Minimal Injective Resolutions and Auslander-Gorenstein Property for Path Algebras

Abstract

Let $R$ be a ring and $\mathcal{Q}$ be a finite and acyclic quiver. We present an explicit formula for the injective envelopes and projective precovers in the category $\mathrm{Rep}(\mathcal{Q},\mathrm{R})$ of representations of $\mathcal{Q}$ by left $R$-modules. We also extend our formula to all terms of the minimal injective resolution of $R\mathcal{Q}$. Using such descriptions, we study the Auslander-Gorenstein property of path algebras. In particular, we prove that the path algebra $R\mathcal{Q}$ is $k$-Gorenstein if and only if $\mathcal{Q}=\overrightarrow{A_n}$ and $R$ is a $k$-Gorenstein ring, where $n$ is the number of vertices of $\mathcal{Q}$.

arXiv commentAccepted for publication in Comm. Algebra

References in corpus (1)

• Dominant dimensions of two classes of finite dimensional algebras