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.
Categories: math.RT math.CT math.RA