Finite generation of symmetric ideals

By Matthias Aschenbrenner, Christopher J. Hillar. 2004

101 citations recorded by the source corpus.

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 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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/42345c069c9be89be70a9ee2d2cd47942e7d5583

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In Algebra and Number Theory

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Citations in the corpus

• Noetherianity up to Symmetry (2013, unknown): “For much more on this theme see [ Coh67, Coh87, AH07, AH08, HS12, HMdC13 ].”
• Topological Noetherianity of polynomial functors (2017, background): “, k, j ∈ N], acted upon by Sym(N) via πyij = yiπ(j), is Sym(N)-Noetherian for any Noetherian ground ring R [5, 8, 19].”