Finite generation of symmetric ideals

• Autores: Matthias Aschenbrenner, Christopher J. Hillar
• Localización: Transactions of the American Mathematical Society, ISSN 0002-9947, Vol. 359, Nº 11, 2007, págs. 5171-5192
• Idioma: inglés
• DOI: 10.1090/s0002-9947-07-04116-5
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• Resumen

  • Let be a commutative Noetherian ring, and let be the polynomial ring in an infinite collection of indeterminates over . Let be the group of permutations of . The group acts on in a natural way, and this in turn gives the structure of a left module over the group ring . We prove that all ideals of invariant under the action of are finitely generated as -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.