Gradings and Derived Categories

Let \({{\mathcal G}}\) be a group, Λ a \({{\mathcal G}}\)-graded Artin algebra and gr(Λ) denote the category of finitely generated \({{\mathcal G}}\)-graded Λ-modules. This paper provides a framework that allows an extension of tilting theory to \({{\mathcal D}}^b(\rm gr(\Lambda))\) and to study connections between the tilting theories of \({{\mathcal D}}^b(\Lambda)\) and \({{\mathcal D}}^b(\rm gr(\Lambda))\). In particular, using that if T is a gradable Λ-module, then a grading of T induces a \({{\mathcal G}}\)-grading on EndΛ(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 \({{\mathcal D}}^b(\rm gr(\Lambda))\) and \({{\mathcal D}}^b(\rm gr(\rm End_{\Lambda}(\textit T)))\).