The first and second Brauer–Thrall conjectures for finite-dimensional algebras

Let $A$ be a finite-dimensional algebra over a field $k$. Suppose that there are infinitely many isomorphism classes of finitely generated indecomposable $A$-modules.

Brauer–Thrall conjecture. The following statements hold:

1. For infinitely many integers $n$, there exists an indecomposable $A$-module of length $n$.
2. If $k$ is infinite, then, for infinitely many integers $n$, there exist infinitely many isomorphism classes of indecomposable $A$-modules of length $n$.

The first Brauer–Thrall conjecture was proved by Roter in 1968, and the paper establishes analogous Brauer–Thrall type theorems for totally reflexive modules over certain local rings.