F-理想及其组合性质

In 1975, Stanley applied methods in commutative algebra to combinatorial problems and settled the famous upper bound conjecture. Since then more and more mathematicians were attracted to study interplays of algebraic, combinatorial and graphical aspects of a subject. In the process, the concept Stanley-Reisner ideal was introduced and played a fundamental role. .F-ideal is a class of squarefree monomial ideal closely related to the Stanley-Reisner ideal. By introducing a combinatorial concept callld perfect sets, we have succeded in describing a class of homogenous f-ideals, proved the existence of the homogenous f-ideals, by making use of Turan’s theorem. In particular, we give a complete characterization of homogenous f-ideals of degree 2..In the research project, we plan to characterize f-ideals of degree n, and more over give further study on general f-ideals, by introducing new combinatorial concepts and by making use of the theory of hypergraph. We also want to study the algebraic properties of f-ideals, such as its hight unmixed properties, by the combinatorial description or graphic properties. By the work of this research project, we are making effort to revealing the combinatorial and algebraic properties of f-ideals, which will enrich the theory of monomial ideals.

自从1975年Stanley使用交换代数的方法解决组合中有名的上界猜想以来，代数与组合、图论之间的紧密联系就成为了很多数学工作者关心的热门课题。在这一课题的发展演变过程中，单纯复形的Stanley-Reisner理想发挥了重要的作用。本项目申请人及其合作者在前期工作中通过引入组合对象“完美集”来刻画一类与Stanley-Reisner理想息息相关的理想------齐次f-理想，并进一步借助图论中的图兰定理证明了齐次f-理想的存在性，且给出了齐2次f-理想的完全分类。本项目拟在此基础上，进一步通过引入新的组合对象并结合“超图”这一图论工具来研究齐n次f-理想，以及一般f-理想的分类问题。与此同时，本项目拟运用得到的组合、图论刻画反过来研究f-理想本身的诸如unmixed性质等代数性质。通过本项目的工作，力求将f-理想这一新型的单项式理想的良好组合、代数属性揭示出来，丰富单项式理想的研究内容。

F-理想是有着丰富的代数和组合内涵的一类新型理想。本项目建立在前期对齐2次f-理想的分类研究的基础之上，实现了对f-理想相关的组合、图论以及代数问题的进一步刻画，主要获得了如下两个方面的成果：（1）在组合图论方面，引入并研究了一类与shellable性质息息相关的组合性质——强shellable性质，并结合超图的特性给出了一类具有某种强shellable性质的超图的组合图论刻画；（2）在代数方面，通过研究f-图的shellable性质，证明了所有齐2次f-理想都具有Cohen-Macaulay这一代数属性。此外，在本项目研究过程中一些新对象和新方法（例如余一维图）的引入，有利于图论技巧在代数、组合问题中发挥重要作用，促进了多个学科之间的交叉融合，也为后续研究提供了新的素材和新的思路。