Equivariant Hilbert series in non-noetherian polynomial rings
Uwe Nagel,Tim Römer +1 more
TLDR
In this article, the authors studied 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.About:
This article is published in Journal of Algebra.The article was published on 2017-09-15 and is currently open access. It has received 52 citations till now. The article focuses on the topics: Hilbert–Poincaré series & Hilbert series and Hilbert polynomial.read more
Citations
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FI- and OI-modules with varying coefficients
Uwe Nagel,Tim Römer +1 more
TL;DR: In this article, it was shown that every submodule of a finitely generated FI-module over a noetherian polynomial FI-algebra has a finite Grobner basis, which translates into stabilization of syzygies in any fixed homological degree.
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Hilbert series of symmetric ideals in infinite polynomial rings via formal languages
TL;DR: In this article, the authors gave a much shorter proof of this theorem using tools from the theory of formal languages and a simple algorithm that computes the Hilbert series H I ( s, t ) of I and proved that it is rational.
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Betti numbers of symmetric shifted ideals
Jennifer Biermann,Hernán de Alba,Federico Galetto,Satoshi Murai,Uwe Nagel,Augustine O'Keefe,Tim Römer,Alexandra Seceleanu +7 more
TL;DR: Symmetric shifted ideals as mentioned in this paper are fixed by the natural action of the symmetric group and, within the class of monomial ideals fixed by this action, they can be considered as an analogue of stable monomial ideal.
Posted Content
The representation theory of the increasing monoid
Sema Gunturkun,Andrew Snowden +1 more
TL;DR: In this paper, the authors study the representation theory of the increasing monoid and provide a fairly comprehensive picture of the representation category: for example, the Grothendieck group (including the effective cone), classify injective objects, establish properties of injective and projective resolutions, construct a derived auto-duality, and so on.
References
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FI-modules and stability for representations of symmetric groups
TL;DR: The theory of FI-modules was introduced and developed in this paper, and it is shown that for any fixed degree the character is given, for n large enough, by a polynomial in the cycle-counting functions that is independent of n.
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Asymptotic Behaviour of the Castelnuovo-Mumford Regularity
TL;DR: In this paper, the asymptotic behavior of the Castelnuovo norm and the Mumford norm of the integral closure of the powers of a homogeneous ideal I is studied.
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Asymptotic behaviour of Castelnuovo-Mumford regularity
TL;DR: In this article, it was shown that the Castelnuovo-Mumford regularity of a polynomial ring over a field is bounded by a linear function with leading coefficient at most P.
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Comparing powers and symbolic powers of ideals
Cristiano Bocci,Brian Harbourne +1 more
TL;DR: In this paper, the authors studied the problem of containment of symbolic powers in a polynomial ring over an algebraically closed field, and showed that the containment theorems of Ein-Lazarsfeld-Smith and Hochster-Huneke are optimal for every fixed dimension and codimension.