Simple module: Difference between revisions

Revision as of 07:14, 10 May 2024

Latest revision as of 04:22, 19 May 2025

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A common approach to [[mathematical proof|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

:<math>\cdots \subset M_2 \subset M_1 \subset M.</math>

In order to prove the fact this way, one needs conditions on this sequence and on the modules ''M''_''i''&thinsp;/''M''_{''i''&thinsp;+&hairsp;1}. One particularly useful condition is that the length of the sequence is finite and each quotient module ''M''_''i''&thinsp;/''M''_{''i''&thinsp;+&hairsp;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 module|finite length]] [[indecomposable module]] is a [[local ring]], so that the strong [[Krull–Schmidt theorem]] holds and the [[category (mathematics)|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 ring]]s, this is no loss as every module is a [[semisimple module]] and so a [[direct sum of modules|direct sum]] of simple modules. [[Ordinary character theory]] provides better arithmetic control, and uses simple '''C'''''G'' modules to understand the structure of [[finite group]]s ''G''. [[Modular representation theory]] uses [[Brauer character]]s 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 [[quiver (mathematics)|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.