Indecomposables live in all smaller lengths

TL;DR

This paper proves that in finite-dimensional algebras over an algebraically closed field, the existence of an indecomposable module of a certain length guarantees the existence of an accessible indecomposable module of that length, extending previous results.

Contribution

It strengthens Bongartz's result by showing that indecomposable modules of any length are accessible, providing a new inductive framework for understanding module lengths.

Findings

Indecomposable modules of a given length imply accessible modules of that length.

Accessible modules are constructed inductively from simple modules.

The result applies to all lengths where indecomposable modules exist.

Abstract

Let $\Lambda$ be a finite-dimensional $k$-algebra with $k$ algebraically closed. Bongartz has recently shown that the existence of an indecomposable $\Lambda$-module of length $n > 1$ implies that also indecomposable $\Lambda$-modules of length $n-1$ exist. Using a slight modification of his arguments, we strengthen the assertion as follows: If there is an indecomposable module of length $n$, then there is also an accessible one. Here, the accessible modules are defined inductively, as follows: First, the simple modules are accessible. Second, a module of length $n \ge 2$ is accessible provided it is indecomposable and there is a submodule or a factor module of length $n-1$ which is accessible.