Representations of quivers over the algebra of dual numbers

JOURNAL OF ALGEBRA

卷:475

Article

Ringel, Claus Michael¹  Zhang, Pu¹ 

[1] Shanghai Jiao Tong Univ, Dept Math, Shanghai 200240, Peoples R China

关键词: Finite acyclic quivers and their;    representations over an algebra;    Dual numbers;    Gorenstein algebras;    Torsionless modules;    Gorenstein-projective modules;    Strongly Gorenstein-projective;    modules Differential modules;    Perfect differential modules;    Derived category;    Perfect complexes;    Frobenius category;    Homotopy category;    Triangulated category of;    singularities;   

DOI  :  10.1016/j.jalgebra.2016.12.001

来源: Elsevier

【 摘 要 】

The representations of a quiver Q over a field k (the kQ-modules, where kQ is the path algebra of Q over k) have been studied for a long time, and one knows quite well the structure of the module category mod kQ. It seems to be worthwhile to consider also representations of Q over arbitrary finite-dimensional k-algebras A. Here we draw the attention to the case when A = kappa[epsilon] is the algebra of dual numbers (the factor algebra of the polynomial ring k[T] in one variable T modulo the ideal generated by T-2), thus to the A-modules, where A = kappa Q[epsilon] = kappa Q[T]/(T-2). The algebra A is a 1-Gorenstein algebra, thus the torsionless A-modules are known to be of special interest (as the Gorenstein-projective or maximal Cohen Macaulay modules). They form a Frobenius category L, thus the corresponding stable category L (L) under bar is a triangulated category. As we will see, the category L is the category of perfect differential kQ-modules and L (L) under bar is the corresponding homotopy category. The category G is triangle equivalent to the orbit category of the derived category D-b(mod kQ) modulo the shift and the homology functor H: mod A -> mod kQ yields a bijection between the indecomposables in L (L) under bar and those in mod kQ. Our main interest lies in the inverse, it is given by the minimal L-approximation. Also, we will determine the kernel of the restriction of the functor H to G and describe the Auslander Reiten quivers of L and L (L) under bar. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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