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Exact Morphism Category and Gorenstein-projective Representations

Published online by Cambridge University Press:  20 November 2018

Xiu-Hua Luo
Xiu-Hua Luo*
Affiliation:
Department of Mathematics, Nantong University, Nantong 226019, P. R. China e-mail: xiuhualuo2014@163.com
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Abstract

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Let $Q$ be a finite acyclic quiver, let $J$ be an ideal of $kQ$ generated by all arrows in $Q$, and let $A$ be a finite-dimensional $k$ -algebra. The category of all finite-dimensional representations of $\left( Q,\,{{J}^{2}} \right)$ over $A$ is denoted by $\text{rep}\left( Q,\,{{J}^{2}},\,A \right)$ . In this paper, we introduce the category $\text{exa}\left( Q,{{J}^{2}},A \right),$ which is a subcategory of $\text{rep}\left( Q,\,{{J}^{2}},\,A \right)$ of all exact representations. The main result of this paper explicitly describes the Gorenstein-projective representations in $\text{rep}\left( Q,\,{{J}^{2}},\,A \right)$, via the exact representations plus an extra condition. As a corollary, $A$ is a self-injective algebra if and only if the Gorenstein-projective representations are exactly the exact representations of $\left( Q,\,{{J}^{2}} \right)$ over $A$.

Keywords

18G25representation of a quiver over an algebraexact representationGorenstein-projective module
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 4 , 01 December 2015 , pp. 824 - 834
Copyright
Copyright © Canadian Mathematical Society 2015

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