Folding derived categories with Frobenius functors

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 F on an algebra A induces naturally a functor F on the (bounded) derived category Db(A) of mod-A, and we further prove that the derived category Db(AF) of mod-AF for the F-fixed point algebra AF is naturally embedded as the triangulated subcategory Db(A)F of F-stable objects in Db(A). When applying the theory to an algebra with finite global dimension, we discover a folding relation between the Auslander-Reiten triangles in Db(AF) and those in Db(A). Thus, the AR-quiver of Db(AF) can be obtained by folding the AR-quiver of Db(A). Finally, we further extend this relation to the root categories ℛ(AF) of AF and ℛ(A) of A, and show that, when A is hereditary, this folding relation over the indecomposable objects in ℛ(AF) and ℛ(A) results in the same relation on the associated root systems as induced from the graph folding relation.