Finite generation of symmetric ideals

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 ${\mathfrak S}_{X}$ be the group of permutations of $X$. The group ${\mathfrak S}_{X}$ acts on $R$ in a natural way, and this in turn gives $R$ the structure of a left module over the left group ring $R[{\mathfrak S}_{X}]$. We prove that all ideals of $R$ invariant under the action of ${\mathfrak S}_{X}$ are finitely generated as $R[{\mathfrak S}_{X}]$-modules. The proof involves introducing a certain well-quasi-ordering on monomials and developing a theory of Gr\"obner bases and reduction in this setting. We also consider the concept of an invariant chain of ideals for finite-dimensional 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.