Cohomology theory of finite groups

2019 Fiscal Year Final Research Report

• Project/Area Number: 15K04777
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Shinshu University
• Principal Investigator: SASAKI HIROKI 信州大学, 学術研究院教育学系, 教授 (60142684)
• Project Period (FY): 2015-04-01 – 2020-03-31
• Keywords: 有限群 / ブロック・イデアル / コホモロジー環 / デフェクト群 / ソース多元環 / 部分対

Outline of Final Research Achievements

Block ideals of finite groups are objects in representation theory of finite groups; nowadays their source algebras are in main concern. Cohomology rings of block ideals are a kind of invariants of the source algebras. We expect that the cohomology rings of block ideals are the images of the maps from the cohmohology rings of defect groups which is induced by the source algebras. In this study we have established that this conjecture holds for blocks of tame representation type and with defect groups isomorphic with extraspecial p-groups; we have also verified for block with defect groups isomorphic with wreathed 2-groups that their cohomology ring is the image of maps induced by a direct summand of the source algebras. In course of investigation of these block we have proved theorems concerning direct summands of the source algebras, which are so impotant.

Free Research Field

代数学

Academic Significance and Societal Importance of the Research Achievements

有限群のブロック（イデアル）のコホモロジー環に対してはホモロジー代数的手法が適用できず、表現論の深い理解が必要である。本研究は、ブロックのコホモロジー環同士にも、通常のコホモロジー環と同様に、然るべき写像が存在するべきであるという予想の証明に向かっての第一歩を印したものである。その過程で、ブロックのソース多元環の直和因子についての存在と重複度についての定理を提出したが、これはソース多元環の加群構造に関して停滞していた研究に新たな重要な進展をもたらした。