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TitleFinite Generation of Symmetric Ideals
Author(s) Matthias Aschenbrenner, Christopher Hillar
TypeArticle in Journal
AbstractLet $A$ 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.
KeywordsInvariant ideal, well-quasi-ordering, symmetric group, symmetric Grobner basis
URL http://arxiv.org/abs/math.AC/0411514
JournalTrans. Amer. Math. Soc.
Translation No
Refereed No