Koszul Duality for Coherent Sheaves

Abstract

We establish a bounded derived Koszul duality for infinite-dimensional Koszul algebras and derive the corresponding singular Koszul duality. We then specialize this framework to two classes of Koszul algebras, namely quadratic monomial algebras and absolutely Koszul algebras satisfying an additional homological condition, for which the resulting dualities admit particularly well-behaved forms. As an application to algebraic geometry, let \(\Lambda\) be a commutative noetherian Koszul algebra generated in degree \(1\), and set \(X=\operatorname{Proj}(\Lambda)\). We obtain a Koszul-dual description of \(\mathsf{D}^{b}\!\bigl(\operatorname{coh}(X)\bigr)\), yielding a BGG-type correspondence for projective schemes defined by such algebras. As a second application, in noncommutative projective geometry, we consider generalized Artin--Schelter regular Koszul algebras \(\Lambda^{!}\) arising as Koszul duals of finite-dimensional self-injective Koszul algebras \(\Lambda\). We show that \(\mathsf{D}^{b}\!\bigl(\operatorname{qgr}(\Lambda^{!})\bigr)\) is triangulated equivalent to the bounded derived category of finite-dimensional modules over a finite-dimensional Koszul algebra of finite global dimension. This yields a Beilinson-type description of \(\mathsf{D}^{b}\!\bigl(\operatorname{qgr}(\Lambda^{!})\bigr)\), extending the classical description of coherent sheaves on projective space to this noncommutative setting.