权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

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

计算交换代数的基础，着眼于凸多面体的组合学

基本信息

批准号：
09440013
负责人：
HIBI Takayuki
金额：
$ 2.18万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09440013/
关键词：
componentwise linear generic initial ideal Cohen-Macaulay ring squaretree inonomial lexsegment ideal graded Betti number polynomial ring Kruskal-Katona thearem Componentwise linear h-三角形単体的複体 Stanley-Reisner ideal Cohen-Macaulay /

项目摘要

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.

在本研究项目期间的重要活动是，首先，提出分量线性理想的概念，并建立其基本理论，其次，研究一般的初始理想的单纯复形，并讨论其具体和有效的应用组合。首先，我们得到了单纯复形的无平方项理想是分量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，在对多项式环的上半线性理想的通有初始理想的基础研究上，得到了多项式环的齐次理想具有稳定Betti数的充要条件是该理想是分量线性的.这一定理保证了分支线性理想在计算交换代数中的重要作用。第三，为了得到经典组合学中Kruskal-Katona定理在有限集上的精细推广，通过讨论多项式环中与无平方理想的一般初始理想具有相同分次Betti数的无平方强稳定理想的存在性，我们确实成功地对著名的猜想得到了肯定的回答，即具有固定Hubert函数的无平方理想的分次Betti数小于大于或等于lexsegment理想的那些。