$τ$-exceptional sequences for representations of quivers over local algebras

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) $τ$-exceptional sequences over the algebra $Λ= R\otimes kQ$, which is isomorphic to $RQ$. We show there is a bijection between the set of complete (signed) $τ$-exceptional sequences in $\text{mod }kQ$ and the set of complete (signed) $τ$-exceptional sequences in $\text{mod }Λ$. Moreover, we prove that every $τ$-perpendicular subcategory of $\text{mod }Λ$ is equivalent to the module category of $R\otimes kQ'$, for some quiver $Q'$. As a consequence, we prove that the $τ$-cluster morphism categories of $kQ$ and $Λ$ are equivalent.