Folding derived categories with Frobenius functors

Abstract: Following the work [B. Deng, J. Du, Frobenius morphisms and representations of algebras, Trans. Amer. Math. Soc. 358 (2006) 3591–3622], we show that a Frobenius morphism on an algebra induces naturally a functor on the (bounded) derived category of , and we further prove that the derived category of for the -fixed point algebra is naturally embedded as the triangulated subcategory of -stable objects in . When applying the theory to an algebra with finite global dimension, we discover a folding relation between the Auslander–Reiten triangles in and those in . Thus, the AR-quiver of can be obtained by folding the AR-quiver of . Finally, we further extend this relation to the root categories of and of , and show that, when is hereditary, this folding relation over the indecomposable objects in and results in the same relation on the associated root systems as induced from the graph folding relation. [Copyright &y& Elsevier]