 
Summary: FINITE GENERATION OF SYMMETRIC IDEALS
MATTHIAS ASCHENBRENNER AND CHRISTOPHER J. HILLAR
In memoriam Karin Gatermann (19652005).
Abstract. Let A be a commutative Noetherian ring, and let R = A[X] be
the polynomial ring in an infinite collection X of indeterminates over A. Let
SX be the group of permutations of X. The group SX acts on R in a natural
way, and this in turn gives R the structure of a left module over the left group
ring R[SX ]. We prove that all ideals of R invariant under the action of SX are
finitely generated as R[SX ]modules. The proof involves introducing a certain
wellquasiordering on monomials and developing a theory of Gršobner bases
and reduction in this setting. We also consider the concept of an invariant
chain of ideals for finitedimensional polynomial rings and relate it to the
finite generation result mentioned above. Finally, a motivating question from
chemistry is presented, with the above framework providing a suitable context
in which to study it.
1. Introduction
A pervasive theme in invariant theory is that of finite generation. A fundamen
tal example is a theorem of Hilbert stating that the invariant subrings of finite
dimensional polynomial algebras over finite groups are finitely generated [6, Corol
lary 1.5]. In this article, we study invariant ideals of infinitedimensional polynomial
