Archive | 2021
The Jordan-Hölder property, Grothendieck monoids and Bruhat inversions
Abstract
The Jordan-Hölder theorem for modules says that the ways in which a module can be built up from simple modules are essentially unique. We may say that the category of modules (with finite length) satisfies the Jordan-Hölder property, abbreviated by (JHP). The aim of my talk is to investigate (JHP) in the setting of Quillen’s exact categories. As in the case of module categories, we can define simple objects, composition series and (JHP) in exact categories. Typical examples are extension-closed subcategories of modΛ for an artin algebra Λ, and in this case, all objects have at least one composition series. However, it turns out that there exists many categories which does not satisfies (JHP), as well as those which does. It is known that (JHP) implies the free-ness of the Grothendieck group, but the converse does not hold: for “nice” categories such as functorially finite torsion(-free) classes, their Grothendieck groups are free of finite rank, but (JHP) fails in some cases. Thus it is natural to consider a more sophisticated invariant than Grothendieck groups. Then I define Grothendieck monoids, which is a commutative monoid subject to the same universal property as the Grothendieck group. Then we have the following result: