Skip to main content
SpringerLink
Log in

Square-free Gröbner degenerations

  • Published:
Inventiones mathematicae Aims and scope

Abstract

Let I be a homogeneous ideal of \(S=K[x_1,\ldots , x_n]\) and let J be an initial ideal of I with respect to a term order. We prove that if J is radical then the Hilbert functions of the local cohomology modules supported at the homogeneous maximal ideal of S/I and S/J coincide. In particular, \({\text {depth}} (S/I)={\text {depth}} (S/J)\) and \({\text {reg}} (S/I)={\text {reg}} (S/J)\).

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. Abbott, J., Bigatti, A.M., Robbiano, L.: CoCoA: a system for doing computations in commutative algebra. available at http://cocoa.dima.unige.it

  2. Adiprasito, K., Benedetti, B.: The Hirsch conjecture holds for normal flag complexes. Math. Oper. Res. 39, 1340–1348 (2014)

    Article  MathSciNet  Google Scholar 

  3. Baclawski, K.: Rings with lexicographic straightening law. Adv. Math. 39, 185–213 (1981)

    Article  MathSciNet  Google Scholar 

  4. Bayer, D., Charalambous, H., Popescu, S.: Extremal Betti numbers and applications to monomial ideals. J. Algebra 221, 497–512 (1999)

    Article  MathSciNet  Google Scholar 

  5. Bayer, D., Stillman, M.: A criterion for detecting \(m\)-regularity. Invent. Math. 87, 1–12 (1987)

    Article  MathSciNet  Google Scholar 

  6. Benedetti, B., Varbaro, M.: On the dual graph of Cohen–Macaulay algebras. Int. Math. Res. Not. 17, 8085–8115 (2015)

    Article  MathSciNet  Google Scholar 

  7. Brion, M.: Multiplicity-free subvarieties of flag varieties. Contemp. Math. 331, 13–23 (2003)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  9. Bruns, W., Schwänzl, R.: The number of equations defining a determinantal variety. Bull. Lond. Math. Soc. 22, 439–445 (1990)

    Article  MathSciNet  Google Scholar 

  10. Bruns, W., Vetter, U.: Determinantal Rings. Lecture Notes Mathematics, vol. 1327. Springer, Berlin (1988)

    Book  Google Scholar 

  11. Caviglia, G., Constantinescu, A., Varbaro, M.: On a conjecture by Kalai. Isr. J. Math. 204, 469–475 (2014)

    Article  MathSciNet  Google Scholar 

  12. 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 

  13. Chardin, M.: Some results and questions on Castelnuovo–Mumford regularity, Syzygies and Hilbert functions. Lect. Not. Pure Appl. Math. 254, 1–40 (2007)

    MATH  Google Scholar 

  14. Conca, A.: Linear spaces, transversal polymatroids and ASL domains. J. Alg. Combin. 25, 25–41 (2007)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  16. Conca, A., De Negri, E., Gorla, E.: Multigraded generic initial ideals of determinantal ideals. In: Conca, A., Gubeladze, J., Römer, T. (eds.) Homological and Computational Methods in Commutative Algebra. INdAM Series, vol. 20, pp. 81–96. Springer, Heidelberg (2017)

    Chapter  Google Scholar 

  17. Conca, A., De Negri, E., Gorla, E.: Universal Gröbner bases and Cartwright-Sturmfels ideals, to appear in Int. Math. Res. Not. (2016). arXiv:1608.08942

  18. 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 

  19. Constantinescu, A., De Negri, E., Varbaro, M.: Singularities and radical initial ideals (2019). arXiv:1906.03192

  20. Dao, H., De Stefani, A., Ma, L.: Cohomologically full rings, To appear in IMRN (2018). arXiv:1806.00536

  21. De Concini, D., Eisenbud, D., Procesi, C.: Hodge algebras. Astérisque 91, 510 (1982)

    MathSciNet  MATH  Google Scholar 

  22. Di Marca, M., Varbaro, M.: On the diameter of an ideal. J. Algebra 511, 471–485 (2018)

    Article  MathSciNet  Google Scholar 

  23. Eisenbud, D.: Introduction to algebras with straightening laws. Lect. Not. Pure Appl. Math. 55, 243–268 (1980)

    MathSciNet  MATH  Google Scholar 

  24. Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics. Springer, Berlin (1994)

