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