ON SCOTT MODULES AND p-PERMUTATION MODULES: AN APPROACH THROUGH THE BRAUER MORPHISM
ON SCOTT MODULES AND p-PERMUTATION MODULES: AN APPROACH THROUGH THE BRAUER MORPHISM
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DOI：
10.1090/s0002-9939-1985-0773988-9
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发表时间：
1985-03
期刊：
影响因子：
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通讯作者：
M. Broué
中科院分区：
文献类型：
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作者：
M. Broué
在Lluis Puig之后，我们通过系统地使用广义Brauer态射给出了p-置换模（也称为“平凡源模”）的理论。本文的目的是系统地利用L. Puig（私人通信）。所谓“自包含”，我们的意思是只需要关于表示论的基本事实的知识：（5）中提出的顶点和源的初等理论（也见（4）），以及关于提升幂等元的经典结果。Burry关于低缺陷群系数的模理论解释的结果是作为该演示的副产品获得的。设G是一个有限群，0是一个交换环，对一个离散赋值是完备的，极大理想p和剩余域F = 0/p.注意，0可以等于F。(0.1)定义.设M是一个无0的OG-模。我们称M是p-置换模，如果当P是G的p-子群时，存在M的0-基被P稳定化。
Following Lluis Puig we give a presentation of the theory of p- permutation modules (also called "trivial source modules") by a systematic use of the generalized Brauer morphism. The aim of this paper is to present a somewhat new and self-contained treatment of p-permutation modules and Scott modules by a systematic use of the Brauer morphism, as suggested by L. Puig (private communication). By "self-contained" we mean that only knowledge of basic facts of representation theory is needed: elementary theory of vertices and sources as presented in (5) (see also (4)), as well as classical results about lifting idempotents. Burry's result about the module-theoretic interpretation of the coefficients of lower defect groups is obtained as a by-product of that presentation. Let G be a finite group, and let 0 be a commutative ring, complete for a discrete valuation, with maximal ideal p and residual field F = 0/p. We assume F has characteristic p > 0. Note that 0 may be equal to F. (0.1) DEFINITION. Let M be an 0-free OG-module. We say that M is a p- permutation module if, whenever P is a p-subgroup of G, there is an 0-basis of M which is stabilized by P.