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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 (seedoi:10.1090/s0002-9939-1987-0866454-5 fatcat:qbtw46qbcfgibkoztvuplc3uli