Gorenstein-Projective Modules over the Ring of Dual Integers

Abstract

The ring of dual integers is the bounded polynomial ring $\mathbb Z[\eps]=\mathbb Z[T]/(T^2)$ with integer coefficients. We describe the (finitely generated) Gorenstein-projective $\mathbb Z[\eps]$-modules as the torsionless $\mathbb Z[\eps]$-modules, while the stable category of $\Gproj\mathbb Z[\eps]$ modulo projectives is shown to be equivalent to the orbit category $\mathcal D^b(\mathbb Z)/[1]$ of the derived category of the integers. It follows that the latter carries the structure of a triangulated category. \smallskip The category $\Gproj\mathbb Z[\eps]$ is related to the embeddings of a subgroup in a free abelian group and has a quotient which is equivalent to the category of finite abelian groups. In fact, we present a cube which has as vertices eight related categories and as edges in each of the three directions functors which are related to push-down functors modulo the shift; canonical functors to stable categories; and homology functors, respectively. We note that in $\Gproj\mathbb Z[\eps]$ uniqueness of direct sum decomposition fails.