Foundation of computational Commutative algebra with a view toward combinatorics on convex polytopes

在本研究项目期间的重要活动是，第一，提出分量线性理想的概念并建立其基本理论；第二，研究单纯复形的一般初始理想，并讨论它们在组合学中的具体而有效的应用。首先，我们得到了单纯复形的无平方单项理想是分支线性的当且仅当它的对偶复形是序列Cohen-Macaulay复形，并解释了序列Cohen-Macaulay复形及其h-三角形的代数性质。其次，在对逆线性理想的一般初始理想的基本研究的基础上，建立了多项式环的齐次理想具有稳定Betti数的重要结果当且仅当该理想是分支线性的。这样的定理保证了分支线性理想将在计算交换代数中发挥重要作用。第三，为了得到经典组合学中Kruskal-Katona定理在有限集上的复杂推广，通过讨论多项式环上无平方理想的分次Betti数与多项式环上的无平方理想的一般初始理想的分次Betti数相同的存在性，我们确实成功地得到了如下猜想的肯定答案：具有固定Hubert函数的无平方理想的分次Betti数小于或等于词段理想的分次Betti数。

The important activity during the period of the present research project is, first, to present the concept of componentwise linear ideals and to establish its fundamental theory and, second, to study generic initial ideals of simplicial complexes and to discuss their concrete and effective applications to combinatorics. First of all, we obtained the theorem that the squarefree monomial ideal associated with a simplicial complex is componentwise linear if and only if its dual complex is sequentially Cohen-Macaulay, and explained the algebraic aspect of sequentially Cohen-Macaulay complexes and their h-triangles. Second, based on fundamental study about generic initial ideals of coruponentwise linear ideals, the important result that a homogeneous ideal of the polynomial ring possesses the stable Betti numbers if and only if the ideal is componentwise linear was established. Such the theorem guarantees that componentwise linear ideals will play an important role in computational commutative algebra. Third, in order to obtain sophisticated generalization of Kruskal-Katona theorem in classical combinatorics on finite sets, via the discussion on the existence of a squarefree strongly stable ideal having the same graded Betti numbers as those of the generic initial ideal of a squarefree ideal in the polynomial ring, we did succeed in obtaining the affirmative answer to the outstanding conjecture that the graded Betti numbers of a squarefree ideal with a fixed Hubert function are less than or equal to those of the lexsegment ideal.