Minimal injective resolutions and Auslander-Gorenstein property for path algebras

ABSTRACT

Let R be a ring and 𝒬 be a finite and acyclic quiver. We present an explicit formula for the injective envelopes and projective precovers in the category Rep(𝒬,R) of representations of 𝒬 by left R-modules. We also extend our formula to all terms of the minimal injective resolution of R𝒬. Using such descriptions, we study the Auslander-Gorenstein property of path algebras. In particular, we prove that the path algebra R𝒬 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 𝒬.

KEYWORDS:

• Auslander-Gorenstein property
• injective envelope
• representations of quivers

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16G20
• 16B50
• 18A40
• 18G05

Acknowledgments

The authors would like to thank the referee for useful comments and hints that improved our exposition. The authors also thank the Center of Excellence for Mathematics (University of Isfahan).