Stable representation homology and Koszul duality

Abstract

This paper is a sequel to [BKR], where we studied the derived affine scheme DRep_n(A) of the classical representation scheme Rep_n(A) for an associative k-algebra A. In [BKR], we have constructed canonical trace maps Tr_n(A): HC(A) -> H[DRep_n(A)]^GL extending the usual characters of representations to higher cyclic homology. This raises a question whether a well known theorem of Procesi [P] holds in the derived setting: namely, is the algebra homomorphism Sym[Tr_n(A)]: Sym[HC(A)] -> H[DRep_n(A)]^GL defined by Tr_n(A) surjective ? In the present paper, we answer this question for augmented algebras. Given such an algebra, we construct a canonical dense DG subalgebra DRep_\infty(A)^Tr of the topological DG algebra DRep_\infty(A)^{GL_\infty}. It turns out that on passing to the inverse limit (as n -> \infty), the family of maps Sym[Tr_n(A)] "stabilizes" to an isomorphism Sym[\bar{HC}(A)] = H[DRep_\infty(A)^Tr]. The derived version of Procesi's theorem does therefore hold in the limit. However, for a fixed (finite) n, there exist homological obstructions to the surjectivity of Sym[Tr_n(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 relative Chevalley-Eilenberg complex C(gl_\infty(A), gl_\infty(k); k) equipped with the natural coalgebra structure is Koszul dual to the DG algebra DRep_\infty(A)^Tr. We also extend our main results to bigraded DG algebras, in which case we show that DRep_{\infty}(A)^Tr = DRep_{\infty}(A)^GL_{\infty}. As an application, we compute the (bigraded) Euler characteristics of DRep_\infty(A)^GL_{\infty} and \bar{HC}(A) and derive some interesting combinatorial identities.