Simple Modules and Composition Series
If M is a module which has a non-zero proper submodule N, then there is a short exact sequence
A common approach to proving a fact about M is to show that the fact is true for the center term of a short exact sequence when it is true for the left and right terms, then to prove the fact for N and M/N. If N has a non-zero proper submodule, then this process can be repeated. This produces a chain of submodules
In order to prove the fact this way, one needs conditions on this sequence and on the modules Mi/Mi + 1. One particularly useful condition is that the length of the sequence is finite and each quotient module Mi/Mi + 1 is simple. In this case the sequence is called a composition series for M. In order to prove a statement inductively using composition series, the statement is first proved for simple modules, which form the base case of the induction, and then the statement is proved to remain true under an extension of a module by a simple module. For example, the Fitting lemma shows that the endomorphism ring of a finite length indecomposable module is a local ring, so that the strong Krull-Schmidt theorem holds and the category of finite length modules is a Krull-Schmidt category.
The Jordan–Hölder theorem and the Schreier refinement theorem describe the relationships amongst all composition series of a single module. The Grothendieck group ignores the order in a composition series and views every finite length module as a formal sum of simple modules. Over semisimple rings, this is no loss as every module is a semisimple module and so a direct sum of simple modules. Ordinary character theory provides better arithmetic control, and uses simple CG modules to understand the structure of finite groups G. Modular representation theory uses Brauer characters to view modules as formal sums of simple modules, but is also interested in how those simple modules are joined together within composition series. This is formalized by studying the Ext functor and describing the module category in various ways including quivers (whose nodes are the simple modules and whose edges are composition series of non-semisimple modules of length 2) and Auslander–Reiten theory where the associated graph has a vertex for every indecomposable module.
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