Hilbert-Chow Morphism for Non Commutative Hilbert Schemes and Moduli Spaces of Linear Representations

Let k be a commutative ring and let R be a commutative k- algebra. The aim of this paper is to define and discuss some connection morphisms between schemes associated to the representation theory of a ( non necessarily commutative) R- algebra A. We focus on the scheme Rep(A)(n)//GL(n) of the n-dimensional representations of A, on the Hilbert scheme Hilb(A)(n) parameterizing the left ideals of codimension n of A and on the affine scheme Spec Gamma(n)(R) (A)(ab) of the abelianization of the divided powers of order n over A. We give a generalization of the Grothendieck-Deligne norm map from Hilb(A)(n) to Spec Gamma(n)(R) (A)(ab). This map specializes to the Hilbert Chow morphism on the geometric points when A is commutative and k is an algebraically closed field. Describing the Hilbert scheme as the base of a principal bundle we shall factor this map through the moduli space Rep(A)(n)// GL(n) giving a nice description of this Hilbert- Chow morphism, and consequently proving that it is projective.