极性与戈林斯坦联络
Polarization and Gorenstein liaison
AI总结：
研究通过极化技术探讨G-联络构造，证明极化稳定Cohen-Macaulay单项式理想的Stanley-Reisner复形可顶点分解，并建立极化与G-联络的关系。
摘要：
在戈林斯坦联络理论中，一个主要的开放问题是：是否每个算术Cohen-Macaulay子方案都能与一个完全交集G-联络。Migliore和Nagel证明，如果这样的方案是普遍戈林斯坦（例如，约简），则在重新嵌入使其被视为P^{n+1}的子方案后，确实可以与完全交集G-联络。受此结果启发，我们考虑从与之密切相关的约简方案的G-联络构造G-联络的技术。极性是一种从任意单项式理想产生平方自由单项式理想工具。基本双G-联络在平方自由单项式理想上可以诱导自其Stanley-Reisner复形的顶点分解。给定一个单项式理想I和其极化P(I)的Stanley-Reisner复形的顶点分解，我们给出条件，允许将P(I)相关的基本双G-联络提升到I本身的相应基本双G-联络。我们利用在此过程中建立的关系，证明极化稳定Cohen-Macaulay单项式理想的Stanley-Reisner复形是顶点可分解的。然后我们引入并研究任意齐次理想Gröbner基的极化，并给出极化几何顶点分解与基本双G-联络之间的关系，这类似于我们关于顶点分解和基本双G-联络的结果。
英文摘要：
A major open question in the theory of Gorenstein liaison is whether or not every arithmetically Cohen--Macaulay subscheme of $\mathbb{P}^n$ can be G-linked to a complete intersection. Migliore and Nagel showed that, if such a scheme is generically Gorenstein (e.g., reduced), then, after re-embedding so that it is viewed as a subscheme of $\mathbb{P}^{n+1}$, indeed it can be G-linked to a complete intersection. Motivated by this result, we consider techniques for constructing G-links on a scheme from G-links on a closely related reduced scheme. Polarization is a tool for producing a squarefree monomial ideal from an arbitrary monomial ideal. Basic double G-links on squarefree monomial ideals can be induced from vertex decompositions of their Stanley--Reisner complexes. Given a monomial ideal $I$ and a vertex decomposition of the Stanley--Reisner complex of its polarization $P(I)$, we give conditions that allow for the lifting of an associated basic double G-link of $P(I)$ to a basic double G-link of $I$ itself. We use the relationship we develop in the process to show that the Stanley--Reisner complexes of polarizations of stable Cohen--Macaulay monomial ideals are vertex decomposable. We then introduce and study polarization of a Gröbner basis of an arbitrary homogeneous ideal and give a relationship between geometric vertex decomposition of a polarization and elementary G-biliaison that is analogous to our result on vertex decomposition and basic double G-linkage.
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