Details:

Title	Finite Generation of Symmetric Ideals
Author(s)	Matthias Aschenbrenner, Christopher Hillar
Type	Article in Journal
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 symmetric group 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 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 partial order on monomials and showing that it is a well-quasi-ordering. 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.
Keywords	Invariant ideal, well-quasi-ordering, symmetric group, symmetric Grobner basis
Length	21
File
URL	http://arxiv.org/abs/math.AC/0411514
Language	English
Journal	Trans. Amer. Math. Soc.
Year	2006
Edition	0
Translation	No
Refereed	No