Moduli of linear representations, symmetric products and the non commutative Hilbert scheme

Francesco Vaccarino (1)

1 Polito - Politecnico di Torino = Polytechnic of Turin

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 $A=k\{x_1,\dots,x_m\}$ is the free $k-$associative algebra on $m$ letters, we give a simple description of this norm map.

Mots clés: 14A15; 14C05; 16G99

Domaines: Théorie des représentations [math.RT]

Francesco Vaccarino. Moduli of linear representations, symmetric products and the non commutative Hilbert scheme. 2008. ⟨hal-00441319⟩