Homomorphisms, localizations and a new algorithm to construct invariant rings of finite groups

Let G be a finite group acting on a polynomial ring A over the field K and let AG denote the corresponding ring of invariants. Let B be the subalgebra of AG generated by all homogeneous elements of degree less than or equal to the group order |G|. Then in general B is not equal to AG if the characteristic of K divides |G|. However we prove that the field of fractions Quot(B) coincides with the field of invariants Quot(AG)=QuotG(A). We also study various localizations and homomorphisms of modular invariant rings as tools to construct generators for AG. We prove that there is always a nonzero transfer c∈AG of degree <|G|, such that the localization (AG)c can be generated by fractions of homogeneous invariants of degrees less than 2⋅|G|−1. If A=Sym(V⊕FG) with finite-dimensional FG-module V, then c can be chosen in degree one and 2⋅|G|−1 can be replaced by |G|. Let N denote the image of the classical Noether-homomorphism (see the definition in the paper). We prove that N contains the transfer ideal and thus can be used to calculate generators for AG by standard elimination techniques using Gröbner-bases. This provides a new construction algorithm for AG.