Study on minimal free resolution of Stanley-Reisner rings

本研究的目的是研究Stanley-Reisner环的最小自由分解的代数和组合性质，并考虑它的组合应用。2004学年，我们研究了Buchsbaum Stanley-Reisner环的线性分解。我们确定了Stanley-Reisner环的重数的下界。并且证明了如果它们具有最小重数，则它们具有线性分辨。在2005学年，我们主要研究了Buchsbaum Stanley-Reisner环的重数与其Castelnuovo-Mumford正则性之间的关系。我们证明了如果Stanley-Reisner理想的重数小于等于d，则其Castelnuovo-Mumford正则性小于或等于d。此外，我们还证明了，如果Stanley-Reisner理想的重数小于等于2d-1，则其Castelnuovo-Mumford正则性小于等于d，如果Stanley-Reisner理想的生成元数小于或等于DWE，我们还在线性分解的Stanley-Reisner环中深入研究了d-线性分辨的Stanley-Reisner环。利用上述结果，我们证明了如果Stanley-Reisner环的重数小于或等于d，且Stanley-Reisner理想的生成元的次数大于或等于d，则它有d-线性归结。此外，我们还证明了，如果Stanley-Reisner环的重数小于或等于2d-1，且Stanley-Reisner理想的生成元阶为d，则Stanley-Reisner环有d-线性归结。

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.