Semigroup Forum

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

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

RESEARCH ARTICLE

Abstract

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 \(k> k_{a,b}\).

Keywords

Numerical semigroup ring Tangent cone Periodicity Equations Complete intersection Betti numbers 

References

  1. 1.
    Ramírez Alfonsín, J.L.: The Diophantine Frobenius Problem. Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2005)Google Scholar
  2. 2.
    Bruns, W., Herzog, J.: Cohen–Macaulay Rings, Revised ed. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1998)MATHGoogle Scholar
  3. 3.
    Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3–1-6—A computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2012)
  4. 4.
    Delorme, C.: Sous-monoïdes d’intersection complète de N. Ann. Sci. Ecole Norm. Sup. (4) 9(1), 145–154 (1976)MathSciNetMATHGoogle Scholar
  5. 5.
    Herzog, J.: Generators and relations of Abelian semigroups and semigroup rings. Manuscr. Math. 3, 175–193 (1970)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Herzog, J.: When is a regular sequence super regular? Nagoya Math. J. 83, 183–195 (1981)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Herzog, J., Stamate, D.I.: On the defining equations of the tangent cone of a numerical semigroup ring. J. Algebra 418, 8–28 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Jayanthan, A.V., Srinivasan, H.: Periodic occurence of complete intersection monomial curves. Proc. Am. Math. Soc. 141(12), 4199–4208 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ohsugi, H., Hibi, T.: Indispensable binomials of finite graphs. J. Algebra Appl. 4(4), 421–434 (2005)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Robbiano, L., Valla, G.: On the equations defining tangent cones. Math. Proc. Camb. Philos. Soc. 88, 281–297 (1980)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Vu, T.: Periodicity of Betti numbers of monomial curves. J. Algebra 418, 66–90 (2014)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

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

Personalised recommendations