Budget Amount *help 
¥1,600,000 (Direct Cost : ¥1,600,000)
Fiscal Year 1999 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1998 : ¥1,100,000 (Direct Cost : ¥1,100,000)

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 (ArtinTate 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D11ィエD1 (G, H, A) is needed to be determined and I found out that it is described using the twosisded 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 Gmodule 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.
