Exceptional versus $\tau$-exceptional sequences for the Auslander algebra of $K[x]/(x^t)$

TL;DR

This paper demonstrates that for the Auslander algebra of a truncated polynomial ring, all complete exceptional sequences are also complete τ-exceptional sequences, and it explores how their mutation generalizes classical mutation concepts.

Contribution

It establishes the equivalence of complete exceptional and τ-exceptional sequences for the Auslander algebra of K[x]/(x^t) and extends mutation theory to τ-exceptional sequences.

Findings

Complete exceptional sequences are τ-exceptional in this algebra.

Mutation of τ-exceptional sequences generalizes classical mutation.

Results deepen understanding of module category structures.

Abstract

For $\mathcal{A}_t$, the Auslander algebra of $K[x]/(x^t)$, it is shown that every complete exceptional sequence of $\mathcal{A}_t$-modules is a complete $\tau$-exceptional sequence. Moreover, it is established that the mutation of complete $\tau$-exceptional sequences generalises the mutation of complete exceptional sequences in the category of $\mathcal{A}_t$-modules.