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FOGARTY'S PROOF OF THE FINITE GENERATION OF CERTAIN SUBRINGS
 

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 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
invariant theory.
Proposition 1.2. ([GIT, Theorem 1.1]) Let G be a linearly reductive group

  

Source: Alper, Jarod - Department of Mathematics, Columbia University

 

Collections: Mathematics