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

本研究项目期间的重要活动是：首先，提出组成线性理想的概念并建立其基本理论；其次，研究简单复合体的一般初始理想并讨论其在组合学中的具体有效应用。首先，我们得到了与简单复合体相关的无平方单项式理想当且仅当其对偶复合体是顺序Cohen-Macaulay时是分量线性的定理，并解释了顺序Cohen-Macaulay复合体及其h三角形的代数方面。其次，在对各分量线性理想的一般初始理想进行基本研究的基础上，得到了多项式环的齐次理想具有稳定的Betti数当且仅当该理想为各分量线性的重要结果。这一定理保证了分量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定理在有限集合上的复杂推广，通过讨论多项式环上具有与无平方理想的一般初始理想相同的梯度Betti数的无平方强稳定理想的存在性，对于具有固定Hubert函数的无平方理想的分级Betti数小于等于lexsegment理想的分级Betti数这一突出猜想，我们确实获得了肯定的答案。

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.