Skip to main content
SpringerLink
Log in

Local cohomology of binomial edge ideals and their generic initial ideals

  • Published:
Collectanea Mathematica Aims and scope Submit manuscript

Abstract

We provide a Hochster type formula for the local cohomology modules of binomial edge ideals. As a consequence we obtain a simple criterion for the Cohen–Macaulayness and Buchsbaumness of these ideals and we describe their Castelnuovo–Mumford regularity and their Hilbert series. Conca and Varbaro (Square-free Groebner degenerations, 2018) have recently proved a conjecture of Conca, De Negri and Gorla (J Comb Algebra 2:231–257, 2018) relating the graded components of the local cohomology modules of Cartwright–Sturmfels ideals and their generic initial ideals. We provide an alternative proof for the case of binomial edge ideals.

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

Similar content being viewed by others

References

  1. Àlvarez Montaner, J., García López, R., Zarzuela Armengou, S.: Local cohomology, arrangements of subspaces and monomial ideals. Adv. Math. 174, 35–56 (2003)

    Article  MathSciNet  Google Scholar 

  2. Àlvarez Montaner, J., Boix, A.F., Zarzuela, S.: On some local cohomology spectral sequences. Int. Math. Res. Not. (2018). https://doi.org/10.1093/imrn/rny186

  3. Banerjee, A., Núñez-Betancourt, L.: Graph connectivity and binomial edge ideals. Proc. Am. Math. Soc. 145, 487–499 (2017)

    Article  MathSciNet  Google Scholar 

  4. Bolognini, D., Macchia, A., Strazzanti, F.: Binomial edge ideals of bipartite graphs. Eur. J. Comb. 70, 1–25 (2018)

    Article  MathSciNet  Google Scholar 

  5. Bouchiba, S., Kabbaj, S.: Tensor products of Cohen–Macaulay rings: solution to a problem of Grothendieck. J. Algebra 252, 65–73 (2002)

    Article  MathSciNet  Google Scholar 

  6. Bruns, W., Herzog, J.: Cohen–Macaulay Rings, Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  7. Cartwright, D., Sturmfels, B.: The Hilbert scheme of the diagonal in a product of projective spaces. Int. Math. Res. Not. 9, 1741–1771 (2010)

    MathSciNet  MATH  Google Scholar 

  8. Conca, A., De Negri, E., Gorla, E.: Universal Gröbner bases for maximal minors. Int. Math. Res. Not. 11, 3245–3262 (2015)

    MATH  Google Scholar 

  9. Conca, A., De Negri, E., Gorla, E.: Universal Gröbner bases and Cartwright–Sturmfels ideals. Int. Math. Res. Not. (2018). https://doi.org/10.1093/imrn/rny075

    Article  MATH  Google Scholar 

  10. Conca, A., De Negri, E., Gorla, E.: Multigraded generic initial ideals of determinantal ideals. In: Homological and Computational Methods in Commutative Algebra Springer INdAM Series 20, Springer (2017)

  11. Conca, A., De Negri, E., Gorla, E.: Cartwright-Sturmfels ideals associated to graphs and linear spaces. J. Comb. Algebra 2, 231–257 (2018)

    Article  MathSciNet  Google Scholar 

  12. Conca, A., Herzog, J.: On the Hilbert function of determinantal rings and their canonical module. Proc. Am. Math. Soc. 122, 677–681 (1994)

    Article  MathSciNet  Google Scholar 

  13. Conca, A., Varbaro, M.: Square-free Groebner degenerations. (2018). Preprint arXiv:1805.11923

  14. de Alba, H., Hoang, D.T.: On the extremal Betti numbers of the binomial edge ideal of closed graphs. Math. Nachr. 291, 28–40 (2018)

    Article  MathSciNet  Google Scholar 

  15. Ene, V., Herzog, J., Hibi, T.: Cohen-Macaulay binomial edge ideals. Nagoya Math. J. 204, 57–68 (2011)

    Article  MathSciNet  Google Scholar 

  16. Ene, V., Zarojanu, A.: On the regularity of binomial edge ideals. Math. Nachr. 288, 19–24 (2015)

    Article  MathSciNet  Google Scholar 

  17. Grothendieck, A.: Eléments de géométrie algébrique, Institut des Hautes Etudes Sci. Publ. Math., vol. 24, Bures-sur-yvette (1965)

  18. Herzog, J., Hibi, T., Hreinsdottir, F., Kahle, T., Rauh, J.: Binomial edge ideals and conditional independence statements. Adv. Appl. Math. 45, 317–333 (2010)

    Article  MathSciNet  Google Scholar 

  19. Herzog, J., Rinaldo, G.: On the extremal Betti numbers of binomial edge ideals of block graphs. Electron. J. Combin. 25(1), 1.63 (2018)

    Article  MathSciNet  Google Scholar 

  20. Hochster, M.: Cohen–Macaulay rings, combinatorics, and simplicial complexes. In: Proceedings of the 2nd Conference on Ring theory, II Univ. Oklahoma, Norman, Okla., 1975, pp. 171-223. Lecture Notes in Pure and Appl. Math., vol. 26, Dekker, New York (1977)

