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

TL;DR

This paper develops a theory of minimal semiinjective resolutions within the $Q$-shaped derived category, generalizing classical concepts and applying to various complex types, including differential modules and $N$-complexes.

Contribution

It introduces a framework for minimal semiinjective resolutions in $ ext{D}_Q(A)$, extending classical homological algebra to a broader class of diagram categories.

Findings

Established a theory of minimal semiinjective resolutions in $ ext{D}_Q(A)$.

Generalized a theorem by Ringel--Zhang on differential modules.

Showed that $ ext{D}_Q(A)$ shares key properties with classical derived categories.

Abstract

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