Finite generation of certain subrings

John Fogarty
1987 Proceedings of the American Mathematical Society  
A more geometric approach can be used to prove finite generation of certain subrings, notably invariants under reductive group actions. 1. Introduction. To some extent, this note purveys old wine in new bottles. It looks at the problem of finite generation of subrings from a more geometric viewpoint than usual. For example, the standard proofs of finite generation of rings of invariants of reductive groups all argue via reduction to the "graded case" (cf. [6, 2]). A simple argument (see
more » ... ion 1) allows one to bypass the graded case. For the rest, if one confines attention to "characterless" algebraic groups G-and this is always possible in practice-then one has, for example, the following: Let A: be a field, let p: G -» GL(7s) be a rational representation of G, and let : E -* Y = Spec(k[E]c) be the invariant map. Let E0 = £~fy(0) and let A = H°(0Eo)G. If [A:k)< oo, then k[E]G is finitely generated.
doi:10.1090/s0002-9939-1987-0866454-5 fatcat:qbtw46qbcfgibkoztvuplc3uli