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Transactions of the American Mathematical Society

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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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  • 1. W. Adams, P. Loustaunau, An Introduction to Gröbner Bases, Graduate Studies in Mathematics, vol. 3, American Mathematical Society, Providence, RI, 1994. MR 1287608 (95g:13025)
  • 2. G. Ahlbrandt, M. Ziegler, Quasi-finitely axiomatizable totally categorical theories, Stability in Model Theory (Trento, 1984), Ann. Pure Appl. Logic 30 (1986), no. 1, 63-82. MR 831437 (87k:03026)
  • 3. A. R. Camina, D. M. Evans, Some infinite permutation modules, Quart. J. Math. Oxford Ser. (2) 42 (1991), no. 165, 15-26. MR 1094338 (92a:20003)
  • 4. M. Drton, B. Sturmfels and S. Sullivant, Algebraic factor analysis: Tetrads, pentads and beyond, Probability Theory and Related Fields, to appear.
  • 5. P. Erdos, R. Rado, A theorem on partial well-ordering of sets of vectors, J. London Math. Soc. 34 (1959), 222-224. MR 0103841 (21:2604)
  • 6. D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 203, Springer-Verlag, New York, 1995. MR 1322960 (97a:13001)
  • 7. G. Higman, Ordering by divisibility in abstract algebras, Proc. London Math. Soc. (3) 2 (1952), 326-336. MR 0049867 (14:238e)
  • 8. T. A. Jenkyns, C. St. J. A. Nash-Williams, Counterexamples in the theory of well-quasi-ordered sets. in: F. Harary (ed.), Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968), pp. 87-91, Academic Press, New York, 1969. MR 0253943 (40:7156)
  • 9. J. B. Kruskal, The theory of well-quasi-ordering: A frequently discovered concept, J. Combinatorial Theory Ser. A 13 (1972), 297-305. MR 0306057 (46:5184)
  • 10. A. Mead, E. Ruch, A. Schönhofer, Theory of chirality functions, generalized for molecules with chiral ligands. Theor. Chim. Acta 29 (1973), 269-304.
  • 11. R. Michler, Gröbner bases of symmetric quotients and applications, in: C. Christensen et al. (eds.), Algebra, Arithmetic and Geometry with Applications (West Lafayette, IN, 2000), pp. 627-637, Springer-Verlag, Berlin, 2004. MR 2037115 (2004m:13072)
  • 12. E. Milner, Well-quasi-ordering of sequences of ordinal numbers, J. London Math. Soc. 43 (1968), 291-296. MR 0224479 (37:78)
  • 13. C. St. J. A. Nash-Williams, On well-quasi-ordering finite trees, Proc. Cambridge Philos. Soc. 59 (1963), 833-835. MR 0153601 (27:3564)
  • 14. -, On well-quasi-ordering transfinite sequences, Proc. Cambridge Philos. Soc. 61 (1965), 33-39. MR 0173640 (30:3850)
  • 15. R. Rado, Partial well-ordering of sets of vectors, Mathematika 1 (1954), 89-95. MR 0066441 (16:576b)
  • 16. E. Ruch, A. Schönhofer, Theorie der Chiralitätsfunktionen, Theor. Chim. Acta 19 (1970), 225-287.
  • 17. E. Ruch, A. Schönhofer, I. Ugi, Die Vandermondesche Determinante als Näherungsansatz für eine Chiralitätsbeobachtung, ihre Verwendung in der Stereochemie und zur Berechnung der optischen Aktivität, Theor. Chim. Acta 7 (1967), 420-432.
  • 18. B. Sturmfels, Gröbner Bases and Convex Polytopes, AMS University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR 1363949 (97b:13034)

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Additional Information

Matthias Aschenbrenner
Affiliation: Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607

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

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

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