Semigroup Forum

, Volume 93, Issue 2, pp 225–246 | Cite as

Asymptotic properties in the shifted family of a numerical semigroup with few generators



Let \(a<b\) be positive integers and for any integer \(k\) consider the semigroup \(H_k= \langle k, a+k, b+k \rangle \). If \(K\) is any field, we study the defining relations of the semigroup ring \(K[H_k]\) and its tangent cone \({\text {gr}}_{\mathfrak {m}}K[H_k]\), for \(k\gg 0\). Recent results in Herzog and Stamate (J Algebra 418:8–28, 2014), Jayanthan and Srinivasan (Proc Am Math Soc 141(12):4199–4208, 2013), and Vu (J Algebra 418:66–90, 2014), show that their Betti numbers are eventually periodic in \(k\). We give a better threshold \(k_{a,b}\) than the one already known for which this happens and we describe how the defining equations are periodically changing. We explicitly find all the shifts \(k> k_{a,b}\) that produce complete intersections, completing a result in Jayanthan and Srinivasan (2013). We write the minimal free resolution of \({\text {gr}}_{\mathfrak {m}}K[H_k]\) and we show that its regularity is a quasilinear function for \(k> k_{a,b}\).


Numerical semigroup ring Tangent cone Periodicity Equations Complete intersection Betti numbers 


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceUniversity of BucharestBucharestRomania

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