Characteristic p method approaches to generalized Cohen-Macaulay rings

对于域上多项式环上的广义Cohen-Macaulay单项理想类的寻找问题，我们找到了部分解。也就是说，在与S.Goto和M.Okudaira的联合论文中，我们引入了广义完全交的概念，广义完全交是一类幂都是广义Cohen-Macaulay的Stanley-Reisner理想，并给出了这些理想的完全组合刻画，包括这些理想有线性分解的情况。作为上述工作的一个副产品，我们找到了稳定单项理想有最小Taylor解的充要条件。此外，我们还成功地给出了此类理想的最小生成元集的完整描述。见第三篇论文，这是与M.Okudaira共同完成的工作。通过与J.Herzog和其他人的共同工作，上述条件被证明适用于分量线性理想的更一般的设置。关于孤立非F-有理奇点的局部上同调，我们也得到了一些新的结果。即利用S.Goto和P.Schenzel的美元序列理论和C.Huneke和K.E.关于Kodaira消失的一些结果。Smith，我们用参数的紧闭包和极限闭包给出了局部上同调模的简洁表示。除了上同调消失和不消失的一些直接后果外，我们还考虑了最高局部上同调中的零的紧闭合如何控制下局部上同调的消失和不消失。我们成功地得到了孤立奇点的次高局部上同调的一些结果。

We found a partial solution to the problem of finding classes of generalized Cohen-Macaulay monomial ideals in a polynomial ring over a filed. Namely, in the joint papers with S. Goto and M. Okudaira, we introduce the notion of generalized complete intersection, which is a class of Stanley-Reisner ideals whose powers are all generalized Cohen-Macaulay, and gave the complete combinatorial characterization of these ideals, including the case that these ideals have liner resolutions. See the first and the second paper in the references.As a byproduct of the above mentioned work, we found a necessary and sufficient condition for stable monomial ideals to have minimal Taylor resolutions. Furthermore we succeeded in giving the complete description of minimal sets of generators of such ideals. See the third paper, which is a joint work with M. Okudaira. Through a joint work with J. Herzog and others, the above mentioned condition turned out to hold in much more general setting of component wise linear ideals. See the fourth paper.We also obtained some new results on local co homologies of isolated non F-rational singularities. Namely, by using the theory of USD-sequences by S. Goto and P. Schenzel and some results on Kodaira-vanishing by C. Huneke and K. E.. Smith, we gave a clean representation of the local cohomology modules in terms of tight closures and limit closures of parameters. Apart from some immediate consequences on vanishing and non-vanishing of the cohomollogies, we consider how the tight closure of zero in the highest local co homology controls the vanishing and non-vanishing of the lower cohomologies. We succeeded in obtaining some result on the next highest local co homology of an isolated singularity.