Equivariant Hilbert series in non-noetherian polynomial rings

Uwe Nagel, Tim Römer

Research output: Contribution to journalArticlepeer-review

22 Scopus citations

Abstract

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(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(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 Gröbner 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.

Original languageEnglish
Pages (from-to)204-245
Number of pages42
JournalJournal of Algebra
Volume486
DOIs
StatePublished - Sep 15 2017

Bibliographical note

Publisher Copyright:
© 2017 Elsevier Inc.

Keywords

  • Gröbner basis
  • Hilbert function
  • Krull dimension
  • Monoid
  • Multiplicity
  • Orbit
  • Symmetric group

ASJC Scopus subject areas

  • Algebra and Number Theory

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