A derived equivalence between cluster equivalent algebras

Let Q be an acyclic quiver. Associated with any element w of the Coxeter group of Q, triangulated categories were introduced in Buan et al. (2009) [BIRS09], . For any reduced expression w of w, the categories are shown to be triangle equivalent to generalized cluster categories CΓw associated to algebras Γw of global dimension ⩽2 in Amiot et al. (2011) [ART11], . For w satisfying a certain property, called co-c-sortable, other algebras Aw of global dimension ⩽2 are constructed in Amiot (2009) [Ami09], and Amiot et al. (2011) [AIRT11], with a triangle equivalence . The main result of this paper is that the algebras Γw and Aw are derived equivalent when w is co-c-sortable. The proof constructs explicitly a tilting module using the 2-APR-tilting theory introduced in Iyama and Oppermann (2011) [IO09].