The Jordan-Hölder property and Grothendieck monoids of exact categories

Abstract

We investigate the Jordan-Hölder 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).

Schur's lemma for exact categories implies abelian

“…Proof. This immediately follows from the Noether isomorphism theorem in exact categories, see [Eno,Proposition 2.5] for example. Now we will prove that Filt S is wide for a semibrick S. The following is a key lemma.…”

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

“…In [Eno,Theorem 5.10], the author gives a numerical criterion for (JHP). To rephrase his result in our context, we introduce the support of modules or torsion-free classes.…”

“…We give a proof using τ -tilting theory and the Grothendieck group K 0 (F ) of the exact category F , for which we refer to [AIR] and [Eno] respectively. It is shown in [Eno,Theorem 4.12, Corollary 5.14] that the following are equivalent:…”

“…To see that ( 2) and (2 ′ ) are also equivalent, let us consider K 0 (F ). By using [Eno,Lemma 5.7], one can show that the natural map K 0 (F ) → K 0 (mod Λ) is an injection, and that its image is precisely Z (supp F ) . Thus all the conditions are equivalent.…”

Composition series of arbitrary cardinality in abelian categories

“…The importance of the Jordan-Hölder Theorem in the study of groups, modules, and abelian categories has also motivated a large volume work devoted to establishing when a "Jordan-Hölderlike theorem" will hold in different contexts. Some recent examples include exact categories [BHT21,E19+] and semimodular lattices [Ro19,P19+]. In both of these examples, the "composition series" in question are assumed to be of finite length, as is the case for the classical Jordan-Hölder Theorem.…”