The \(Q\)-shaped derived category of a ring -- compact and perfect objects (Q6567159)

"For any ring \(A\) and a self-injective quiver \(Q\) with relations, the authors considered the \(Q\)-shaped derived category \(\mathcal{D}_Q(A)\), whose objects are the representations of \(Q\) with values in \(\mathrm{Mod}\, A\), see [\textit{H. Holm} and \textit{P. Jorgensen}, ``The $Q$-shaped derived category of a ring'', Preprint, \url{arXiv:2101.06176}]. If \(Q\) is the following quiver\N\[\N\begin{tikzcd}\N\cdots \ar[r, ""\alpha""] &\bullet \ar[r, ""\alpha""] &\bullet \ar[r, ""\alpha""] &\cdots\N\end{tikzcd}\N\]\Nwith the relations \(\alpha^2=0\), then the representations of \(Q\) with values in \(\mathrm{Mod}\, A\) is just the complex of \(\mathrm{Mod}\, A\), and the \(Q\)-shaped derived category \(\mathcal{D}_Q(A)\) is the usual derived category \(\mathcal{D}(A)\). Therefore, the \(Q\)-shaped derived category \(\mathcal{D}_Q(A)\) generalises \(\mathcal{D}(A)\) when \(Q\) is an arbitrary self-quiver.\N\NIt is shown that \(\mathcal{D}_Q(A)\) and \(\mathcal{D}(A)\) share some common properties, for example, both of them are triangulated categories.\NHowever, there are some key properties of \(\mathcal{D}(A)\) which do not generalise to \(\mathcal{D}_Q(A)\), and the purpose of this paper is to compare and contrast these categories by investigating several key classes of objects. \NFirstly, they show that a well-known result, which says that a bounded complex of finitely generated projective \(A\)-modules is compact in \(\mathcal{D}(A)\), can not be generalized to \(\mathcal{D}_Q(A)\). \NSecondly, they introduce perfect objects in \(\mathcal{D}_Q(A)\) and compare them to compact objects in \(\mathcal{D}_Q(A)\).\NAlso, they prove that \(\mathcal{D}_Q(A)\) is a compactly generated triangulated category whose compact generators can be described completely.\NFinally, they investigate the class of cofibrant (resp. fibrant) objects in the injective (resp. projective) model structure on the representations category of \(Q\) with values in \(\mathrm{Mod}\, A\)."