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
wellquasiordering. 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.  Keywords  Invariant ideal, wellquasiordering, 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 
