Representations and cohomology of algebras and categories

代数和范畴的表示和上同调

• 批准号: 0400951
• 负责人: Markus Linckelmann
• 金额: $ 10.5万
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-01 至 2007-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400951&HistoricalAwards=false
• 关键词: Representations; cohomology; algebras; categories

The main focus of the present proposal is the study of representationsand cohomology of EI-categories - that is, categories all of whoseendomorphisms are isomorphisms. The motivations for this proposal havetheir roots in long standing conjectures in modular representation theory aswell as open problems in the homotopy theory of p-local groups, a certaintype of p-complete topological spaces whose homotopy theory has beendeveloped in work by Broto, Levi and Oliver.The principal investigator has recently shown that one of the mostwell-known conjectures in modular representation theory, Alperin's weightconjecture, (which is originally a numerical equality involving the number of simple modules of a p-block of a finite group), has a structuralreformulation in terms of the cohomology of a certain functor on a suitableEI-category. One of the directions the principal investigator proposes to explore isthat there should be similar structural results - exploiting work ofG. R. Robinson - regarding Dade's conjectures (which are, very roughly,refinements of Alperin's weight conjecture taking into account moresubtle invariants of characters). The problem is to find the ``right"EI-category.Two important open problems in block theory can be formulated in acompletely general way as ``gluing problems" of 2-cocycles of certain functors onappropriate categories: the question whether one can associate aclassifying space with each p-block and the question whether the K"ulshammer-Puig2-cocycles on automorphism groups of centric subgroups of a block arethe restrictions of a 2-cocycle defined on the fusion system of theblock.Another recent result of the principal investigator is that there isa spectral sequence relating the cohomology of a functoron a regular EI-category to the cohomology of functors on the poset ofisomorphism classes of objects of that category.This spectral sequence should be useful for addressingthe two mentioned gluing problems; thisis another direction the principal investigator proposes to take. One way to look at a - not even necessarily mathematical - object is toconsider all maps of that object to itself which preserve itsstructure (the group of its "symmetries") and deduce from those maps properties of the considered object -this is, in a very simplified way, the general principle ofrepresentationtheory.

本提案的主要焦点是 EI 范畴的表示和上同调的研究 - 即所有内同态都是同构的范畴。这个提议的动机源于模表示理论中长期存在的猜想以及 p-局域群同伦理论中的开放问题，p-局部群是一种 p-完全拓扑空间，其同伦理论是由 Broto、Levi 和 Oliver 在工作中发展起来的。首席研究员最近表明，模表示理论中最著名的猜想之一，Alperin 的权重猜想， （最初是涉及有限群的 p 块的简单模数的数值等式），根据适当 EI 类别上某个函子的上同调进行结构重构。 首席研究员建议探索的方向之一是应该有类似的结构结果——利用 G 的工作。 R. Robinson - 关于 Dade 的猜想（非常粗略地讲，这是对 Alperin 重量猜想的改进，考虑到了字符的更微妙的不变量）。问题是找到“正确的”EI-范畴。块理论中的两个重要的开放问题可以用一种完全通用的方式表述为某些函子的 2-余循环在适当类别上的“粘合问题”：是否可以将分类空间与每个 p-块相关联的问题以及块的中心子群的自同构群上的 K"ulshammer-Puig2-余循环是否存在的问题 是在块的融合系统上定义的2-余循环的限制。主要研究者的另一个最近结果是，存在一个谱序列，将正则 EI 类别上的函子的上同调与该类别的对象的同构类偏序集上的函子的上同调联系起来。这个谱序列应该有助于解决提到的两个粘合问题；这是校长的另一个方向 调查员建议采取。 看待一个对象（甚至不一定是数学对象）的一种方法是考虑该对象与其自身的所有映射，这些映射保留其结构（其“对称性”组），并从这些映射中推断出所考虑对象的属性——以一种非常简单的方式，这就是表示论的一般原理。