发表机构
University of Missouri（密苏里大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究Amao重数与Epsilon重数的极限，证明在域上有限型的任意局部环中，相关极限恒存在，完善了Epsilon重数的极限相关结论。
AI 中文摘要
设R是具有极大理想m_R的d维Noether局部环，I是R的一个理想。Stephen Landsittel近期的一篇论文表明，若R的完备化".hat R"的幂零根基维数小于d，则I的Epsilon重数".epsilon(I)"是Amao重数的极限，即".epsilon(I)= lim_{m→∞} a(I^m,(I^m)^{sat})/m^d"。本文证明，对任意在域上有限型的局部环R，极限".lim_{m→∞} a(I^m,(I^m)^{sat})/m^d"总是存在的。
英文摘要
Let $R$ be a $d$-dimensional Noetherian local ring with maximal ideal $m_R$ and let $I$ be an ideal of $R$. The epsilon multiplicity $ε(I)$ of $I$ is defined by Ulrich and Validashti in \cite{UV} as a limsup of normalized lengths. We show in this paper that the epsilon multiplicity is always a limit of these normalized lengths. Specifically, $$ ε(I)= \lim_{n\rightarrow\infty} \frac{\ell_R((I^n)^{\rm sat}/I^n)}{n^d/d!}. $$ We prove that the epsilon multiplicity satisfies an associativity formula, analogous to that of the ordinary multiplicity. We prove these results by comparing the epsilon multiplicity to a suitable limsup of normalized Amao multiplicities, the Amao epsilon multiplicity $aε(I)$. We show that the epsilon multiplicity is always equal to the Amao epsilon multiplicity, which is itself a limit.
CommentsThis version has significant more material. It proves that the epsilon multiplicity exists as a limit for ideals in an arbitrary Noetherian local ring
