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 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 $\Gamma_R^n(A)^{ab} \,$ 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 $\Gamma_R^n(A)^{ab} \,$ which 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 giving a nice description of this Hilbert-Chow morphism, and consequently proving that it is projective.