1999 Fiscal Year Final Research Report Summary
Research of relative cohomology theory of finite groups (Periodicity and spectral sequences)
| Project/Area Number |
10640040
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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 | Science University of Tokyo |
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Principal Investigator |
SANADA Katsunori Science University of Tokyo, Faculty of Science, Associate Professor, 理学部, 助教授 (50196292)
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| Project Period (FY) |
1998 – 1999
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| Keywords | cohomology / cup product / spectral sequence |
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| Research Abstract |
My research objectives were the completion of the theory of relative complete cohomology of finite groups, that is, "the theorem about periodicity" and "the theory of spectral sequences", and I have continued the research.
First, I tried to extend the methods of absolute theory of periodic cohomology (Artin-Tate Theorem) to relative case and I introduced the cup product ∪ : HィイD4^ィエD4ィイD1mィエD1 (G, H, A)【cross product】 HィイD4^ィエD4ィイD1nィエD1 (G, H, B) → HィイD4^ィエD4ィイD1m+nィエD1 (G, H, A【cross product】 B).
"The theorem about periodicity" is essentially equivalent to "the duality theorem". For the solution of this theorem, HィイD4^ィエD4ィイD1-1ィエD1 (G, H, A) is needed to be determined and I found out that it is described using the two-sisded residue class decomposition of the finite group G by the subgroup H (there is a related results by Adamson). By using this result, I have been studying "the duality theorem". To investigate the possibility of the proof of the theorem, I have been studying the programming of GAP, Maple and Mathematica etc.
On the other hand, another objective of my research was to extend the spectral sequence HィイD1pィエD1 (G/N, HィイD1qィエD1(N, A)) ⇒ィイD2pィエD2 HィイD1p+qィエD1 (G, A) to the case N was not necessarily normal in G, but I have not been able to solve the problem how I should regard HィイD1qィエD1 (N, A) as a G-module and I have been studying this.
For the generalized quaternion group of order of an exponent of 2, I determined the ring structure of the cohomology ring of the group with an order ring of a simple component of the group ring as the coefficient module.
I have plans that in general I will determine the ring structure of the cohomology ring of a finite group with an order ring of a simple component of the group ring as the coefficient module, and I will investigate the existence of a ring isomorphism between the cohomology rings with Morita equivalent orders as coefficient modules.
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