Equivariant Hilbert series in non-noetherian polynomial rings

Autor(en): Nagel, Uwe
Roemer, Tim 
Stichwörter: Grobner basis; Hilbert function; IDEALS; Krull dimension; Mathematics; MODULES; Monoid; Multiplicity; Orbit; Symmetric group; SYZYGIES; VARIETIES
Erscheinungsdatum: 2017
Herausgeber: ACADEMIC PRESS INC ELSEVIER SCIENCE
Journal: JOURNAL OF ALGEBRA
Volumen: 486
Startseite: 204
Seitenende: 245
Zusammenfassung: 
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 Grobner 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. (C) 2017 Elsevier Inc. All rights reserved.
ISSN: 00218693
DOI: 10.1016/j.jalgebra.2017.05.011

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