Equivariant Hilbert Series in non-Noetherian Polynomial Rings
By Uwe Nagel, Tim Roemer. 2015
59 citations recorded by the source corpus.
We introduce and study equivariant Hilbert series of ideals in polynomial rings in countably many variables that are invariant under a suitable action of a symmetric group or the monoid $Inc(\mathbb{N})$ of strictly increasing functions. Our first main result states that these series are rational functions in two variables. A key is to introduce also suitable submonoids of $Inc(\mathbb{N})$ and to compare invariant filtrations induced by their actions. Extending a result by Hillar and Sullivant, we show that any ideal that is invariant under these submonoids admits a Gr\"obner basis consisting of finitely many orbits. As our second main result we prove that the Krull dimension and multiplicity of ideals in an invariant filtration grow eventually linearly and exponentially, respectively, and we determine the terms that dominate this growth.
Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/c388d52fd99f3059355ebf7a030f1286536d372d
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Citations in the corpus
- FI- and OI-modules with varying coefficients (2017, background): “(i) In the case A = P, the above colimit has been studied, for example, in [7, 14, 18] by using the monoid of increasing maps Inc(N) = {π : N → N | π(i) < π(i+ 1) for all i ≥ 1}.”