发表机构
Universidade Tecnológica Federal do Paraná; Universidade de São Paulo – ICMC（巴拉那联邦技术大学; 圣保罗大学 - 计算机与数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究诺特局部环中分次族的渐近重数序列，刻画了低阶分量的收敛性，并发现高阶分量可任意变化；对诺特滤过和复解析情形给出了收敛条件，并发展了对角线混合理论。
AI 中文摘要
我们研究了诺特局部环中任意理想的分次族的渐近重数序列。我们刻画了第零分量的收敛性，并证明了在自然等维性和链状性假设下，第一个不被强制消失的正分量总是收敛的。相反，高阶分量可能表现出任意的渐近行为：即使在 $k[[x,y]]$ 中，具有固定根、高度和解析展形的整闭滤过也能实现任意给定的下极限和上极限。然而，对于诺特分次滤过，所有渐近分量都存在、有限且有理，并满足 Rees 型和赋值型的整依赖准则。在约化纯维复解析情形中，一个共同的去奇化修正提供了第二种收敛机制，无需有限生成假设。我们还发展了对角线混合理论：对于分次 $\mathfrak m$-准素族，顶部对角线分量是 Cutkosky 混合重数之和，而在共同解析模型上，所有对角线混合分量都收敛。后一结果推广了单族共同模型定理。
英文摘要
We study asymptotic multiplicity sequences of graded families of arbitrary ideals in Noetherian local rings. We characterize convergence of the zeroth component and prove that the first positive component not forced to vanish always converges under natural equidimensionality and catenarity assumptions. In contrast, higher components may exhibit arbitrary asymptotic behavior: even in $k[[x,y]]$, integrally closed filtrations with fixed radical, height, and analytic spread can realize any prescribed lower and upper limits. For Noetherian graded filtrations, however, all asymptotic components exist, are finite and rational, and satisfy Rees-type and valuative criteria for integral dependence. In the reduced pure-dimensional complex analytic setting, a common principalizing modification provides a second convergence mechanism without finite-generation assumptions. We also develop a diagonal mixed theory: for graded $\mathfrak m$-primary families, the top diagonal components are sums of Cutkosky's mixed multiplicities, while on a common analytic model all diagonal mixed components converge. The latter result extends the one-family common-model theorem.
