分量线性性、Fröberg 类比及线性支撑二单项式理想的分类
Componentwise linearity, Fröberg's analogue, and classification of linear support-two monomial ideals
- Indian Institute of Technology Madras（印度理工学院马德拉斯分校）
- SRM University-AP（SRM大学安得拉邦校区）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究支撑二单项式理想的分量线性性，证明其根理想具有线性分解，并完整分类了所有线性支撑二单项式理想及其幂，揭示该性质不依赖基域特征。
AI中文摘要：
本文研究了支撑二单项式理想的分量线性性质。我们的第一个主要结果表明，如果 $I$ 是一个支撑二单项式理想，则其底层简单图 $G_I$ 是共弦图；等价地，由 Fröberg 定理，$\sqrt{I}$ 具有线性分解。这一现象对于一般单项式理想而言相当罕见。事实上，存在具有线性分解的单项式理想，其根理想却不具有线性分解，即使根理想是图的边理想也是如此。对于任何单项式理想 $I$，总有 $\mu(I)\geq \mu(\sqrt{I})$。在文献中，满足 $\mu(I)=\mu(\sqrt{I})$ 的支撑二单项式理想具有特殊意义，因为它们包括简单图、加权有向图、边加权图和顶点加权图的边理想。我们将此类理想称为最小支撑二单项式理想。我们明确刻画了所有具有线性分解的最小支撑二单项式理想及其幂。接下来，我们对非最小情形的线性性进行分类。由此，我们获得了线性支撑二单项式理想的完整分类，这表明对于支撑二单项式理想，线性性质不依赖于基域的特征。
英文摘要：
In this paper, we investigate the componentwise linear property of support-two monomial ideals. Our first main result shows that if $I$ is a support-two monomial ideal, then its underlying simple graph $G_I$ is co-chordal; equivalently, by Fröberg's theorem, $\sqrt{I}$ admits a linear resolution. This phenomenon is quite rare for general monomial ideals. In fact, there exist monomial ideals with linear resolutions whose radicals fail to have linear resolutions, even when the radical is the edge ideal of a graph. For any monomial ideal $I$, one always has $μ(I)\geq μ(\sqrt{I})$. In the literature, support-two monomial ideals satisfying $μ(I)=μ(\sqrt{I})$ are of special interest, as they include edge ideals of simple graphs, weighted oriented graphs, edge-weighted graphs, and vertex-weighted graphs. We refer to such ideals as minimal support-two monomial ideals. We explicitly characterize all minimal support-two monomial ideals, as well as their powers, that admit linear resolutions. Next, we classify the linearity of non-minimal ones. Consequently, we obtain a complete classification of linear support-two monomial ideals, which shows that the property of being linear does not depend on the characteristics of the base field for support-two monomial ideals.
