The Jordan-H\"older property and Grothendieck monoids of exact categories

Abstract

We investigate the Jordan-H\"older property (JHP) in exact categories. First we introduce a new invariant of exact categories, the Grothendieck monoids, and show that (JHP) holds if and only if the Grothendieck monoid 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 projectives is equal to that of simples. We study torsion-free classes in type A quiver in detail using the combinatorics of symmetric groups. In particular, we show that simples correspond to Bruhat inversions of a \(c\)-sortable element, and give the combinatorial criterion for (JHP).