Finite Generation of Symmetric Ideals

Abstract:  The Hilbert Basis Theorem says that for a commutative Noetherian ring $A$ and for a finite collection of variables $X$, every ideal of $A[X]$ is finitely generated over $A[X]$. This certainly is not true if $X$ is an infinite collection of variables. However, we can say that an ideal invariant under the action of the permutation group $\mathfrak{S}_X$ is finitely generated over the group ring $A[X][\mathfrak{S}_X]$. In this talk, we will discuss sketch of proof of this result. It involves introducing a certain well-partial-ordering on monomials of $X$ and developing a theory of Grobner bases and reduction in this setting.