Gradings and Derived Categories

Let G{{\mathcal G}} be a group, Λ a G{{\mathcal G}}-graded Artin algebra and gr(Λ) denote the category of finitely generated G{{\mathcal G}}-graded Λ-modules. This paper provides a framework that allows an extension of tilting theory to D^b(gr(L)){{\mathcal D}}^b(\rm gr(\Lambda)) and to study connections between the tilting theories of D^b(L){{\mathcal D}}^b(\Lambda) and D^b(gr(L)){{\mathcal D}}^b(\rm gr(\Lambda)). In particular, using that if T is a gradable Λ-module, then a grading of T induces a G{{\mathcal G}}-grading on End_Λ(T), we obtain conditions under which a derived equivalence induced by a gradable Λ-tilting module T can be lifted to a derived equivalence between the derived categories D^b(gr(L)){{\mathcal D}}^b(\rm gr(\Lambda)) and D^b(gr(End_L(T))){{\mathcal D}}^b(\rm gr(\rm End_{\Lambda}(\textit T))).