First Brauer–Thrall conjecture for finite-dimensional algebras

Let $A$ be an algebra, and let $\tau$-tilting theory be considered over a field $\mathbb{K}$. An algebra is of finite representation type when it has only finitely many isomorphism classes of indecomposable modules. First Brauer–Thrall conjecture. $A$ is of finite representation type if and only if there is a positive integer $d$ such that

$$\dim_\mathbb{K}(M)\leq d$$

for every indecomposable $A$-module $M$. This conjecture was proved by Roter and Auslander, so the statement is now a theorem in the standard finite-dimensional setting.