Gorenstein-Projective Modules over the Ring of Dual Integers

The ring of dual integers is the bounded polynomial ring $\mathbb{Z}[\epsilon ]=\mathbb{Z}[T]\mathrm{/}(T^2)$ with integer coefficients. We describe the (finitely generated) Gorenstein-projective $\mathbb{Z}[\epsilon ]$-modules as the torsionless $\mathbb{Z}[\epsilon ]$-modules, or equivalently, as the perfect differential structures of abelian groups. Moreover, the stable category of G-proj$\mathbb{Z}[\epsilon ]$ modulo projectives is shown to be equivalent to the orbit category ${\mathcal{D}}^b(\mathbb{Z})\mathrm{/}[1]$ of the derived category of the integers and to the homotopy category of perfect differential structures. We note that the category G-proj$\mathbb{Z}[\epsilon ]$ 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 functors which are such that the faces of the cube give rise to commutative diagrams. Among interesting properties in G-proj$\mathbb{Z}[\epsilon ]$, we note that uniqueness of direct sum decomposition fails.