Indecomposables live in all smaller lengths

TL;DR

This paper proves that in finite-dimensional associative algebras over algebraically closed fields, indecomposable modules exist at every length without gaps, extending to certain abelian categories.

Contribution

It establishes the absence of gaps in the lengths of indecomposable modules for finite-dimensional algebras and related categories, generalizing previous results.

Findings

No gaps in lengths of indecomposable modules for finite-dimensional algebras.

The result applies to abelian categories with simple objects having endomorphism algebra k.

Extends classical module theory results to broader categorical contexts.

Abstract

Let k be an algebraically closed field and A a finite dimensional associative k-algebra. We prove that there is no gap in the lengths of indecomposable A-modules of finite length. The analogous result holds for an abelian k-linear category C if the endomorphism algebras of the simples are k.