$\tau$-exceptional sequences for representations of quivers over local algebras

TL;DR

This paper explores $ au$-exceptional sequences over algebras formed by tensoring a local algebra with a quiver algebra, establishing bijections and category equivalences that connect their representation theories.

Contribution

It introduces a bijection between $ au$-exceptional sequences in $kQ$ and $ au$-exceptional sequences in $ ext{mod }ig(R ensor kQig)$, and shows their $ au$-cluster morphism categories are equivalent.

Findings

Bijection between complete $ au$-exceptional sequences in $kQ$ and $ ext{mod }ig(R ensor kQig)$.

Every $ au$-perpendicular subcategory of $ ext{mod }ig(R ensor kQig)$ is equivalent to a module category of a similar form.

Equivalence of $ au$-cluster morphism categories for $kQ$ and $ig(R ensor kQig)$.

Abstract

Let $k$ be an algebraically closed field. Let $R$ be a finite dimensional commutative local $k$-algebra and let $Q$ be a quiver with no oriented cycles. In this paper, we study (signed) $\tau$-exceptional sequences over the algebra $\Lambda = R\otimes kQ$, which is isomorphic to $RQ$. We show there is a bijection between the set of complete (signed) $\tau$-exceptional sequences in $\text{mod }kQ$ and the set of complete (signed) $\tau$-exceptional sequences in $\text{mod }\Lambda$. Moreover, we prove that every $\tau$-perpendicular subcategory of $\text{mod }\Lambda$ is equivalent to the module category of $R\otimes kQ'$, for some quiver $Q'$. As a consequence, we prove that the $\tau$-cluster morphism categories of $kQ$ and $\Lambda$ are equivalent.