Summary: FOGARTY'S PROOF OF THE FINITE GENERATION OF
Abstract. This is an expository note covering Fogarty's geometric ap-
proach to proving finite generation of certain subrings, including in-
variants under linearly reductive group actions. We offer a very mild
generalization which allows one to conclude that good moduli spaces
are finite type.
In [Fog87], John Fogarty proves the following remarkable result:
Proposition 1.1. [Fog87, Proposition p. 203] Let R be an excellent ring
and : X Y a surjective R-morphism. If X is irreducible and finite type
over R and Y is normal and noetherian, then Y is finite type over R.
This proposition has applications toward Hilbert's 14th problem. Recall
that a smooth, affine group scheme G over a field k is linearly reductive if
the functor V V G from the category of G-representations to the category
of k-vector spaces is exact. This is equivalent to requiring that representa-
tions are completely reducible. We have the following result from geometric
Proposition 1.2. ([GIT, Theorem 1.1]) Let G be a linearly reductive group