2007 Fiscal Year Final Research Report Summary
Cohomology Theory of Finite Groups
| Project/Area Number |
17540032
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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 | Shinshu University (2006-2007)
Ehime University (2005) |
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Principal Investigator |
SASAKI Hiroki Shinshu University, School of General Education, Professor (60142684)
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| Co-Investigator(Kenkyū-buntansha) |
WATANABE Atumi Kumamoto University, Faculty of Science, Professor (90040120)
SANADA Katsunori Tokyo University of Science, Faculty of Sciense, Professor (50196292)
KAWAI Hiroaki Sajo Univeisity, Faculty of Engineering, Associate Professor (10222431)
NIWASAKI Takashi Ehime University, Center for General Education, Associate Professor (50218252)
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| Project Period (FY) |
2005 – 2007
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| Keywords | block ideal / Brauer correspondence / block cohomology / block variety / Green correspondence |
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| Research Abstract |
Many problems of fundamental importance are left unsolved in the theory of cohomology of block ideals of finite groups. Let G be a finite groups and k an algebraically closed field of prime characteristic dividing the order of G. Let B be a block ideal of kG and let D be a defect group. Let P be a subgroup of D and let H be a subgroup of G containing DC _G (D) and N_G (P). Assume that a block ideal C of kH and the block B are in Brauer correspondence and D is also a defect group of C. It is significantly important to investigate relationships between the block cohomologies H^* (G, B) and H^* (H,C). For example, when H^* (G, B) ⊆ H^* (H,C) does hold, this inclusion map should be understood through transfer maps between Hochshild cohomology rings of the blocks B and C. We showed that the (B, C) -bimodule L which is the Green correspondent of C to G x H has many nice properties which are useful not only for applications for the cohomology theory but also for modular representation theory of finite groups. Under some additional conditions the module L defines the transfer map L:HH^* (B)→ HH^* (C) which induce the inclusion map l:H^* (G,B)→ H^* (H,C) through embeddings of H^* (G, B) into HH^* (B) and of H^* (H,C) into HH^* (c). It is very interesting from a view point of modular representation theory to determine the blocks of kH in which the Green correspondent V of an indecomposable module U lying in the block B lies. We showed that, using the module L above that under some condition the module V lies in the Brauer correspondent C and when H^* (G,B)⊆ H^* (H,C) the block varieties of the modules coincides in the sense that V _(G,B)(U)= l^* V _(H,C)(V).
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Research Products
(22 results)
