Characteristic p method approaches to generalized Cohen-Macaulay rings
| Project/Area Number |
17540049
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | Ritsumeikan University |
|---|
Principal Investigator |
TAKAYAMA Yukihide Ritsumeikan University, Department of Mathematical Sciences, Professor, 理工学部, 教授 (20247810)
|
|---|
| Project Period (FY) |
2005 – 2006
|
| Project Status |
Completed (Fiscal Year 2006)
|
| 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)
|
|---|
| Keywords | commutative ring theory / combinatorics / minimal free resolution / monomial ideal / tight closure / 代数学 / 代数幾何学 / generalized Cohen-Macaulay環 / F-特異点 / 小平消滅定理 / 局所コホモロジー |
|---|
| 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.
|
Report
(3 results)
Research Products
(13 results)
