The Noether bound in invariant theory of finite groups

摘要

Let R be a commutative ring, V a finitely generated free R-module and G less than or equal to GL(R)(V) a finite group acting naturally on the graded symmetric algebra A = Sym(V). Let beta (A(G)) denote the minimal number m, such that the ring A(G) of invariants can be generated by finitely many elements of degree at most m. Furthermore, let H vertical bar> G be a normal subgroup such that the index G : H is invertible in R. In this paper we prove the inequality beta (A(G)) less than or equal to beta (A(H)) . G : H. For H = 1 and G invertible in R we obtain Noether's bound beta (A(G)) less than or equal to G, which so far had been shown for arbitrary groups only under the assumption that the factorial of the group order, G!, is invertible in R. (C) 2000 Academic Press. [References: 17]