Stable representation homology and Koszul duality

Abstract

Abstract This paper is a sequel to [Adv. Math. 245 (2013), 625–689], where we study the derived affine scheme DRep n ( A ) ${{\mathrm {DRep}}_n(A)}$ parametrizing the n-dimensional representations of an associative k-algebra A. In [Adv. Math. 245 (2013), 625–689], we have constructed canonical trace maps Tr n ( A ) • : HC • ( A ) → H • [ DRep n ( A ) ] GL n ${\operatorname{Tr}_n(A)_{\bullet }: \mathrm {HC}_{\bullet }(A) \rightarrow \mathrm {H}_{\bullet }[{\mathrm {DRep}}_n(A)]^{{\mathrm {GL}}_n} }$ extending the usual characters of representations to higher cyclic homology. This raises the natural question whether a well-known theorem of Procesi [Adv. Math. 19 (1976), 306–381.] holds in the derived setting: namely, is the algebra homomorphism Λ Tr n ( A ) • : Λ k [ HC • ( A ) ] → H • [ DRep n ( A ) ] GL n ${\Lambda \mathrm {Tr} _n(A)_{\bullet } : \Lambda _k[\mathrm {HC}_{\bullet }(A)] \rightarrow \mathrm {H}_{\bullet }[{\mathrm {DRep}}_n(A)]^{{\mathrm {GL}}_n}}$ defined by Tr n ( A ) • ${ \operatorname{Tr}_n(A)_{\bullet } }$ 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 ${{\mathrm {DRep}}_{\infty }(A)^{\operatorname{Tr}}}$ of the topological DG algebra lim ← DRep n ( A ) GL n ${\varprojlim {\mathrm {DRep}}_n(A)^{{\mathrm {GL}}_n} }$ . Our main result is that on passing to the inverse limit, the family of maps Λ Tr n ( A ) • ${\Lambda \mathrm {Tr} _n(A)_{\bullet }}$ `stabilizes' to an isomorphism Λ k ( HC ¯ • ( A ) ) ≅ H • [ DRep ∞ ( A ) Tr ] ${\Lambda _k(\overline{\mathrm {HC}}_{\bullet }(A)) \cong \mathrm {H}_{\bullet }[{\mathrm {DRep}}_{\infty }(A)^{\operatorname{Tr}}]}$ . 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 Λ Tr n ( A ) • ${ \Lambda \mathrm {Tr} _n(A)_{\bullet }}$ , 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 ) ${{\mathcal {C}}_\bullet ({\mathfrak {gl}}_\infty (A), {\mathfrak {gl}}_\infty (k); k)}$ equipped with a natural coalgebra structure is Koszul dual to the DG algebra DRep ∞ ( A ) Tr ${{\mathrm {DRep}}_{\infty }(A)^{\operatorname{Tr}}}$ . We also extend our main results to bigraded DG algebras, in which case we show the equality DRep ∞ ( A ) Tr = DRep ∞ ( A ) GL ∞ ${ {\mathrm {DRep}}_{\infty }(A)^{\operatorname{Tr}} = {\mathrm {DRep}}_{\infty }(A)^{{\mathrm {GL}}_{\infty }}}$ . As an application, we compute the Euler characteristics of DRep ∞ ( A ) GL ∞ ${{\mathrm {DRep}}_{\infty }(A)^{{\mathrm {GL}}_{\infty }}}$ and HC ¯ • ( A ) ${\overline{\mathrm {HC}}_{\bullet }(A)}$ and derive some interesting combinatorial identities.