混合重数和分次族的Rees定理及其逆定理

中文摘要 AI 辅助

设$(R,\mathfrak m)$为Noetherian局部环，并设$\mathcal I^{(1)},\ldots,\mathcal I^{(s)}$为分次$\mathfrak m$-primary族。我们引入弱渐近和强齐次联合约化，并证明Rees混合重数定理的相应推广，该定理将联合约化与混合重数联系起来。实际上，对于弱渐近联合约化，混合重数可实现为Hilbert--Samuel重数的归一化极限，从而得到渐近Rees定理。对于多重分次容许族，类型$\mathbf d=(d_1,\ldots,d_s)$的强齐次联合约化$\{x_{ij}\}$（其中$x_{ij}\in I_{a_{ij}}^{(i)}$）满足公式\\[ e((x_{ij});R) = \left(\prod_{i,j}a_{ij}\right) e\\!\left( \mathcal I^{(1)[d_1]},\ldots, \mathcal I^{(s)[d_s]} \right). \\] 我们还在形式等维性条件下证明了局部化逆定理，在adic情形下恢复了Rees的经典逆定理。作为应用，我们推导出分次族的Böger型定理，其adic特化恢复了Böger的经典定理。相反，弱渐近Rees定理的逆定理即使在一维正则局部环上的adic族中也不成立。

英文摘要

Let $(R,\mathfrak m)$ be a Noetherian local ring and let $\mathcal I^{(1)},\ldots,\mathcal I^{(s)}$ be graded $\mathfrak m$-primary families. We introduce weak asymptotic and strong homogeneous joint reductions and prove corresponding extensions of Rees's mixed multiplicity theorem relating joint reductions and mixed multiplicities. Actually, for weak asymptotic joint reductions, mixed multiplicities are realized as normalized limits of Hilbert--Samuel multiplicities, yielding an asymptotic Rees theorem. For multigraded admissible collections, a strong homogeneous joint reduction $\{x_{ij}\}$ of type $\mathbf d=(d_1,\ldots,d_s)$, with $x_{ij}\in I_{a_{ij}}^{(i)}$, satisfies the formula \[ e((x_{ij});R) = \left(\prod_{i,j}a_{ij}\right) e\!\left( \mathcal I^{(1)[d_1]},\ldots, \mathcal I^{(s)[d_s]} \right). \] We also prove a localized converse under formal equidimensionality, recovering the classical converse of Rees in the adic case. As an application, we derive a Böger-type theorem for graded families, whose adic specialization recovers Böger's classical theorem. In contrast, the converse to the weak asymptotic Rees theorem fails even for an adic family on a one-dimensional regular local ring.

发表机构

• Universidade Tecnológica Federal do Paraná（巴拉那联邦技术大学）
• Universidade de São Paulo – ICMC（圣保罗大学-圣卡洛斯理工学院）

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