Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. William W. Adams and Philippe Loustaunau, An introduction to Gröbner bases, Graduate Studies in Mathematics, vol. 3, American Mathematical Society, Providence, RI, 1994. MR 1287608
  • 2. Gisela Ahlbrandt and Martin Ziegler, Quasi-finitely axiomatizable totally categorical theories, Ann. Pure Appl. Logic 30 (1986), no. 1, 63–82. Stability in model theory (Trento, 1984). MR 831437, 10.1016/0168-0072(86)90037-0
  • 3. Alan R. Camina and David M. Evans, Some infinite permutation modules, Quart. J. Math. Oxford Ser. (2) 42 (1991), no. 165, 15–26. MR 1094338, 10.1093/qmath/42.1.15
  • 4. M. Drton, B. Sturmfels and S. Sullivant, Algebraic factor analysis: Tetrads, pentads and beyond, Probability Theory and Related Fields, to appear.
  • 5. P. Erdős and R. Rado, A theorem on partial well-ordering of sets of vectors, J. London Math. Soc. 34 (1959), 222–224. MR 0103841
  • 6. David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960
  • 7. Graham Higman, Ordering by divisibility in abstract algebras, Proc. London Math. Soc. (3) 2 (1952), 326–336. MR 0049867
  • 8. T. A. Jenkyns and C. St. J. A. Nash-Williams, Counterexamples in the theory of well-quasi-ordered sets, Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968) Academic Press, New York, 1969, pp. 87–91. MR 0253943
  • 9. Joseph B. Kruskal, The theory of well-quasi-ordering: A frequently discovered concept, J. Combinatorial Theory Ser. A 13 (1972), 297–305. MR 0306057
  • 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. Ruth I. Michler, Gröbner bases of symmetric quotients and applications, Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000), Springer, Berlin, 2004, pp. 627–637. MR 2037115
  • 12. E. C. Milner, Well-quasi-ordering of sequences of ordinal numbers, J. London Math. Soc. 43 (1968), 291–296. MR 0224479
  • 13. C. St. J. A. Nash-Williams, On well-quasi-ordering finite trees, Proc. Cambridge Philos. Soc. 59 (1963), 833–835. MR 0153601
  • 14. C. St. J. A. Nash-Williams, On well-quasi-ordering transfinite sequences, Proc. Cambridge Philos. Soc. 61 (1965), 33–39. MR 0173640
  • 15. R. Rado, Partial well-ordering of sets of vectors, Mathematika 1 (1954), 89–95. MR 0066441
  • 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. Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR 1363949

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 13E05, 13E15, 20B30, 06A07

Retrieve articles in all journals with MSC (2000): 13E05, 13E15, 20B30, 06A07

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