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Cohomology theory of finite groups

有限群上同调理论

基本信息

批准号：
11640033
负责人：
SASAKI Hiroki
金额：
$ 2.24万
依托单位：
Ehime University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640033/
关键词：
finite group cohomology representation theory relative projectivity パラメーター系ノルム写像

项目摘要

As a continuation of our research on mod p cohomology algebras of finite groups with extraspecial Sylow p-subgroups, which was done under Grant-in-Aid for Scientific Research during 1997〜1998 (project number 096400460), we calculated the mod 7 cohomolgy algebra of Held simple group. In this work we completed the theoreical frame work on the cohomology algebras of finite groups of this kind. We also calculated the mod p cohomology algebra of the special linear group of degree 3 over the prime field of characteristic p.We improved a theorem of Carlson on system of parameters. Namely if a finite group G has p-rank r, then the mod p cohomology algebra has a system of parameters ζ_1,...,ζ_r with the following properties: (1) for each i = 1,...,r, the element ζ_i is a sum of transfers from the centralizers of elementary abelian p=subgrpups of rank i; (2) for each i = 1,...,r, the restriction of {ζ_1,...,ζ_i} to an elementary abelian p-subgroup of rank i is a system of paramters of the cohomology algebra of this elementary abelian p-subgroup. From this fact we, in particular, showed that if a finite group G has p-rank less than or equal to 3, then the trivival kG-module k has index zero.Using transfer maps of extension groups introduced by Carlson, Peng, Wheeler, we showed that an element ρ in the mod p cohomology algebra of a finite group G is regular if and only if the transfer map Tr^<L_p> : Ext^*_<kG>(L_ρ,L_ρ) → Ext^*_<kG>, (k,k) defined by the Carlson module L_ρ is the zero map. Relating to this result we proved that if a finitely generated kG-module W is protective over a cyclic shifted subgroup in the center of a Sylow p-subgroup, then the transfer map Tr^W : Ext^*_<kG>(W,W) → Ext^*_<kG>(k,k) defined by the module W is the zero map.

作为1997 ~ 1998年国家科学研究资助项目（项目号096400460）对具有特殊Sylow p-子群的有限群的mod p上同调代数研究的延续，我们计算了hold单群的mod 7上同调代数。在此工作中，我们完成了这类有限群上同代数的理论框架。我们还计算了特征p的素域上特殊的3次线性群的模p上同调代数，改进了卡尔森关于参数系统的一个定理。也就是说，如果一个有限群G具有p秩r，那么模p上同调代数有一个参数系ζ_1，…，ζ_r具有以下性质：(1)对于每个I = 1，…，r，元素ζ_i是由秩i的初等阿贝尔群p=子群的中心中心迁移的和；(2)当I = 1时，…，r, {ζ_1，…，对于秩为I的初等阿贝尔p子群，ζ_i}是这个初等阿贝尔p子群的上同调代数的参数系统。从这个事实，我们特别地证明了如果一个有限群G的p-秩小于或等于3，那么平凡的kg -模k的指标为0。利用Carlson， Peng， Wheeler引入的扩展群的迁移映射，证明了有限群G的mod p上同调代数中的元素ρ是正则的当且仅当Carlson模L_ρ定义的迁移映射Tr^<L_p>: Ext^*_<kG>(L_ρ,L_ρ)→Ext^*_<kG>, （k,k）是零映射。由此证明了如果有限生成的kG-模W在Sylow p-子群中心的循环移位子群上是保护的，则由模W定义的迁移映射Tr^W: Ext^*_<kG>（W，W）→Ext^*_<kG>(k,k)是零映射。