Some Formulas for Epsilon Multiplicity in Local Rings
Stephen Landsittel
TL;DR
The paper proves that the epsilon multiplicity $\\varepsilon(I)$ exists as a limit for Noetherian local rings whose completion has nilradical of nonmaximal dimension, unifying existence with a volume formula. It develops a framework that reduces to the complete case, analyzes saturations and filtrations, and uses exact sequences to relate $\\varepsilon(I)$ to the Amao multiplicity $a(I^m,(I^m)^{sat})$, yielding $\\varepsilon(I) = \lim_{m\to\infty} a(I^m,(I^m)^{sat})/m^d$. A generalized volume-equals-multiplicity formula is established in this setting, broadening prior results and connecting to saturation behavior under completion. The results rely on structural reductions (to $R/N$ and to $\\widehat{R}$), filtration techniques, and Swanson-type theorems to handle non-finitely generated Rees algebras, with implications for multiplicities of graded families of ideals in local rings.
Abstract
We prove that the epsilon multiplicity exists in a Noetherian local ring whenever the nildradical of the completion of R has nonmaximal dimension. We also extend the volume equals multiplicity formula for the epsilon multiplicity to this setting.