A Gorenstein numerical semi-group ring having a transcendental series of Betti numbers
type: Item
instance of: scholarly article
title: A Gorenstein numerical semi-group ring having a transcendental series of Betti numbers
zbMATH Open document ID: 1303.13017
author: Clas Löfwall
author: Samuel Lundqvist
author: Jan-Erik Roos
published in: Journal of Pure and Applied Algebra
publication date: 2014-11-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/1212.0720
review text: A numerical semigroup \(S\) is an additive semigroup of natural numbers generated by a set of numbers \(0<g_1<g_2<\dots<g_n\) such that \(gcd(g_1,\dots,g_n)=1\). Let \(k\) be a field of characteristic zero. The numerical semigroup ring \(k[S]\) of \(S\) over \(k\) is by definition the subring of the polynomial ring \(k[t]\) generated by the monomials \(t^{g_1},t^{g_2},\dots,t^{g_n}\). The Poincaré-Betti series of \(k[S]\) is by definition the generating series \(P_{k[S]}(z)=\sum_{i\geq 0}\mid\mathrm{Tor}_i^{k[s]}(k,k)\mid z^i\). According to \textit{J. Tate} [Ill. J. Math. 1, 14--27 (1957; Zbl 0079.05501)] a complete intersection always has a rational Poincaré-Betti series. In the paper under review, the authors prove that there are semigroups \(S\) such that \(G=k[S]\) is Gorenstein ring and such that \(P_G(z)\) is a transcendental function. More precisely, let \(R\) is the subring of the polynomial ring \(k[t]\), generated by the twelve monomials \(t^{36},t^{48},t^{50},t^{52},t^{56},t^{60},t^{66},t^{67},t^{107},t^{121},t^{129},t^{135}\). Then \(R\) is a Gorenstein ring and the generating series \(P_R(z)\) is rationally related to the infinite product \(\prod_{n=1}^{\infty}\frac{(1+z^{2n-1})^2}{(1-z^{2n})^2}\) and thus transcendental.
reviewed by: Amir Mafi
Mathematics Subject Classification ID: 13D40
zbMATH DE Number: 6371714
describes a project that uses: Macaulay2
describes a project that uses: GradedLieAlgebras
describes a project that uses: Liedim
MaRDI profile type: Publication
OpenAlex ID: W2132307022
arXiv ID: 1212.0720
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DOI: 10.1016/J.JPAA.2014.05.016
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description: scientific article