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