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Cohomology Theory of Finite Groups

有限群上同调理论

基本信息

批准号：
17540032
负责人：
SASAKI Hiroki
金额：
$ 2.39万
依托单位：
Shinshu University (2006-2007)Ehime University (2005)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540032/
关键词：
block ideal Brauer correspondence block cohomology block variety Green correspondence 有限群コホモロジー

项目摘要

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).

在有限群的块理想的上同调理论中，有许多重要的问题没有得到解决。设G是有限群，k是除G阶的素特征的代数闭域。设B是kG的块理想，D是亏群.设P是D的一个子群，H是G的一个包含DC _G（D）和NG（P）的子群.设kH的块理想C与块B具有Brauer对应关系，D也是C的亏群。研究块上同调H^*（G，B）和H^*（H，C）之间的关系是非常重要的。例如，当H^*（G，B）H^*（H，C）成立时，这个包含映射应该通过块B和C的Hochshild上同调环之间的转移映射来理解。我们证明了C到G x H的绿色对应的（B，C）-双模L具有许多优良的性质，这些性质不仅对上同调理论的应用，而且对有限群的模表示理论的应用都是有用的.在某些附加条件下，模L定义了转移映射L：HH^*（B）→ HH^*（C），通过H^*（G，B）嵌入HH^*（B）和H^*（H，C）嵌入HH^*（c），导出包含映射1：H^*（G，B）→ H^*（H，C）.从模表示论的观点来看，确定kH中位于块B中的不可分解模U的绿色对应V所在的块是非常有趣的。利用上述模L，我们证明了在一定条件下模V位于Brauer对应C中，当H^*（G，B）<$H^*（H，C）时，模的块簇在V _（G，B）（U）= l^* V _（H，C）（V）的意义下重合.