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Study on minimal free resolution of Stanley-Reisner rings

Stanley-Reisner环最小自由分辨率研究

基本信息

批准号：
16540028
负责人：
TERAI Naoki
金额：
$ 2.3万
依托单位：
Saga University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2005
项目状态：
已结题

项目摘要

The purpose of this research is to study algebraic and combinatorial properties of minimal free resolution of Stanley-Reisner rings and to consider its combinatorial applications.In the academic year 2004 we studied Buchsbaum Stanley-Reisner rings with linear resolution. We determined the lower bound for the multiplicity of Stanley-Reisner rings. And we showed that they have linear resolution if they possess the minimal multiplicity. We also showed a necessary and sufficient condition for Buchsbaum Stanley-Reisner rings to have linear resolution in terms of the reduced homology groups of the corresponding simplicial complex and its links.In the academic year 2005 we mainly studied the relation between the multiplicity of Stanley-Reisner rings and their Castelnuovo-Mumford regularity. We proved that the Castelnuovo-Mumford regularity of a Stanley-Reisner ideal is less than or equal to the dimension d if its multiplicity is less than or equal to d. Moreover we verified that the Castelnuovo-Mumford regularity of a Stanley-Reisner ideal is less than or equal to d if its multiplicity is less than or equal to 2d-1, and if the degree of generators of the Stanley-Reisner ideal is less than or equal to dWe also investigated Stanley-Reisner rings with d-linear resolution among those with linear resolution intensively. Using the above result we showed that a Stanley-Reisner ring has d-linear resolution if its multiplicity is less than or equal to d and if the degree of generators of the Stanley-Reisner ideal is more than or equal to d. Moreover we showed that a Stanley-Reisner ring has d-linear resolution if its multiplicity is less than or equal to 2d-1 and if the degree of generators of the Stanley-Reisner ideal is d. By Alexander duality, we also verified that a Stanley-Reisner ring is Cohen-Macaulay if its multiplicity is large enough.

本研究的目的是研究Stanley-Reisner环的最小自由分辨率的代数和组合性质，并考虑其组合应用。在2004学年，我们用线性分辨率研究了Buchsbaum Stanley-Reisner环。我们确定了Stanley-Reisner环多重性的下界。我们证明了如果它们具有最小的多重性它们就具有线性分辨率。我们还证明了Buchsbaum Stanley-Reisner环具有线性分辨的一个充分必要条件，即对应的简单配合物及其连杆的约化同调群。在2005学年，我们主要研究了Stanley-Reisner环的多重性与其Castelnuovo-Mumford正则性之间的关系。我们证明了Stanley-Reisner理想的Castelnuovo-Mumford正则性小于或等于d维，如果它的多重性小于或等于d维。我们进一步证明了Stanley-Reisner理想的Castelnuovo-Mumford正则性小于或等于d维，如果它的多重性小于或等于2d-1，当Stanley-Reisner理想的产生子的度小于或等于d时，我们还对线性分辨率的Stanley-Reisner环进行了深入的研究。使用以上结果我们发现Stanley-Reisner环维线性决议如果其多样性小于或等于d如果Stanley-Reisner发电机的理想程度超过或等于d。此外我们表明Stanley-Reisner环维线性决议如果其多样性小于或等于2 d 1如果Stanley-Reisner发电机的程度由亚历山大二元性,理想是d。我们还验证了如果一个Stanley-Reisner环的多重度足够大，它就是Cohen-Macaulay环。

项目成果

期刊论文数量（29）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

代数系と符号理論(2版)

代数系统和编码理论（第二版）

DOI：
发表时间：
2005
期刊：
影响因子：
0
作者：
N.Terai;K.-I.Yoshida;Y.Taguchi;[U] 上原健
通讯作者：
[U] 上原健

Castelnuovo--Mumford regularity and initial ideals with no embedded prime ideal

Castelnuovo--芒福德正则性和没有嵌入素理想的初始理想

DOI：
发表时间：
2004
期刊：
Acta Math.Vietnam 29
影响因子：
0
作者：
Higo;Shoji.;緒方明;Naoki Terai;河田将一他;Aldo Conca;Jurgen Herzog;高原朗子;高原朗子他;Jurgen Herzog;肥後祥治;H.Ohsugi;N.Terai
通讯作者：
N.Terai

Stanley-Reisner rings with large multiplicity are Cohen-Macalay

重数较大的 Stanley-Reisner 环是 Cohen-Macalay 环