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By Bernhard (f-paris-imj) Scherotzke


Cluster categories were introduced in [A. B. Buan et al., Adv. Math. 204 (2006), no. 2, 572–618; MR2249625 (2007f:16033)] for acyclic quivers and independently in [P. Caldero, F. Chapoton and R. Schiffler, Trans. Amer. Math. Soc. 358 (2006), no. 3, 1347–1364; MR2187656 (2007a:16025)] for Dynkin quivers of type A. They have played an important rôle in the study of Fomin and Zelevinsky’s cluster algebras. Integral cluster categories appear naturally in the study of quantum cluster algebras as defined and studied in [A. Berenstein and A. Zelevinsky, Adv. Math. 195 (2005), no. 2, 405–455; MR2146350 (2006a:20092); V. V. Fock and A. B. Goncharov, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 6, 865–930; MR2567745 (2011f:53202)]. Indeed, one would like to interpret the quantum parameter q as the cardinality of a finite field [D. Rupel, Int. Math. Res. Not. IMRN 2011, no. 14, 3207– 3236; MR2817677 (2012g:13042)] and in order to study the cluster categories over all prime fields simultaneously [F. Qin, J. Reine Angew. Math. 668 (2012), 149–190; MR2948875], one considers the cluster category over the ring of integers; cf. the appendix to [F. Qin, op. cit.]. In this paper, the authors continue the study begun there: For an acyclic quiver Q and a principal ideal domain R, they construct the cluster category CRQ using C. Amiot’s method [Ann. Inst. Fourier (Grenoble) 59 (2009), no. 6, 2525–2590; MR2640929 (2011c:16026)] as a triangle quotien

Topics: This shows in particular that the orbit category is tria
Year: 2013
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