Koszul Duality for Multigraded Algebras

Abstract

Classical Koszul duality sets up an adjoint pair of functors, establishing an equivalence $  F: D^b(A) \leftrightarrows D^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 $\Lambda$-graded algebra, where $\Lambda$ is any abelian group (not just $\Z$). Second, we will allow filtered algebras. In fact we are considering filtered quadratic algebras with an (internal) $\Lambda$-grading.