A Non-graded Koszul Duality and Its Applications

Abstract

Let \(\Lambda\) be a finite-dimensional Koszul algebra with Koszul dual \(\Lambda^!\). We establish derived Koszul dualities at the level of bounded derived categories, both in the graded setting \(\mathsf{D}^{b}(\Lambda\textup{-gmod})\) and in the ungraded setting \(\mathsf{D}^{b}(\Lambda\textup{-mod})\), without imposing finiteness conditions on \(\Lambda^!\). We first prove a graded derived Koszul duality for every finite-dimensional Koszul algebra, with no Noetherian or coherence assumptions on the Koszul dual. We then show that the bounded derived category \(\mathsf{D}^{b}(\Lambda\textup{-mod})\) can be reconstructed from the graded theory as the triangulated hull of an orbit category. This yields a genuinely non-graded derived Koszul duality. We further establish singular and dg refinements of these dualities. For Iwanaga--Gorenstein Koszul algebras, this gives a stable Koszul duality for graded Gorenstein-projective modules and their ungraded counterparts, providing a non-graded form of the Bernstein--Gel'fand--Gel'fand correspondence. As applications, we obtain new descriptions of the bounded derived categories \(\mathsf{D}^{b}(\mathcal{O}_{\lambda})\) for all integral blocks of category \(\mathcal{O}\), including singular blocks, thereby closing a gap left open in the work of Beilinson, Ginzburg, and Soergel. We also establish analogous dualities for certain categories of perverse sheaves arising in geometric representation theory. Finally, we formulate conjectural descriptions of bounded derived and singularity categories of finite-dimensional graded algebras in terms of dg orbit categories.