Representations of quivers over the algebra of dual numbers

Abstract

The representations of a quiver Q over a field k have been studied for a long time. 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[\epsilon] is the algebra of dual numbers, thus to the \Lambda-modules with \Lambda = kQ[\epsilon]. The algebra \Lambda{} is a 1-Gorenstein algebra, therefore the torsionless \Lambda-modules are known to be of special interest (as the Gorenstein-projective or maximal Cohen-Macaulay modules). They form a Frobenius category \Cal L, thus the corresponding stable category is a triangulated category. As we show, the category \Cal L is the category of perfect differential kQ-modules and the stable category of \Cal L is triangle equivalent to the orbit category of the derived category D^b(mod kQ) modulo the shift. The homology functor H from mod \Lambda{} to \mod kQ yields a bijection between the indecomposables in the stable category of \Cal L and those in mod kQ. Our main interest lies in the inverse, it is given by the minimal \Cal L-approximation. Also, we determine the kernel of the restriction of the functor H to \Cal L and we describe the Auslander-Reiten quiver of \Cal L.