A New Proof Concerning Quasi-Tilted Algebras

TL;DR

This paper offers a concise new proof that quasi-tilted algebras which are representation-finite must be tilted, clarifying a key classification result in algebra.

Contribution

It presents a shorter, more straightforward proof of a known theorem relating representation-finite quasi-tilted algebras to tilted algebras.

Findings

Representation-finite quasi-tilted algebras are tilted.

The new proof simplifies understanding of the classification.

Reinforces the connection between representation-finiteness and tilting.

Abstract

Let $\Lambda$ be a quasi-tilted algebra. If $\Lambda$ is representation-finite, it was shown by Happel, Reiten, and Smal{\o} that $\Lambda$ is tilted. We provide a new, short proof of this result.