Finite generation of symmetric ideals
| dc.creator | Aschenbrenner, Matthias | |
| dc.creator | Hillar, Christopher J. | |
| dc.date | 2004-11-23 | |
| dc.date | 2007-05-02 | |
| dc.date.accessioned | 2026-07-07T07:58:59Z | |
| dc.date.available | 2026-07-07T07:58:59Z | |
| dc.description | 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 Gröbner 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. | |
| dc.description | 22 pages, print version, Transactions of the AMS | |
| dc.identifier | https://arxiv.org/abs/math/0411514 | |
| dc.identifier | http://arxiv.org/abs/math/0411514 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/128206 | |
| dc.subject | Commutative Algebra | |
| dc.subject | Combinatorics | |
| dc.subject | 13E05, 13E15, 20B30, 06A07 | |
| dc.title | Finite generation of symmetric ideals | |
| dc.type | text |