Exact Morphism Category and Gorenstein-projective Representations

Canadian mathematical bulletin

Xiu-Hua Luo¹ 

[1] Department of Mathematics, Nantong University , Nantong 226019, P. R. China

关键词: representations of a quiver over an algebra;    exact representations;    Gorenstein-projective modules;   

DOI  :  10.4153/CMB-2015-051-6

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Let $Q$ be a finite acyclic quiver, $J$ be an ideal of $kQ$ generatedby all arrows in $Q$, $A$ be a finite-dimensional $k$-algebra. Thecategory of all finite-dimensional representations of $(Q, J^2)$ over$A$ is denoted by $operatorname{rep}(Q, J^2, A)$. In this paper, weintroduce the category $operatorname{exa}(Q,J^2,A)$, which is asubcategory 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, ifand only if the Gorenstein-projective representations are exactly theexact representations of $(Q, J^2)$ over $A$.

【 授权许可】

Unknown