The integral cluster category (Q2895807)

As the title of the paper indicates, the authors continue the study of integral cluster categories. The underlying algebras are quiver algebras \(RQ\), where \(R\) is a commutative ring and \(Q\) is an acyclic quiver. In the case \(R\) is a field, the cluster category \(\mathcal{C}^{\mathrm{orb}}_{{RQ}}\), as originally defined by Buan et al., is the category of orbits of the bounded derived category of \(RQ\) under the action of the auto-equivalence \(\Sigma^{-2}S\), where \(\Sigma\) is the suspension functor and \(S\) is the appropriately defined Serre functor. An alternative construction was given by Amiot: her cluster category \(\mathcal{C}_{RQ}\) is a triangulated quotient of the perfect derived category of the Ginzburg dg algebra \(\Gamma\) associated with \(RQ\). (For a quick refresh on the Ginzburg algebra and its derived category, the authors refer to the paper by the first author and \textit{D. Yang} [Adv. Math. 226, No. 3, 2118--2168 (2011; Zbl 1272.13021), Theorems 2.6 and 2.11], with the former pointer being inadvertently omitted from the text, see p. 2875 of the reviewed paper.) It is a result of Amiot that the two definitions are equivalent.NEWLINENEWLINEReturning to a general commutative ring \(R\), the authors show in Theorem~4.1 that the derived tensor product with \(\Gamma\) induces a fully faithful embedding \(\mathcal{C}^{\mathrm{orb}}_{{RQ}} \to \mathcal{C}_{RQ}\). (In the original statement of the theorem, the domain category is written without the subscript \(RQ\). The subscript is used in the introduction.) The first main result of the paper, Theorem 5.10, says that when \(R\) is hereditary, the above embedding is actually an equivalence. (This is even stronger that what is claimed in the introduction, where \(R\) is assumed to be a PID.)NEWLINENEWLINEThe second main result concerns rigid indecomposables in \(\mathcal{C}_{RQ}\) when \(R\) is a PID. It is shown, in Theorem 5.9, that each rigid object in the integral cluster category is either (the image of) a rigid indecomposable \(RQ\)-module or the suspension of (the image of) an indecomposable projective \(RQ\)-module. In the second part of the theorem, the authors pass from \(R\) to a residue field~\(\mathbb{F}\), and show that the corresponding reduction functor induces a bijection between isoclasses of rigid indecomposables in \(\mathcal{C}_{RQ}\) and those in \(\mathcal{C}_{\mathbb{F}Q}\).