2007 Fiscal Year Final Research Report Summary
Study on the Lefschetz property for complete intersections
| Project/Area Number |
18540003
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Hokkaido University of Education |
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Principal Investigator |
HARIMA Tadahito Hokkaido University of Education, Faculty of Education, Associate Professor (30258313)
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| Co-Investigator(Kenkyū-buntansha) |
WATANABE Junzo Tokai University, Science, Professor (40022727)
WACHI Akihito Hokkaido Institute of Technology, Comprehensive Education, Associate Professor (30337018)
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| Project Period (FY) |
2006 – 2007
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| Keywords | algebra / commutative ring / Gorenstein algebra / complete intersection / Artinian algebra / Lefschetz property / Hilbert function / generic initial ideal |
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| Research Abstract |
The original purpose of this research was to study the following problems on the strong Lefschetz property for complete intersections.
Problem 1: Do all complete intersections have the strong Lefschetz property?
Problem 2: Does every Gorenstein algebra (which is in the linkage class of a complete intersection) have the strong Lefschetz property?
First, we introduced the "central simple modules" for Artinian Gorenstein algebras and obtained some results in the study of Problem 1. Second, we introduced the "k-strong Lefschetz property" for Artinian algebras and studied the generic initial ideals of complete intersection ideals whose quotient algebras have the n-strong Lefschetz property. The main results obtained in this research project are as follows:
1. An Artinian Gorenstein algebra has the strong Lefschetz property if and only if all central simple modules of some linear form have the strong Lefschetz property. As an application of the above result, we proved that a finite free extensio
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n of an Artinian algebra with the strong Lefschetz property has the strong Lefschetz property if the fiber does, Furthermore, we showed some examples of complete intersections with the strong, Lefschetz property, for example, the complete intersection defined by power sums of consecutive degrees has the strong Lefschetz property. This is a joint work with Junzo Watanabe.
2. Let R be the polynomial ring in n variables over a field of characteristic zero, and I a graded ideal of R whose quotient ring R/I has the n-strong Lefschetz property. Suppose that all k-th differences of the Hilbert function of R/I are quasi-symmetric. Then the generic initial ideal of I is the unique almost revlex ideal with the same Hilbert function as R/I. Furthermore, using the above result, we found some examples of complete intersections whose generic initial ideals are the unique almost revlex ideals. This is a joint work with Akihito Wachi.
The following problem is very interesting and a coming theme.
Problem Do all complete intersections defined by symmetric polynomials have the strong Lefschetz property? Less
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