No abstract

Abstract. We describe the derived functor DRep V (A) of the affine representation scheme Rep V (A) parametrizing the representations of an associative k-algebra A on a finite-dimensional vector space V . We construct the characteristic maps Tr V (A)n : HCn (

This paper is a sequel to [BKR], where we study the derived affine scheme DRep

Abstract. Some 15 years ago M. Kontsevich and A. Rosenberg [KR] proposed a heuristic principle according to which the family of schemes {Rep n (A)} parametrizing the finite-dimensional

Let X be a smooth scheme over a field of characteristic 0. Let D • poly (X) be the complex of polydifferential operators on X equipped with Hochschild co-boundary. Let L(D 1 poly (X)) be the free Lie algebra generated over O X by D 1 poly (X) concentrated in degree 1 equipped with Hochschild co-boundary. We have a symmetrization map I : ⊕ k Sym k (L(D 1 poly (X))) → D • poly (X). Theorem 1 of this paper measures how the map I fails to commute with multiplication.We recall from Kapranov [6] that the Atiyah class of the tangent bundle yields a map α TX :a Lie algebra object in D + (X). Theorem 2 enables us to realize the Atiyah class of T X as a honest map of complexes. Theorem 2 of this paper says that T X [−1] is quasi-isomorphic to L(D 1 poly (X)). It further states that the natural Lie bracket on L(D 1 poly (X)) represents the Atiyah class of T X . An immediate consequence of Theorems 1 and 2 is Corollary 1, a result "dual" to Theorem 1 of Markarian [3] that measures how the Hochschild-Kostant-Rosenberg quasi-isomorphism fails to commute with multiplication.In order to understand Theorem 1 conceptually, we prove a theorem (Theorem 3) stating that D • poly (X) is the universal enveloping algebra of T X [−1] in D + (X). At this juncture, we recall from Kapranov [6] that if E is a vector bundle of E, the Atiyah class of E equips E with the structure of a module over the Lie algebra T X [−1] in D + (X). An easy consequence of Theorem 3 is Theorem 4, which interprets the Chern character E as the "character of the representation E of T X [−1]" and gives a description of the big Chern classes of E.Finally, Theorem 4 along with Theorem 1 is used to give an explicit formula (Theorem 5) expressing the big Chern classes of E in terms of the components of the Chern character of E.