  21. Jayanthan, A.V., Kumar, A.: Regularity of binomial edge ideals of Cohen-Macaulay bipartite graphs. Commun. Algebra 47, 4797–4805 (2019)

    Article  MathSciNet  Google Scholar 

  22. Jayanthan, A.V., Narayanan, N., Raghavendra Rao, B.V.: Regularity of binomial edge ideals of certain block graphs. Proc. Indian Acad. Sci. Math. Sci. 129(3), 36 (2019)

    Article  MathSciNet  Google Scholar 

  23. Jayanthan, A.V., Narayanan, N., Raghavendra Rao, B.V.: An upper bound for the regularity of binomial edge ideals of trees. J. Algebra Appl. 18(9), 1950170 (2019)

    Article  MathSciNet  Google Scholar 

  24. Kumar, A., Sarkar, R.: Hilbert series of binomial edge ideals. Commun. Algebra 47, 3830–3841 (2019)

    Article  MathSciNet  Google Scholar 

  25. Mascia, C., Rinaldo, G.: Krull dimension and regularity of binomial edge ideals of block graphs. J. Algebra Appl. (2019). https://doi.org/10.1142/S0219498820501339

    Article  Google Scholar 

  26. Mascia, C., Rinaldo, G.: Extremal Betti numbers of some Cohen-Macaulay binomial edge ideals. Preprint arXiv:1809.03423

  27. Matsuda, K., Murai, S.: Regularity bounds for binomial edge ideals. J. Commun. Algebra 5, 141–149 (2013)

    Article  MathSciNet  Google Scholar 

  28. Miller, E., Sturmfels, B.: Combinatorial commutative algebra. Graduate Texts in Mathematics, p. 417. Springer, New York (2005)

    MATH  Google Scholar 

  29. Mohammadi, F., Sharifan, L.: Hilbert function of binomial edge ideals. Commun. Algebra 42, 688–703 (2014)

    Article  MathSciNet  Google Scholar 

  30. Ohtani, M.: Graphs and ideals generated by some 2-minors. Commun. Algebra 39, 905–917 (2011)

    Article  MathSciNet  Google Scholar 

  31. Rauf, A., Rinaldo, G.: Construction of Cohen–Macaulay binomial edge ideals. Commun. Algebra. 42, 238–252 (2014)

    Article  MathSciNet  Google Scholar 

  32. Rinaldo, G.: Cohen–Macaulay binomial edge ideals of small deviation. Bull. Math. Soc. Sci. Math. Roumanie 56(104), 497–503 (2013)

    MathSciNet  MATH  Google Scholar 

  33. Rinaldo, G.: Cohen-Macaulay binomial edge ideals of cactus graphs. J. Algebra Appl. 18(4), 1950072 (2019)

    Article  MathSciNet  Google Scholar 

  34. Saeedi Madani, S., Kiani, D.: Binomial edge ideals of graphs. Electron. J. Comb. 19, P44 (2012)

    MathSciNet  MATH  Google Scholar 

  35. Schenzel, P., Zafar, S.: Algebraic properties of the binomial edge ideal of a complete bipartite graph. An. Stiint Univ. “Ovidius” Constanza Ser. Mat 22, 217–237 (2014)

    MathSciNet  MATH  Google Scholar 

  36. Stanley, R.P.: Combinatorics and Commutative Algebra. Progress in Mathematics, 41, 2nd edn. Birkhäuser Boston Inc, Boston, MA (1996)

    Google Scholar 

  37. Watanabe, K.I., Ishikawa, T., Tachibana, S., Otsuka, K.: On tensor products of Gorenstein rings. J. Math. Kyoto Univ. 9, 413–423 (1969)

    Article  MathSciNet  Google Scholar 

  38. Zahid, Z., Zafar, S.: On the Betti numbers of some classes of binomial edge ideals. Electron. J. Comb. 20, P37 (2013)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We greatly appreciate Aldo Conca, Emanuela de Negri and Elisa Gorla for bringing to our attention their work on Cartwright–Sturmfels ideals and, in particular, Conjecture 1.1 during a workshop in honor of Peter Schenzel held in Osnabrück. We also want to thank Alberto F. Boix and Santiago Zarzuela for so many discussions about local cohomology spectral sequences we had over the last few years. We also thank the anonymous referee for the helpful comments on the manuscript.

Author information

Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Politècnica de Catalunya, Av. Diagonal 647, Barcelona, 08028, Spain

    Josep Àlvarez Montaner

Authors
  1. Josep Àlvarez Montaner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Josep Àlvarez Montaner.

Additional information

Publisher's Note

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

Partially supported by Generalitat de Catalunya 2017 SGR-932 project and Spanish Ministerio de Economía y Competitividad MTM2015-69135-P. the author is a member of the Barcelona Graduate School of Mathematics (BGSMath)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Àlvarez Montaner, J. Local cohomology of binomial edge ideals and their generic initial ideals. Collect. Math. 71, 331–348 (2020). https://doi.org/10.1007/s13348-019-00268-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13348-019-00268-z

Keywords

Mathematics Subject Classification

Search

Navigation