The Jordan-Hölder property and Grothendieck monoids of exact categories

Abstract

We investigate the Jordan-Hölder property (JHP) in exact categories. First, we show that (JHP) holds in an exact category if and only if the Grothendieck monoid introduced by Berenstein and Greenstein is free. Moreover, we give a criterion for this which only uses the Grothendieck group and the number of simple objects. Next, we apply these results to the representation theory of artin algebras. For a large class of exact categories including functorially finite torsion(-free) classes, (JHP) holds precisely when the number of indecomposable projectives is equal to that of simples. We study torsion-free classes in a quiver of type A in detail using the combinatorics of symmetric groups. We introduce Bruhat inversions of permutations and show that simples in a torsion-free class are in bijection with Bruhat inversions of the corresponding $c$-sortable element. We use this to give a combinatorial criterion for (JHP).

49 pages, revised version following the referee's suggestions, no major changes

References in corpus (1)

• Bruhat inversions in Weyl groups and torsion-free classes over preprojective algebras

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• Bruhat inversions in Weyl groups and torsion-free classes over preprojective algebras
• Examples and non-examples of integral categories and the admissible intersection property
• Composition series of arbitrary cardinality in modular lattices and abelian categories
• Stratifying systems and Jordan-Hölder extriangulated categories