Skip to main content
SpringerLink
Log in

Fiber Invariants of Projective Morphisms and Regularity of Powers of Ideals

  • Published:
Acta Mathematica Vietnamica Aims and scope Submit manuscript

Abstract

We introduce an invariant, associated to a coherent sheaf of graded modules over a projective morphism of schemes, which controls when sheaf cohomology can be passed through the given morphism. We then use this invariant to estimate the stability indexes of the regularity and a∗-invariant of powers of homogeneous ideals. Specifically, for an equigenerated homogeneous ideal I in a standard graded algebra over a Noetherian ring, we give bounds for the smallest values of power q starting from which a∗(Iq) and reg(Iq) become linear functions.

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. Alilooee, A., Banerjee, A.: Powers of edge ideals of regularity three bipartite graphs. J. Commut. Algebra 9(4), 441–454 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bagheri, A., Chardin, M., Hà, H.T.: The eventual shape of Betti tables of powers of ideals. Math. Res. Lett. 20(6), 1033–1046 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Banerjee, A., Beyarslan, S., Hà, H.T.: Regularity of powers of edge ideals: from local properties to global bounds. Preprint, arXiv:1805.01434(2018)

  4. Berlekamp, D.: Regularity defect stabilization of powers of an ideal. Math. Res. Lett. 19(1), 109–119 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Beyarslan, S., Hà, H.T., Trung, T.N.: Regularity of powers of forests and cycles. J. Algebraic Combin. 42(4), 1077–1095 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Borna, K.: On linear resolution of powers of an ideal. Osaka J. Math. 46(4), 1047–1058 (2009)

    MathSciNet  MATH  Google Scholar 

  7. Brodmann, M.P., Sharp, R.Y.: Local Cohomology: An Algebraic Introduction with Geometric Applications Cambridge Studies in Advanced Mathematics, vol. 60. Cambridge University Press, Cambridge (1998)

    Book  Google Scholar 

  8. Bruns, W.: Algebras defined by powers of determinantal ideals. J. Alg. 142(1), 150–163 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bruns, W., Conca, A., Varbaro, M.: Maximal minors and linear powers. J. Reine Angew. Math. 702, 41–53 (2015)

    MathSciNet  MATH  Google Scholar 

  10. Chardin, M.: Powers of ideals and the cohomology of stalks and fibers of morphisms. Algebra Number Theory 7(1), 1–18 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chardin, M.: Powers of ideals: Betti numbers, cohomology and regularity. Commutative Algebra, pp.l 317–333. Springer, New York (2013)

    Google Scholar 

  12. Chardin, M.: Regularity stabilization for the powers of graded \(\mathfrak {m}\)-primary ideals. Proc. Am. Math. Soc. 143(8), 3343–3349 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cutkosky, S.D., Herzog, J., Trung, N.V.: Asymptotic behaviour of the Castelnuovo-Mumford regularity. Composito Mathematica 118, 243–261 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Eisenbud, D., Harris, J.: Powers of ideals and fibers of morphisms. Math. Res. Lett. 17(2), 267–273 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Eisenbud, D., Huneke, C.: Cohen-Macaulay Rees algebras and their specialization. J. Alg. 81, 202–224 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  16. Eisenbud, D., Ulrich, B.: Notes on regularity stabilization. Proc. Am. Math. Soc. 140(4), 1221–1232 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. Available at https://faculty.math.illinois.edu/Macaulay2/

  18. Grothendieck, A.A.: Éléments de géométrie algébrique. I. Le langage des schémas. Inst. Hautes Études Sci. Publ. Math. No. 4, pp. 228 (1960)

  19. Gu, Y.: Regularity of powers of edge ideals of some graphs. Acta Math. Vietnam. 42(3), 445–454 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hà, H.T.: Asymptotic linearity of regularity and a∗-invariant of powers of ideals. Math. Res. Lett. 18(1), 1–9 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hà, H.T., Trung, N.V.: Asymptotic behaviour of arithmetically Cohen-Macaulay blow-ups. Trans. Am. Math. Soc. 357(9), 3655–3672 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hà, H.T., Trung, N.V., Trung, T.N.: Depth and regularity of powers of sums of ideals. Math. Z. 282(3–4), 819–838 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hang, N.T., Trung, T.N.: Regularity of powers of cover ideals of unimodular hypergraphs. J. Algebra 513, 159–176 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hartshorne, R.: Algebraic Geometry Graduate Texts in Mathematics, No. 52. Springer, New York (1977)

