TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 358, Number 6, Pages 2373–2398 S 0002-9947(05)04035-3 Article electronically published on December 20, 2005
KOSZUL DUALITY AND EQUIVALENCES OF CATEGORIES
GUNNAR FLØYSTAD
Abstract. Let A and A! be dual Koszul algebras. By Positselski a filtered algebra U with gr U = A is Koszul dual to a differential graded algebra (A!,d). We relate the module categories of this dual pair by a ⊗−Hom adjunction. This descends to give an equivalence of suitable quotient categories and generalizes work of Beilinson, Ginzburg, and Soergel.
Introduction Koszul algebras are graded associative algebras which have found applications in many different branches of mathematics. A prominent feature of Koszul algebras is that there is a related dual Koszul algebra. Many of the applications of Koszul algebras concern relating the module categories of two dual Koszul algebras and this is the central topic of this paper. Let A and A! be dual Koszul algebras which are quotients of the tensor algebras T (V )andT (V ∗) for a finite-dimensional vector space V and its dual V ∗.The classical example being the symmetric algebra S(V ) and the exterior algebra E(V ∗). Bernstein, Gel’fand, and Gel’fand [3] related the module categories of S(V )and E(V ∗), and Beilinson, Ginzburg, and Soergel [2] developed this further for general pairs A and A!. Now Positselski [21] considered filtered deformations U of A such that the asso- ciated graded algebra gr U is isomorphic to A. He showed that this is equivalent to giving A! the structure of a curved differential graded algebra (cdga), i.e. A! has ! 2 an anti-derivation d and a distinguished cycle c in (A )2 such that d (a)=[c, a]. When U is augmented then c =0andA! is a differential graded algebra. The typ- ical example of this is when U is the enveloping algebra of a Lie algebra V .ALie algebra structure on V is equivalent to giving E(V ∗) the structure of a differential graded algebra. We take this situation as our basic setting and relate the module categories of U and the cdga (A!,d,c). More precisely let Kom(U) be the category of complexes of left U-modules. We define a natural category Kom(A!,d,c)ofcurved differen- tial graded (cdg) left modules over the cdga (A!,d,c)(whenc = 0 these are the differential graded modules), and relate them by a pair of adjoint functors F (1) Kom(A!,d,c) Kom(U). G The vector space V may be equipped with a grading of an abelian group Λ such that U and (A!,d,c) have Λ-gradings. Then these categories consist of Λ-graded
Received by the editors January 26, 2004. 2000 Mathematics Subject Classification. Primary 16S37, 16D90.