The finite-group Noether theorem does not supply invariant finite generation for positive-dimensional groups

Noether's finiteness theorem for invariants of a finite group (Noether's finiteness theorem: the invariants of a finite group acting on a finite-type algebra over a Noetherian ring form an algebra of finite type) requires the acting group to be finite. It therefore does not supply the finite generation of invariant rings of positive-dimensional groups: applied to the action of ${\mathbf{G}}_m={\mathbb{C}}^{\times }$ on ${\mathbb{C}}^2$ of Invariants of the hyperbolic action of the multiplicative group on the plane the hypothesis fails, since ${\mathbf{G}}_m$ is not finite (Reductive and linearly reductive complex algebraic groups), even though the invariant ring there is $\mathbb{C}[xy]$; and no argument on this page reduces a reductive group action to a finite-group action. The finite-group theorem is thus not a substitute for the Reynolds-operator proof of finite generation for complex reductive groups given by the bridge theorem of this page.

The remark is a hypothesis comparison, not a proof step: it records the design caveat that the finite-group Noether bound is not invoked anywhere in the finite-generation argument for reductive $G$, where the Reynolds operator and the graded ideal argument do all the work. No reduction of a ${\mathbf{G}}_m$- or reductive action to a finite-group action is asserted or used.