Finite generation of symmetric ideals

@article{Aschenbrenner2004FiniteGO,
  title={Finite generation of symmetric ideals},
  author={Matthias Aschenbrenner and Christopher J. Hillar},
  journal={Transactions of the American Mathematical Society},
  year={2004},
  volume={359},
  pages={5171-5192}
}
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 Grobner bases and reduction in… 

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References

SHOWING 1-10 OF 31 REFERENCES

Gröbner Bases of Symmetric Quotients and Applications

In this paper, we define the universal Σ-Grobner basis. This Grobner basis allows for an enumeration of elements in Σ-orbits and hence computes a Grobner basis for symmetric quotients of the

The Structure of Some Permutation Modules for the Symmetric Group of Infinite Degree

Abstract Suppose that Ω is an infinite set andkis a natural number. Let [Ω]kdenote the set of allk-subsets of Ω and letFbe a field. In this paper we study theFSym(Ω)-submodule structure of the

An introduction to Gröbner bases

TLDR
This book discusses rings, Fields, and Ideals, and applications of Grobner Bases, as well as improvements to Buchberger's Algorithm and other topics.

On well-quasi-ordering transfinite sequences

  • C. Nash-Williams
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1965
Abstract . Let Q be a well-quasi-ordered set, i.e. a set on which a reflexive and transitive relation ≤ is defined and such that, for every infinite sequence q1,q2,… of elements of Q, there exist i

On well-quasi-ordering infinite trees

  • C. Nash-Williams
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1965
Abstract Let A be the set of all ascending finite sequences (with at least one term) of positive integers. Let s, t ∈ A. Write s ⊲ t if there exist m, n, x1, …, xn such that m < n and x1 < … < xn and

Ideals, Varieties, and Algorithms

(here, > is the Maple prompt). Once the Groebner package is loaded, you can perform the division algorithm, compute Groebner bases, and carry out a variety of other commands described below. In

The Theory of Well-Quasi-Ordering: A Frequently Discovered Concept

Gröbner bases and convex polytopes

Grobner basics The state polytope Variation of term orders Toric ideals Enumeration, sampling and integer programming Primitive partition identities Universal Grobner bases Regular triangulations The

Theory of chirality functions, generalized for molecules with chiral ligands

The theory of chirality functions described in a previous publication is generalized to allow for chiral ligands. In the earlier theory, all symmetry operations of the molecular frame could be

Algebraic factor analysis: tetrads, pentads and beyond

Factor analysis refers to a statistical model in which observed variables are conditionally independent given fewer hidden variables, known as factors, and all the random variables follow a