Koszul duality for non-graded derived categories

Abstract

We are concerned with relating derived categories of all modules of two dual Koszul algebras defined by a locally bounded quiver. We first generalize the well known Acyclic Assembly Lemma and formalize an old method of extending a functor from an additive category into a complex category to its complex category. Applying this to the Koszul functor associated with a Koszul algebra defined by a gradable quiver, we obtain a Koszul complex functor, that descends to an equivalence of a continuous family of pairs of triangulated subcategories of doubly unbounded complexes of the respective derived categories of all modules of the Koszul algebra and its Koszul dual. Under this special setting, this extends Beilinson, Ginzburg and Soegel's Koszul duality. In case the Koszul algebra is right or left locally bounded and its Koszul dual is left or right locally bounded respectively (for instance, the quiver has no right infinite path or no left infinite path), our Koszul duality restricts to an equivalence of the bounded derived categories of finitely supported modules, and an equivalence of the bounded derived categories of finite dimensional modules.