Moduli of linear representations, symmetric products and the non commutative Hilbert scheme in Geometric Methods in Representation Theory, II Michel Brion Editor Séminaires et Congrès 24-II (2012)

Let k be a commutative ring and let R be a commutative k-algebra. Let A be a R-algebra. We discuss the connections between the coarse moduli space of the n-dimensional representations of A, the non-commutative Hilbert scheme on A and the affine scheme which represents multiplicative homogeneous polynomial laws of degree n on A. We build a norm map which specializes to the Hilbert-Chow morphism on the geometric points when A is commutative and k is an algebraically closed field. This generalizes the construction done by Grothendieck, Deligne and others. When k is an infinite field and A=k[x_1,,x_m ]is the free k-associative algebra on m letters, we give a simple description of this norm map