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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：
18540041
负责人：
TERAI Naoki
金额：
$ 2.57万
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：
2006
资助国家：
日本
起止时间：
2006 至 2007
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. We focused on the relation between the multiplicity and the Castelnuovo-Mumford regularity of Stanley-Reisner rings.Before the academic year 2005 we proved that the Castelnuovo-Mumford regularity of a Stanley-Reisner ideal is less than or equal to the dimension d of the Stanley-Reisner rings 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 dIn the academic year 2006, developing these results, we proved 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 3d-2, and if the degree of generators of the first syzygy module of the Stanley-Reisner ideal is less than or equal to d+1. From this result we conjectured 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 (p+2)d- (p-1), and if the degree of generators of the p-th syzygy module of the Stanley-Reisner ideal is less than or equal to d+p. In the academic year 2007 we proved that the above conjecture holds if the dimension of the Stanley-Reisner rings is 2 or 3. We also found that this conjecture is a generalization of the lower bound theorem, that is famous in convex polytope theory, in the facet case.

本文研究了Stanley-Reisner环的极小自由分解的代数和组合性质。我们主要研究了Stanley-Reisner环的重数与Castelnuovo-Mumford正则性之间的关系，在2005学年之前我们证明了当Stanley-Reisner环的重数小于或等于d时，其Castelnuovo-Mumford正则性小于或等于维数d.进一步证明了Stanley-Reisner理想的Castelnuovo-Mumford正则性当其重数小于或等于2d-1时小于或等于d，当Stanley-Reisner理想的生成元次数小于或等于d时。证明了Stanley-Reisner理想的Castelnuovo-Mumford正则性小于等于d，如果它的重数小于等于3d-2，且Stanley-Reisner理想的第一合合模的生成元的次数小于或等于d+1.由这个结果我们证明了Stanley-Reisner理想的Castelnuovo-Mumford正则性小于或等于d，如果它的重数小于或等于（p+2）d-（p-1），如果Stanley-Reisner理想的p次合合模的生成元的次数小于或等于d+在2007学年，我们证明了上述猜想成立，如果Stanley-Reisner环的维数是2或3。我们还发现这个猜想是凸多面体理论中著名的下界定理在小平面情形下的推广。