与强shellable性质相关的组合代数问题研究

As we know, shellability is an important property which attracts so many combinatorial algebraists, since a pure shellable complex is Cohen-Macaulay. By adding a new restriction condition to the definition of shellability, we introduced a special shellable complex, called a strongly shellable complex. Based on a series of previous research about strong shellability, in this research project, we plan to study the combinatorial and algebraic properties of strongly shellable complexes and bi-SS complexes, where a bi-SS complex is a simplicial complex which as well as its Alexander dual are strongly shellable. In this process, we are making use of two important tools. One is the codimension one graph of a strongly shellable complex, while the other is about the relation between the combinatorial properties of a simplicial complex and the algebraic properties of the corresponding Stanley-Reisner ideal..By the work of this research project, we are making effort to revealing the internal properties of the objects related to strong shellability, which will enrich the theory of combinatorial commutative algebra and promote the permeation and integration of algebra, combinatorics and graph theory.

一直以来，纯shellable性质因其作为通往Cohen-Macaulay性质的一条重要途径而吸引着众多组合和代数学家的研究兴趣。本项目申请人及其合作者在前期工作中通过增加约束条件的方法引入了强shellable复形的定义，并围绕强shellable性质展开了系列研究。本项目拟在此基础上，借助余一维图这一新引进的图论工具，结合单纯复形的组合属性与其Stanley-Reisner理想的代数属性之间的对应关系，来探讨强shellable复形、bi-SS复形（其本身与其Alexander对偶都满足强shellable性质的复形）的组合性质，以及其toric理想的代数性质。通过本项目的研究，我们希望结合图论的直观性、组合的技巧性与代数的深刻性，将强shellable复形、bi-SS复形的潜在内涵挖掘出来，进一步丰富和完善与shellable性质、Cohen-Macaulay性质相关的理论体系。

自从Stanley于1975年使用Cohen-Macaulay（简记为CM）理论证明球的上界猜想以来，将组合图论方法与代数问题结合起来的一门新兴学科——组合交换代数便逐渐发展起来。通过研究单纯复形上的shellable、顶点可分解（vertex decomposable）等性质来研究理想的CM性质已经成了组合交换代数中的一个主要课题。本项目主要运用组合交换代数的思想方法，借助图论、组合的技术手段完成了强shellable等与CM相关性质的组合图论刻画和代数刻画，并充实和完善了上一课题关于f-理想与f-图的研究，得到了若干类CM图类。与此同时，我们运用组合交换代数的思想方法和技术手段，协助完成了交换环上的一类零因子图刻画问题，解决了图论中的一类平衡划分问题，以及数论中某些类丢番图方程问题。这些研究都进一步丰富了组合交换代数的研究内容，并让图论组合的技术方法与代数的思想内涵实现了更充分的融合与统一。