Noether bound for invariants in relatively free algebras

Abstract

Let $\mathfrak{R}$ be a weakly noetherian variety of unitary associative algebras (over a field $K$ of characteristic 0), i.e., every finitely generated algebra from $\mathfrak{R}$ satisfies the ascending chain condition for two-sided ideals. For a finite group $G$ and a $d$-dimensional $G$-module $V$ denote by $F({\mathfrak R},V)$ the relatively free algebra in $\mathfrak{R}$ of rank $d$ freely generated by the vector space $V$. It is proved that the subalgebra $F({\mathfrak R},V)^G$ of $G$-invariants is generated by elements of degree at most $b(\mathfrak{R},G)$ for some explicitly given number $b(\mathfrak{R},G)$ depending only on the variety $\mathfrak{R}$ and the group $G$ (but not on $V$). This generalizes the classical result of Emmy Noether stating that the algebra of commutative polynomial invariants $K[V]^G$ is generated by invariants of degree at most $\vert G\vert$.