    MATH  Google Scholar 

  25. Eisenbud, D., Green, M., Harris, J.: Higher castelnuovo theory. Astérisque 218, 187–202 (1993)

    MathSciNet  MATH  Google Scholar 

  26. Herzog, J., Sbarra, E.: Sequentially Cohen-Macaulay modules and local cohomology Algebra, arithmetic and geometry, Part I, II (Mumbai, 2000) Tata Inst. Fund. Res., pp. 327–340 (2002)

  27. Herzog, J., Rinaldo, G.: On the extremal Betti numbers of binomial edge ideals of block graphs (2018). arXiv:1802.06020

  28. Holmes, B.: On the diameter of dual graphs of Stanley–Reisner rings with Serre \((S_2)\) property and Hirsch type bounds on abstractions of polytopes. Electron. J. Combin. 25 (2018)

  29. Hochster, M., Roberts, J.L.: The purity of the Frobenius and local cohomology. Adv. Math. 21, 117–172 (1976)

    Article  MathSciNet  Google Scholar 

  30. Knutson, A.: Frobenius splitting, point-counting, and degeneration (2009). arXiv:0911.4941

  31. Kollár, J., Kovács, S.J.: Deformations of log canonical singularities (2018). arXiv:1803.03325

  32. Kollár, J., Kovács, S.J.: Deformations of log canonical and F-pure singularities (2018). arXiv:1807.07417

  33. Lyubeznik, G.: On the Local Cohomology Modules \(H_a^i(R)\) for Ideals \(a\) Generated by Monomials in an \(R\)-Sequence. Lecture Notes Mathematics, vol. 1092. Springer, Berlin (1983)

    Google Scholar 

  34. Ma, L., Schwede, K., Shimomoto, K.: Local cohomology of Du Bois singularities and applications to families. Compos. Math. 153, 2147–2170 (2017)

    Article  MathSciNet  Google Scholar 

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

  36. Miyazaki, M.: On the discrete counterparts of algebras with straightening laws. J. Commut. Algebra 2, 79–89 (2010)

    Article  MathSciNet  Google Scholar 

  37. Peskine, C., Szpiro, L.: Dimension projective finie et cohomologie locale. IHES Publ. Math. 42, 47–119 (1973)

    Article  Google Scholar 

  38. Schenzel, P.: Zur lokalen Kohomologie des kanonischen Moduls. Math. Z. 165, 223–230 (1979)

    Article  MathSciNet  Google Scholar 

  39. Schenzel, P.: Applications of Dualiziang Complexes to Buchsbaum Rings. Adv. Math. 44, 61–77 (1982)

    Article  Google Scholar 

  40. Schwede, K.: \(F\)-injective singularities are Du Bois. Am. J. Math. 131, 445–473 (2009)

    Article  MathSciNet  Google Scholar 

  41. Singh, A.K.: \(F\)-regularity does not deform. Am. J. Math. 121, 919–929 (1999)

    Article  MathSciNet  Google Scholar 

  42. Sturmfels, B.: Gröbner bases and Stanley decompositions of determinantal rings. Math. Z. 205, 137–144 (1990)

    Article  MathSciNet  Google Scholar 

  43. Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series, vol. 8. American Mathematical Society, Providence (1995)

    MATH  Google Scholar 

  44. Stacks Project Authors, Stacks project. Available at http://stacks.math.columbia.edu

  45. Varbaro, M.: Gröbner deformations, connectedness and cohomological dimension. J. Algebra 322, 2492–2507 (2009)

    Article  MathSciNet  Google Scholar 

  46. Varbaro, M.: Connectivity of hyperplane sections of domains. Commun. Algebra 47, 2540–2547 (2019). arXiv:1802.09445

  47. Yanagawa, K.: Dualizing complex of the face ring of a simplicial poset. J. Pure Appl. Algebra 215, 2231–2241 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We wish to express our gratitude to Linquan Ma, Sándor Kovács and Takayuki Hibi for enlightening discussions. Many examples appearing in the paper have been discovered by means of computations performed with the computer algebra systems [1] and [35].

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Genova, Genoa, Italy

    Aldo Conca & Matteo Varbaro

Authors
  1. Aldo Conca
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matteo Varbaro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matteo Varbaro.

Additional information

Publisher's Note

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

A. Conca, M. Varbaro: Both authors are supported by PRIN 2010S47ARA 003, ‘Geometria delle Varietà Algebriche’ and by INdAM-GNSAGA.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Conca, A., Varbaro, M. Square-free Gröbner degenerations. Invent. math. 221, 713–730 (2020). https://doi.org/10.1007/s00222-020-00958-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-020-00958-7

Search

Navigation