Koszul Duality for Multigraded Algebras

Abstract

Classical Koszul duality sets up an adjoint pair of functors, establishing an equivalence $ÂF:D^b(A)\leftrightarrows ÂD^b(A^{!}):G,$ where $A$ is a quadratic algebra, $A^{!}$ is the quadratic dual, and $D^b$ refers to the bounded derived category of complexes of graded modules over the graded algebra (i.e., $A$ or $A^{!}$). This duality can be extended in many ways. We consider here two extensions: first we wish to allow a $\mathrm{\Lambda }$-graded algebra, where $\mathrm{\Lambda }$ is any abelian group (not just $\text{Z}$). Second, we will allow filtered algebras. In fact we are considering filtered quadratic algebras with an (internal) $\mathrm{\Lambda }$-grading.