| Budget Amount *help |
¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2006: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2005: ¥500,000 (Direct Cost: ¥500,000)
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| Research Abstract |
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.
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