本项目主要构造若干类型交换环，并研究它们的同调性质和组合性质。从简单图（即无重边、无环图）、超图、Poset 、半格等出发借助于单项式理想（以及推广）构造新的交换环类，并研究新交换环类的代数（同调）特性与组合特性及二者之间的联系与制约，研究有关组合理想（边理想等）的代数性质(如极小自由预解式、Betti 数、正则度、深度与同调维数等)。研究由有限抽象单纯复形和零因子单纯复形导出的面环（又叫Stanley-Reisner ring）、边环的代数特性（例如同调与几何不变量、Cohen-Macaulay 性、其单项式理想与子代数结构等）与组合特性(包括组合不变量)，以及这些性质与原环（或者原复形）性质或者结构之间的关系。特别是研究图（以及单纯复形）的shellability. 这一特性在以往涉及到相应单项式理想等Cohen-Macaulay性质的工作中占据着完全核心的主导地位。

In this research item, we plan to study homological and combinatorial properties of some classes of constructed commutative rings. Starting from a simple graph (a hypergraph, a poset or a semilattice respectively), we construct new commutative rings with the aid of monomial ideals as well as their generalizations. We study the algebraic and combinatorial properties of the newly constructed rings, and the relationship between the properties. In particular, we study the properties of the related combinatorial ideals of rings, e.g., the minimal free resolution, the Betti number, the regular degree, the depth and the homological dimensions. We also study the face ring (i.e., Stanley-Reisner ring) and the edge ring, determined by some special class of abstrct simplicial complexes, including zero-divisor simplicial complexes. In particular, we study the Cohen-Macaulay properties, the strcture of monomial ideals and related subalgebras. Special attention will be paid to the further pursuit for deep understanding of shellability of either a graph or a simplicial complex, since it occupies a central position for understanding the Cohen-Mcaulayness of related monomial ideals.