The $Q$-shaped derived category of a ring -- compact and perfect objects

TL;DR

This paper studies the structure of the $Q$-shaped derived category of a ring, establishing its compact generation, defining perfect objects, and analyzing their relationship with compact objects.

Contribution

It proves that the $Q$-shaped derived category is compactly generated and characterizes perfect objects as a triangulated subcategory of compact objects.

Findings

The $Q$-shaped derived category is a compactly generated triangulated category.

Perfect objects form a triangulated subcategory of compact objects.

The subcategories coincide if and only if the perfect objects form a thick subcategory.

Abstract

In a previous work we constructed the $Q$-shaped derived category of any ring $A$ for any suitably nice category $Q$. The $Q$-shaped derived category of $A$, which is denoted by $\mathcal{D}_{Q}(A)$, is a generalization of the ordinary derived category. In this paper we prove that the $Q$-shaped derived category of $A$ is a compactly generated triangulated category. We also define perfect objects in $\mathcal{D}_{Q}(A)$ and prove that these constitute a triangulated subcategory, $\mathcal{D}^\mathrm{perf}_{Q}(A)$, of the category $\mathcal{D}_{Q}(A)^\mathrm{c}$ of compact objects in the $Q$-shaped derived category. The subcategories $\mathcal{D}^\mathrm{perf}_{Q}(A)$ and $\mathcal{D}_{Q}(A)^\mathrm{c}$ coincide if and only if the former is thick.