Composition series of arbitrary cardinality in modular lattices and abelian categories

TL;DR

This paper extends the Jordan-Hölder theorem to a broad class of complete modular lattices and abelian categories, including infinite cases, and explores the structure and uniqueness of composition series in these contexts.

Contribution

It proves a generalized Jordan-Hölder-like theorem for arbitrary cardinalities in modular lattices and abelian categories, including new examples and counterexamples.

Findings

The theorem applies to lattices with no assumptions on cardinality or well-orderedness.

Infinite products of simple modules can have multiple composition series with different sizes.

Objects satisfying the axioms decompose into direct sums of indecomposables, conjecturally uniquely.

Abstract

For a certain family of complete modular lattices, we prove a Jordan--H\"older--Scheier-like" theorem with no assumptions on cardinality or well-orderedness. This family includes both lattices which are both join- and meet-continuous, as well as the lattices of subobjects of any object in an abelian category satisfying properties related to Grothendieck's axioms (AB5) and (AB5*). We then give several examples of objects in abelian categories which satisfy these axioms, including pointwise finite-dimensional persistence modules, presheaves, and certain Pr\"ufer modules. Moreover, we show that, over an arbitrary ring, the infinite product of isomorphic simple modules both fails to satisfy our axioms and admits at least two composition series with distinct cardinalities. We conclude by giving a lattice-theoretic proof that any object which is locally finitely generated and satisfies our axioms can be expressed as a direct sum of indecomposable subobjects. We conjecture that this decomposition is unique.