Skip to main content
SpringerLink
Log in

Affine semigroups of maximal projective dimension-II

  • RESEARCH ARTICLE
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

If the Krull dimension of the semigroup ring is greater than one, then affine semigroups of maximal projective dimension (\(\textrm{MPD}\)) are not Cohen–Macaulay, but they may be Buchsbaum. We give a necessary and sufficient condition for simplicial \(\textrm{MPD}\)-semigroups to be Buchsbaum in terms of pseudo-Frobenius elements. We give certain characterizations of \(\prec \)-almost symmetric \({\mathcal {C}}\)-semigroups. When the cone is full, we prove the irreducible \({\mathcal {C}}\)-semigroups, and \(\prec \)-almost symmetric \({\mathcal {C}}\)-semigroups with Betti-type three satisfy the extended Wilf conjecture. For \(e \ge 4\), we give a class of MPD-semigroups in \({\mathbb {N}}^2\) such that there is no upper bound on the Betti-type in terms of embedding dimension e. Thus, the Betti-type may not be a bounded function of the embedding dimension. We further explore the submonoids of \({\mathbb {N}}^d\), which satisfy the Arf property, and prove that Arf submonoids containing multiplicity are \(\textrm{PI}\)-monoids.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Barucci, V., Fröberg, R.: One-dimensional almost Gorenstein rings. J. Algebra 188(2), 418–442 (1997)

    Article  MathSciNet  Google Scholar 

  2. Bhardwaj, O.P., Goel, K., Sengupta, I.: Affine semigroups of maximal projective dimension. Collect. Math. 74(2), 703–727 (2023)

    Article  MathSciNet  Google Scholar 

  3. Bruns, W., Gubeladze, J., Trung, N.: Problems and algorithms for affine semigroups. Semigroup Forum 64(2), 180–212 (2002)

    Article  MathSciNet  Google Scholar 

  4. Eriksen, E.: Differential operators on monomial curves. J. Algebra 264(1), 186–198 (2003)

    Article  MathSciNet  Google Scholar 

  5. Failla, G., Peterson, C., Utano, R.: Algorithms and basic asymptotics for generalized numerical semigroups in \({\mathbb{N} }^d\). Semigroup Forum 92(2), 460–473 (2016)

    Article  MathSciNet  Google Scholar 

  6. García-García, J.I., Marín-Aragón, D., Vigneron-Tenorio, A.: An extension of Wilf’s conjecture to affine semigroups. Semigroup Forum 96(2), 396–408 (2018)

    Article  MathSciNet  Google Scholar 

  7. García-García, J.I., Ojeda, I., Rosales, J.C., Vigneron-Tenorio, A.: On pseudo-Frobenius elements of submonoids of \({\mathbb{N} }^d\). Collect. Math. 71(1), 189–204 (2020)

    Article  MathSciNet  Google Scholar 

  8. Gimenez, P., Srinivasan, H.: The structure of the minimal free resolution of semigroup rings obtained by gluing. J. Pure Appl. Algebra 223(4), 1411–1426 (2019)

    Article  MathSciNet  Google Scholar 

  9. Gimenez, P., Srinivasan, H.: Gluing semigroups: when and how. Semigroup Forum 101(3), 603–618 (2020)

    Article  MathSciNet  Google Scholar 

  10. Grayson, D. R., Stillman, M. E.: Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/

  11. Jafari, R., Yaghmaei, M.: Type and conductor of simplicial affine semigroups. J. Pure Appl. Algebra, 226(3), Paper No. 106844, 19 pp (2022)

  12. Lipman, J.: Stable ideals and Arf rings. Amer. J. Math. 93, 649–685 (1971)

    Article  MathSciNet  Google Scholar 

  13. Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra, Graduate Texts in Mathematics, vol. 227. Springer, New York (2005)

    Google Scholar 

  14. Rosales, J.C., García-Sánchez, P.A.: Numerical Semigroups, Developments in Mathematics, vol. 20. Springer, New York (2009)

    Book  Google Scholar 

  15. Rosales, J.C., García-Sánchez, P.A., García-García, J.I., Branco, M.B.: Arf numerical semigroups. J. Algebra 276(1), 3–12 (2004)

    Article  MathSciNet  Google Scholar 

  16. Trung, N.: Classification of the double projections of Veronese varieties. J. Math. Kyoto Univ. 22(4), 567–581 (1982)

    MathSciNet  Google Scholar 

  17. Wilf, H.S.: A circle-of-lights algorithm for the money-changing problem. Amer. Math. Monthly 85(7), 562–565 (1978)

    ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IIT Bombay, Powai, Mumbai, 400076, India

    Om Prakash Bhardwaj

  2. IIT Gandhinagar, Palaj, Gandhinagar, Gujarat, 382355, India

    Indranath Sengupta

Authors
  1. Om Prakash Bhardwaj
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Indranath Sengupta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Indranath Sengupta.

Additional information

Communicated by Jorge Almeida.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bhardwaj, O.P., Sengupta, I. Affine semigroups of maximal projective dimension-II. Semigroup Forum 108, 20–42 (2024). https://doi.org/10.1007/s00233-023-10405-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-023-10405-7

Keywords

Search

Navigation