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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Finite generation of symmetric ideals


Authors: Matthias Aschenbrenner and Christopher J. Hillar
Journal: Trans. Amer. Math. Soc. 359 (2007), 5171-5192
MSC (2000): Primary 13E05, 13E15, 20B30, 06A07
Published electronically: June 22, 2007
Erratum: Tran. Amer. Math. Soc. 361 (2009), 5627-5627
MathSciNet review: 2327026
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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 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 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.


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Additional Information

Matthias Aschenbrenner
Affiliation: Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607
Email: maschenb@math.uic.edu

Christopher J. Hillar
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Address at time of publication: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: chillar@math.berkeley.edu, chillar@math.tamu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04116-5
PII: S 0002-9947(07)04116-5
Keywords: Invariant ideal, well-quasi-ordering, symmetric group, Gr\"obner basis
Received by editor(s): July 26, 2004
Received by editor(s) in revised form: April 29, 2005
Published electronically: June 22, 2007
Additional Notes: The first author was partially supported by the National Science Foundation Grant DMS 03-03618.
The work of the second author was supported under a National Science Foundation Graduate Research Fellowship.
Dedicated: In memoriam Karin Gatermann (1965–2005).
Article copyright: © Copyright 2007 American Mathematical Society