Advertisement

Acta Mathematica Vietnamica

, Volume 44, Issue 3, pp 617–638 | Cite as

Homological Invariants of Powers of Fiber Products

  • Hop D. NguyenEmail author
  • Thanh Vu
Article

Abstract

Let R and S be polynomial rings of positive dimensions over a field k. Let IR, JS be non-zero homogeneous ideals none of which contains a linear form. Denote by F the fiber product of I and J in T = RkS. We compute homological invariants of the powers of F using the data of I and J. Under the assumption that either char k = 0 or I and J are monomial ideals, we provide explicit formulas for the depth and regularity of powers of F. In particular, we establish for all s ≥ 2 the intriguing formula depth(T/Fs) = 0. If moreover each of the ideals I and J is generated in a single degree, we show that for all s ≥ 1, reg Fs = maxi[1, s]{reg Ii + si, reg Ji + si}. Finally, we prove that the linearity defect of F is the maximum of the linearity defects of I and J, extending previous work of Conca and Römer. The proofs exploit the so-called Betti splittings of powers of a fiber product.

Keywords

Powers of ideals Fiber product Depth Regularity Castelnuovo–Mumford regularity Linearity defect 

Mathematics Subject Classification (2010)

13D02 13C05 13D05 13H99 

Notes

Acknowledgements

The first named author is grateful to the hospitality and support of the Department of Mathematics, Otto von Guericke Universität Magdeburg, where large parts of this work were finished. The second named author is partially supported by the Vietnam National Foundation for Science and Technology Development under grant number 101.04-2016.21.

