Complexity and varieties for infinitely generated modules
Complexity and varieties for infinitely generated modules
复制标题
无限生成模块的复杂性和多样性
DOI：
10.1017/s0305004100073618
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发表时间：
1995
影响因子：
0.8
通讯作者：
J. Rickard
中科院分区：
文献类型：
--
作者：
D. Benson;J. Carlson;J. Rickard
在过去的十五年里，模的复杂性和多样性理论已经成为有限群的模表示理论的标准工具。此外，该技术已被用于积分表示的研究[8]，并已扩展到对象的表示理论，如有限虚上同调维数的群[1]，代数群的无穷小子群和限制李代数[14，16]。在所有情况下，都需要模范畴上的某种有限性条件来使理论工作。通常，这是以规定所有考虑中的模块都是并行生成的形式出现的。虽然这些限制对大多数应用程序来说是有效的，但有很好的理由希望开发一种理论，以适应无限生成的模块。原因之一可能是将表示法的技巧扩展到其他类的无限群。另一个原因是最近的一些工作揭示了有限性要求的一些缺陷。一个这样的问题可以概括如下。
In the past fifteen years the theory of complexity and varieties of modules has become a standard tool in the modular representation theory of finite groups. Moreover the techniques have been used in the study of integral representations [8] and have been extended to the representation theories of objects such as groups of finite virtual cohomological dimension [1], infinitesimal subgroups of algebraic groups and restricted Lie algebras [14, 16]. In all cases some sort of finiteness condition on the module category has been required to make the theory work. Usually this comes in the form of stipulating that all modules under consideration be finitely generated. While the restrictions have been efficient for most applications to date, there are very good reasons for wanting to develop a theory that will accommodate infinitely generated modules. One reason might be the possibility of extending the techniques of representations to other classes of infinite groups. Another reason is that some recent work has revealed a few of the defects of the finiteness requirement. One such problem can be summarized as follows.