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# Exact Morphism Category and Gorenstein-projective Representations

Published:2015-07-27
Printed: Dec 2015
• Xiu-Hua Luo,
Department of Mathematics, Nantong University , Nantong 226019, P. R. China
 Format: LaTeX MathJax PDF

## Abstract

Let $Q$ be a finite acyclic quiver, $J$ be an ideal of $kQ$ generated by all arrows in $Q$, $A$ be a finite-dimensional $k$-algebra. The category of all finite-dimensional representations of $(Q, J^2)$ over $A$ is denoted by $\operatorname{rep}(Q, J^2, A)$. In this paper, we introduce the category $\operatorname{exa}(Q,J^2,A)$, which is a subcategory of $\operatorname{rep}{}(Q,J^2,A)$ of all exact representations. The main result of this paper explicitly describes the Gorenstein-projective representations in $\operatorname{rep}{}(Q,J^2,A)$, 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 $(Q, J^2)$ over $A$.
 Keywords: representations of a quiver over an algebra, exact representations, Gorenstein-projective modules
 MSC Classifications: 18G25 - Relative homological algebra, projective classes

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