 
Summary: FOGARTY'S PROOF OF THE FINITE GENERATION OF
CERTAIN SUBRINGS
JAROD ALPER
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.
1. Introduction
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 Rmorphism. 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 Grepresentations to the category
of kvector spaces is exact. This is equivalent to requiring that representa
tions are completely reducible. We have the following result from geometric
invariant theory.
Proposition 1.2. ([GIT, Theorem 1.1]) Let G be a linearly reductive group
