Characteristic p method approaches to generalized Cohen-Macaulay rings

我们找到了一个部分解决的问题，发现类的广义Cohen-Macaulay单项理想的多项式环上的一个字段。即在与S.后藤和M. Okudaira的第二章中，我们引入了广义完全交的概念，这是一类Stanley-Reisner理想，它的幂都是广义Cohen-Macaulay，并给出了这类理想的完全组合刻画，包括这类理想有线性分解的情形.作为上述工作的一个副产品，我们得到了稳定单项理想具有极小Taylor分解的一个充要条件。此外，我们成功地给出了这种理想的生成元的极小集的完整描述。见第三篇论文，这是一个联合工作与M。奥平通过与J. Herzog和其他人的合作，上述条件被证明在更一般的分量线性理想的设置中成立。参见第四篇文章，我们还得到了孤立非F-有理奇点的局部上同调的一些新结果。即利用S.后藤和P. Schenzel的结果以及C. Huneke和K. E... Smith的方法，我们用参数的紧闭包和极限闭包给出了局部上同调模的一个简洁的表示。除了一些直接的后果消失和不消失的上同调，我们考虑如何在最高的本地上同调零的紧闭控制的消失和不消失的较低的上同调。我们成功地获得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.