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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi> DRep </m:mi> <m:mi>n</m:mi> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mi>A</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> ${{\mathrm {DRep}}_n(A)}$ parametrizing the n -dimensional representations of an associative k -algebra A . In [Adv. Math. 245 (2013), 625–689], we have constructed canonical trace maps <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mo form="prefix">Tr</m:mo> <m:mi>n</m:mi> </m:msub> <m:msub> <m:mrow> <m:mo>(</m:mo> <m:mi>A</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>•</m:mo> </m:msub> <m:mo>:</m:mo> <m:msub> <m:mi> HC </m:mi> <m:mo>•</m:mo> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mi>A</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>→</m:mo> <m:msub> <m:mi mathvariant="normal">H</m:mi> <m:mo>•</m:mo> </m:msub> <m:msup> <m:mrow> <m:mo>[</m:mo> <m:msub> <m:mi> DRep </m:mi> <m:mi>n</m:mi> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mi>A</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>]</m:mo> </m:mrow> <m:msub> <m:mi> GL </m:mi> <m:mi>n</m:mi> </m:msub> </m:msup> </m:mrow> </m:math> ${\operatorname{Tr}_n(A)_{\bullet }: \mathrm {HC}_{\bullet }(A) \rightarrow \mathrm {H}_{\bullet }[{\mathrm {DRep}}_n(A)]^{{\mathrm {GL}}_n} }$ 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Λ</m:mi> <m:msub> <m:mi> Tr </m:mi> <m:mi>n</m:mi> </m:msub> <m:msub> <m:mrow> <m:mo>(</m:mo> <m:mi>A</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>•</m:mo> </m:msub> <m:mo>:</m:mo> <m:msub> <m:mi>Λ</m:mi> <m:mi>k</m:mi> </m:msub> <m:mrow> <m:mo>[</m:mo> <m:msub> <m:mi> HC </m:mi> <m:mo>•</m:mo> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mi>A</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>]</m:mo> </m:mrow> <m:mo>→</m:mo> <m:msub> <m:mi mathvariant="normal">H</m:mi> <m:mo>•</m:mo> </m:msub> <m:msup> <m:mrow> <m:mo>[</m:mo> <m:msub> <m:mi> DRep </m:mi> <m:mi>n</m:mi> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mi>A</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>]</m:mo> </m:mrow> <m:msub> <m:mi> GL </m:mi> <m:mi>n</m:mi> </m:msub> </m:msup> </m:mrow> </m:math> ${\Lambda \mathrm {Tr} _n(A)_{\bullet } : \Lambda _k[\mathrm {HC}_{\bullet }(A)] \rightarrow \mathrm {H}_{\bullet }[{\mathrm {DRep}}_n(A)]^{{\mathrm {GL}}_n}}$ defined by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mo form="prefix">Tr</m:mo> <m:mi>n</m:mi> </m:msub> <m:msub> <m:mrow> <m:mo>(</m:mo> <m:mi>A</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>•</m:mo> </m:msub> </m:mrow> </m:math> ${ \operatorname{Tr}_n(A)_{\bullet } }$ surjective? In the present paper, we answer this question for augmented algebras. Given such an algebra, we construct a canonical dense subalgebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi> DRep </m:mi> <m:mi>∞</m:mi> </m:msub> <m:msup> <m:mrow> <m:mo>(</m:mo> <m:mi>A</m:mi> <m:mo