Study on minimal free resolution of Stanley-Reisner rings

本研究的目的是研究Stanley-Reisner环最小自由分辨率的代数和组合性质。重点研究了Stanley-Reisner环的多重性与Castelnuovo-Mumford正则性之间的关系。在2005学年之前，我们证明了Stanley-Reisner理想的Castelnuovo-Mumford正则性小于或等于Stanley-Reisner环的维数d，如果它的多重性小于或等于d。此外，我们验证了Stanley-Reisner理想的Castelnuovo-Mumford正则性小于或等于d，如果它的多重性小于或等于2d-1。如果发电机的程度Stanley-Reisner理想是小于或等于dIn 2006学年,发展这些结果,我们证明了Stanley-Reisner Castelnuovo-Mumford规律的理想是小于或等于d如果其多样性小于或等于3 d2的,如果发电机的程度的第一会合模块Stanley-Reisner理想是小于或等于d + 1。由此推测，如果Stanley-Reisner理想的多重性小于或等于（p+2）d- (p-1)，则其Castelnuovo-Mumford正则性小于或等于d，并且如果Stanley-Reisner理想的第p个合模的生成子的度小于或等于d+p。在2007学年，我们证明了当Stanley-Reisner环的维数为2或3时，上述猜想成立。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面情况下的推广。

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.