发表机构
Indian Institute of Technology Kharagpur（印度理工学院卡拉格普尔分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究涉及理想幂的Ext和Tor模的Castelnuovo-Mumford正则性的渐近行为，证明其最终由线性函数的上确界给出，并建立了一种二分法。
AI 中文摘要
设 R 为一个 Noether 标准 N-分次环，并设 I_1,…,I_r 为 R 的齐次理想。设 L、M 和 N 为有限生成的 Z-分次 R-模，且 N⊆M。对于 n:=(n_1,…,n_r)∈N^r，定义 I^n:=I_1^{n_1}…I_r^{n_r}。我们研究两类 Ext 和 Tor 模的（Castelnuovo-Mumford）正则性的渐近行为：一类由商模 I^nM/I^nN 得到，另一类由商模 M/I^nN 得到，且在这两类中均以 L 作为第一个参数。对于每个固定的同调次数，我们证明前一类族的正则性最终由有限多个 n 的线性函数的上确界给出，其中 n_i 的系数属于 I_i 的生成元的次数集合。对于第二类族，当理想 I_i 中至少有一个由正次数的齐次元素生成时，我们在适当条件下建立了一个二分法：正则性要么最终等于 (L,M) 的 Ext 或 Tor 模的正则性，要么表现出与第一类族相同的渐近行为。这些结果源于一个定理，该定理在适当条件下描述了 (U+I^nV)/I^nW 的正则性的渐近行为，其中 U、V 和 W 是有限生成分次 R-模的分次子模，且 W⊆V。
英文摘要
Let \(R\) be a Noetherian standard $\mathbb{N}$-graded ring, and let \(I_1,\ldots,I_r\) be homogeneous ideals of \(R\). Let $L,M$, and $N$ be finitely generated $\mathbb{Z}$-graded \(R\)-modules with \(N\subseteq M\). For \(\underline{n} :=(n_1,\dots,n_r)\in\mathbb N^r\), set \({\bf I}^{\underline{n}}:=I_1^{n_1}\cdots I_r^{n_r}\). We study the asymptotic behaviour of the (Castelnuovo-Mumford) regularity of two families of Ext and Tor modules: those obtained from the quotients ${\bf I}^{\underline{n}}M/{\bf I}^{\underline{n}}N$, and those obtained from the quotients \(M/{\bf I}^{\underline{n}}N\), with \(L\) as the first argument in both the families. For each fixed homological degree, we prove that the regularity of the former family is eventually given by the supremum of finitely many linear functions of $\underline{n}$, where the coefficient of \(n_i\) belongs to the set of degrees of generators of \(I_i\). For the second family, when at least one of the ideals \(I_i\) is generated by homogeneous elements of positive degree, we establish, under suitable conditions, a dichotomy: the regularity is either eventually equal to the regularity of the Ext or Tor module of $(L,M)$, or exhibits the same asymptotic behaviour as in the first family. These results follow from a theorem describing the asymptotic behaviour of the regularity of \((U+{\bf I}^{\underline{n}}V)/{\bf I}^{\underline{n}}W\), under suitable conditions, where \(U\), \(V\), and \(W\) are graded submodules of a finitely generated graded \(R\)-module with \(W\subseteq V\).
Comments14 pages, Comments and suggestions are welcome
