Minimal semiinjective resolutions in the $Q$-shaped derived category

Abstract

Semiinjective resolutions of chain complexes are used for the computation of $\mathrm{Hom}$ spaces in the derived category $\mathcal{D}(A)$ of a ring $A$. Minimal semiinjective resolutions have the additional property of being unique. The $Q$-shaped derived category ${\mathcal{D}}_Q(A)$ consists of $Q$-shaped diagrams for a suitable preadditive category $Q$, and it generalises $\mathcal{D}(A)$. Some special cases of ${\mathcal{D}}_Q(A)$ are the derived categories of differential modules, $m$-periodic chain complexes, and $N$-complexes, and there are many other possibilities. The category ${\mathcal{D}}_Q(A)$ shares some key properties of $\mathcal{D}(A)$; for instance, it is triangulated and compactly generated. This paper establishes a theory of minimal semiinjective resolutions in ${\mathcal{D}}_Q(A)$. As a sample application, it generalises a theorem by Ringel and Zhang on differential modules.