Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Author: Cutkosky, Steven Dale
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W7202354453