Representations of quivers over the algebra of dual numbers

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 modkQ. 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=k[ϵ] is the algebra of dual numbers (the factor algebra of the polynomial ring k[T] in one variable T modulo the ideal generated by T2), thus to the Λ-modules, where Λ=kQ[ϵ]=kQ[T]/〈T2〉. The algebra Λ is a 1-Gorenstein algebra, thus the torsionless Λ-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_ is a triangulated category. As we will see, the category L is the category of perfect differential kQ-modules and L_ is the corresponding homotopy category. The category L_ is triangle equivalent to the orbit category of the derived category Db(modkQ) modulo the shift and the homology functor H:modΛ→modkQ yields a bijection between the indecomposables in L_ and those in modkQ. 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 L and describe the Auslander-Reiten quivers of L and L_.