Derived representation schemes and cyclic homology

Abstract

We describe the derived functor DRepV(A) of the affine representation scheme RepV(A) parametrizing the representations of an associative k-algebra A on a finite-dimensional vector space V. We construct the characteristic maps TrV(A)n:HCn(A)→Hn[DRepV(A)] extending the canonical trace TrV(A):HC0(A)→k[RepV(A)] to the higher cyclic homology of the algebra A, and describe a related derived version of the representation functor introduced recently by M. Van den Bergh. We study various operations on the homology of DRepV(A) induced by known operations on cyclic and Hochschild homology of A.