发表机构
Universidade Tecnológica Federal do Paraná; Universidade de São Paulo – ICMC（巴西联邦技术大学; 圣保罗大学-计算与数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出方向混合重数，细化任意理想的重数序列，兼容经典情形，并建立可加性、结合性、分解及 Rees 型定理，推广 Bhattacharya--Rees--Teissier 公式。
AI 中文摘要
设 $(R,\mathfrak m,k)$ 为 Noether 局部环，$M$ 为非零有限生成 $R$-模，$J_1,\ldots,J_s$ 为 $M$ 上任意正高度的真理想。我们引入方向混合重数的概念，该概念通过保留一个显著方向 $j$，从多分次商 $J^{\mathbf u}M/J^{\mathbf u+\mathbf e_j}M$ 获得。这些不变量细化了任意理想的重数序列，同时与两个经典极端情形兼容：对于单个理想，它们恢复 Achilles--Manaresi 重数序列的正分量；对于 $\mathfrak m$-准素理想，其最高次项分别恢复通常的混合重数。我们建立了可加性、结合性公式、浅层约化，以及对角系数到方向分量的精确分解。在无限剩余域上，每个正方向系数均可由足够一般的方向 $(FC)$-系统实现为最终长度。随后，我们识别了一个附加同步条件，在该条件下方向系数成为参数理想的 Hilbert--Samuel 重数，从而得到 Rees 型定理，并在 $\mathfrak m$-准素情形下恢复经典定理。此外，我们证明了 $J_1^{n_1}\cdots J_s^{n_s}$ 的多齐次乘积公式，该公式特化为经典的 Bhattacharya--Rees--Teissier 公式。
英文摘要
Let $(R,\mathfrak m,k)$ be a Noetherian local ring, let $M$ be a nonzero finitely generated $R$-module, and let $J_1,\ldots,J_s$ be arbitrary proper ideals of positive height on $M$. We introduce the notion of directional mixed multiplicities, obtained from the multigraded quotients $J^{\mathbf u}M/J^{\mathbf u+\mathbf e_j}M$ by retaining one distinguished direction $j$. These invariants refine the mixed multiplicity sequences of arbitrary ideals while remaining compatible with the two classical extremes: for one ideal they recover the positive components of the Achilles--Manaresi multiplicity sequence, and for $\mathfrak m$-primary ideals their top-degree terms recover the usual mixed multiplicities individually. We establish additivity, an associativity formula, superficial reduction, and an exact decomposition of the diagonal coefficients into directional components. Over an infinite residue field, every positive directional coefficient is realized as an eventual length by a sufficiently general directional $(FC)$-system. We then identify an additional synchronization condition under which the directional coefficient becomes the Hilbert--Samuel multiplicity of a parameter ideal, yielding a Rees-type theorem and recovering the classical theorem in the $\mathfrak m$-primary case. In addition, we show a multihomogeneous product formula for $J_1^{n_1}\cdots J_s^{n_s}$, which specializes to the classical Bhattacharya--Rees--Teissier formula.
