Stable representation homology and Koszul duality

Abstract

This paper is a sequel to [2], where we study the derived affine scheme DRepn(A) parametrizing the n-dimensional representations of an associative k-algebra A. In [2], we have constructed canonical trace maps Trn.(A)• : HC•(A) → H•[DRepn(A)]GLn extending the usual characters of representations to higher cyclic homology. This raises the natural question whether a well-known theorem of Procesi [30] holds in the derived setting: namely, is the algebra homomorphism ΛTrn(A)• : Λk[HC•(A)] → H•[DRepn(A)]GLn defined by Trn(A)• surjective? In the present paper, we answer this question for augmented algebras. Given such an algebra, we construct a canonical dense subalgebra DRep∞(A)Tr of the topological DG algebra lim DRepn(A)GLn. Our main result is that on passing to the inverse limit, the family of maps ΛTr←n(A)• 'stabilizes' to an isomorphism λk(HC•(A)) ≅ H•[DRep∞(A)Tr]. The derived version of Procesi's theorem does therefore hold in the limit as n → ∞. However, for a fixed (finite) n, there exist homological obstructions to the surjectivity of ΛTrn(A)•, and we show on simple examples that these obstructions do not vanish in general. We compare our result with the classical theorem of Loday, Quillen and Tsygan on stable homology of matrix Lie algebras. We show that the Chevalley-Eilenberg complex C•(gl∞(A), gl∞(k); k) equipped with a natural coalgebra structure is Koszul dual to the DG algebra DRep∞(A)Tr. We also extend our main results to bigraded DG algebras, in which case we show the equality DRep∞(A)Tr = DRep∞(A)GL∞. As an application, we compute the Euler characteristics of DRep∞(A)GL∞ and HC•(A) and derive some interesting combinatorial identities.