    Google Scholar 

  25. Huneke, C.: The theory of d-sequences and powers of ideals. Adv. in Math. 46, 249–279 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  26. Jayanthan, A.V., Narayanan, N., Selvaraja, S.: Regularity of powers of bipartite graphs. J. Algebraic Combin. 47(1), 17–38 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  27. Jayanthan, A.V., Selvaraja, S.: Asymptotic behavior of Castelnuovo-Mumford regularity of edge ideals of very well-covered graphs. Preprint, arXiv:1708.06883 (2017)

  28. Jayanthan, A.V., Selvaraja, S.: Upper bounds for the regularity of powers of edge ideals of graphs. Preprint, arXiv:1805.01412 (2018)

  29. Kodiyalam, V.: Asymptotic behaviour of Castelnuovo-Mumford regularity. Proc. Am. Math. Soc. 128(2), 407–411 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lu, D.: Geometric regularity of powers of two-dimensional squarefree monomial ideals. Preprint, arXiv:1808.07266 (2018)

  31. Mumford, D.: Lectures on curves on an algebraic surface. With a section by G. M. Bergman Annals of Mathematics Studies, no. 59. Princeton University Press, Princeton (1966)

    Google Scholar 

  32. Raicu, C.C.: Regularity and cohomology of determinantal thickenings. Proc. Lond. Math. Soc. (3) 116(2), 248–280 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  33. Römer, T.: Homological properties of bigraded algebras. Illinois J. Math. 45 (4), 1361–1376 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  34. Seyed Fakhari, S.A.: Symbolic powers of cover ideal of very well-covered and bipartite graphs. Proc. Am. Math. Soc. 146(1), 97–110 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  35. Seyed Fakhari, S.A., Yassemi, S.: Improved bounds for the regularity of powers of edge ideals of graphs. Preprint, arXiv:1805.12508 (2018)

  36. Trung, N.V.: The Castelnuovo regularity of the Rees algebra and the associated graded ring. Trans. Am. Math. Soc. 350(7), 2813–2832 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  37. Trung, N.V., Viet, D.Q., Zarzuela, S.: When is the Rees algebra Gorenstein. J. Alg. 175, 137–156 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  38. Trung, N.V., Wang, H.: On the asymptotic behavior of Castelnuovo-Mumford regularity. J. Pure Appl. Algebra 201(1–3), 42–48 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  39. Whieldon, G.: Stabilization of Betti tables. J. Commut. Algebra 6(1), 113–126 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank Marc Chardin and Ngo Viet Trung for many stimulating discussions on the regularity of powers of ideals over the years. The authors would also like to thank an anonymous referee for a careful read and many useful suggestions making the paper more readable. The second named author is partially supported by Louisiana Board of Regents (grant no. LEQSF(2017-19)-ENH-TR-25).

Author information

Authors and Affiliations

  1. Department of Mathematics, Tulane University, 6823 St. Charles Ave., New Orleans, LA, 70118, USA

    Sankhaneel Bisui, Huy Tài Hà & Abu Chackalamannil Thomas

Authors
  1. Sankhaneel Bisui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huy Tài Hà
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Abu Chackalamannil Thomas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sankhaneel Bisui.

Additional information

In honor of Professor Lê Văn Thiêm’s centenary

Publisher’s Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bisui, S., Hà, H.T. & Thomas, A.C. Fiber Invariants of Projective Morphisms and Regularity of Powers of Ideals. Acta Math Vietnam 45, 183–198 (2020). https://doi.org/10.1007/s40306-019-00352-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40306-019-00352-3

Keywords

Mathematics Subject Classification (2010)

Search

Navigation