closure $\bar{J}$ with J_[k]$J_{[k]}$ being the largest ideal containing J   such that deg⁡f_J,I(n)≤a−k−1$\mathrm{deg}{f}_{J,I}(n)\le a-k-1$. Further results consider the ideal J_[1]$J_{[1]}$ and its relation to the S₂$S_2$-ification of the Rees algebra R[Jt]$R[Jt]$.