References

  1. 1.
    Ahangari Maleki, R.: The Golod property for powers of ideals and Koszul ideals. J. Pure Appl. Algebra 223(2), 605–618 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alilooee, A., Banerjee, A.: Powers of edge ideals of regularity three bipartite graphs. J. Commut. Algebra 9(4), 441–454 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alilooee, A., Beyarslan, S., Selvaraja, S.: Regularity of powers of unicyclic graphs. arXiv:1702.00916 (2017)
  4. 4.
    Ananthnarayan, H., Avramov, L.L., Frank Moore, W.: Connected sums of Gorenstein local rings. J. Reine Angew. Math. 667, 149–176 (2012)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Avramov, L.L.: Infinite free resolution. Six Lectures on Commutative Algebra (Bellaterra, 1996), 1–118, Progr. Math., 166. Birkhäuser (1998)Google Scholar
  6. 6.
    Banerjee, A.: The regularity of powers of edge ideals. J. Algebraic Combin. 41 (2), 303–321 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Banerjee, A., Beyarslan, S., Hà, H.T.: Regularity of edge ideals and their powers. arXiv:1712.00887 (2017)
  8. 8.
    Beyarslan, S., Hà, H.T., Trung, T.N.: Regularity of powers of forests and cycles. J. Algebraic Combin. 42(4), 1077–1095 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Brodmann, M.: The asymptotic nature of the analytic spread. Math. Proc. Camb. Philos. Soc. 86(1), 35–39 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bruns, W., Herzog, J.: Cohen-Macaulay Rings. Rev. ed. Cambridge Studies in Advanced Mathematics 39. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  11. 11.
    Caviglia, G., Hà, H.T., Herzog, J., Kummini, M., Terai, N., Trung, N.V.: Depth and regularity modulo a principal ideal. To appear in J. Algebraic Combin.  https://doi.org/10.1007/s10801-018-0811-9
  12. 12.
    Chardin, M.: Powers of ideals and the cohomology of stalks and fibers of morphisms. Algebra Number Theory 7(1), 1–18 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chardin, M.: Powers of ideals: Betti numbers, cohomology and regularity. Commutative Algebra, 317–333. Springer, Berlin (2013)zbMATHGoogle Scholar
  14. 14.
    Christensen, L.W., Striuli, J., Veliche, O.: Growth in the minimal injective resolution of a local ring. J. Lond. Math. Soc. (2) 81(1), 24–44 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cid-Ruiz, Y., Jafari, S., Nemati, N., Picone, B.: Regularity of bicyclic graphs and their powers. arXiv:1802.07202 (2018)
  16. 16.
    Conca, A., Römer, T.: Generic initial ideals and fibre products. Osaka J. Math. 47, 17–32 (2010)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Conca, A., De Negri, E., Rossi, M.E.: Koszul Algebra and Regularity. Commutative Algebra, pp 285–315. Springer, Berlin (2013)zbMATHGoogle Scholar
  18. 18.
    Conca, A., Iyengar, S.B., Nguyen, H.D., Römer, T.: Absolutely Koszul algebras and the Backelin-Roos property. Acta Math. Vietnam. 40, 353–374 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Cutkosky, S.D., Herzog, J., Trung, N.V.: Asymptotic behaviour of the Castelnuovo-Mumford regularity. Compos. Math. 118(3), 243–261 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Dress, A., Krämer, H.: Bettireihen von Faserprodukten lokaler Ringe. Math. Ann. 215, 79–82 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Eisenbud, D.: Commutative Algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics, vol. 150. Springer, New York (1995)Google Scholar
  22. 22.
    Eisenbud, D., Harris, J.: Powers of ideals and fibers of morphisms. Math. Res. Lett. 17(2), 267–273 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Eisenbud, D., Ulrich, B.: Notes on regularity stabilization. Proc. Am. Math. Soc. 140(4), 1221–1232 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Eisenbud, D., Fløystad, G., Schreyer, F.-O.: Sheaf cohomology and free resolutions over exterior algebras. Trans. Am. Math. Soc. 355, 4397–4426 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Erey, N.: Powers of edge ideals with linear resolutions. Comm. Algebra 46(9), 4007–4020 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Erey, N.: Powers of ideals associated to (C 4, 2K 2)-free graphs. To appear in J. Pure Appl. Algebra.  https://doi.org/10.1016/j.jpaa.2018.10.009
  27. 27.
    Francisco, C., Hà, H.T., Van Tuyl, A.: Splittings of monomial ideals. Proc. Am. Math. Soc. 137, 3271–3282 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Goto, S., Watanabe, K.: On graded rings, I. J. Math. Soc. Japan 30(2), 179–212 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Grayson, D., Stillman, M.: Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2
  30. 30.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Herzog, J., Hibi, T.: Componentwise linear ideals. Nagoya Math. J. 153, 141–153 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Herzog, J., Hibi, T., Zheng, X.: Monomial ideals whose powers have a linear resolution. Math. Scand. 95, 23–32 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Herzog, J., Hibi, T.: The depth of powers of an ideal. J. Algebra 291(2), 534–550 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Herzog, J., Iyengar, S.B.: Koszul modules. J. Pure Appl. Algebra 201, 154–188 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Herzog, J., Puthenpurakal, T.J., Verma, J.K.: Hilbert polynomials and powers of ideals. Math. Proc. Camb. Philos. Soc. 145(3), 623–642 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Iyengar, S.B., Römer, T.: Linearity defects of modules over commutative rings. J. Algebra 322, 3212–3237 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Jayanthan, A.V., Selvaraja, S.: Asymptotic behavior of Castelnuovo-Mumford regularity of edge ideals of very well-covered graphs. arXiv:1708.06883 (2017)
  38. 38.
    Jayanthan, A.V., Narayanan, N., Selvaraja, S.: Regularity of powers of bipartite graphs. J. Algebraic Combin. 47(1), 17–38 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Kodiyalam, V.: Asymptotic behaviour of Castelnuovo-Mumford regularity. Proc. Am. Math. Soc. 128(2), 407–411 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Lescot, J.: La série de Bass d’un produit fibré d’anneaux locaux. C. R. Acad. Sci. Paris Sér. I Math. 293(12), 569–571 (1981)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Martínez-Bernal, J., Morey, S., Villarreal, R.H., Vivares, C.E.: Depth and regularity of monomial ideals via polarization and combinatorial optimization. arXiv:1803.02017 (2018)
  42. 42.
    Moghimian, M., Fakhari, S.A., Yassemi, S.: Regularity of powers of edge ideal of whiskered cycles. Comm. Algebra 45(3), 1246–1259 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Moore, W.F.: Cohomology over fiber products of local rings. J. Algebra 321(3), 758–773 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Nasseh, S., Sather-Wagstaff, S.: Vanishing of Ext and Tor over fiber products. Proc. Am. Math. Soc. 145(11), 4661–4674 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Nguyen, H.D.: Notes on the linearity defect and applications. Illinois J. Math. 59(3), 637–662 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Nguyen, H.D., Vu, T.: Linearity defects of powers are eventually constant. arXiv:1504.04853v1 (2015)
  47. 47.
    Nguyen, H.D., Vu, T.: Powers of sums and their homological invariants. To appear in J. Pure Appl. Algebra.  https://doi.org/10.1016/j.jpaa.2018.10.010
  48. 48.
    Nguyen, H.D., Vu, T.: Homological invariants of powers of fiber products. arXiv:1803.04016 (2018)
  49. 49.
    Peeva, I.: Graded Syzygies. Algebra Appl., vol. 14. Springer, London (2011)CrossRefGoogle Scholar
  50. 50.
    Römer, T.: On Minimal Graded Free Resolutions. Ph.D. dissertation, University of Essen (2001)Google Scholar
  51. 51.
    Şega, L.M.: Homological properties of powers of the maximal ideal of a local ring. J. Algebra 241(2), 827–858 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Şega, L.M.: On the linearity defect of the residue field. J. Algebra 384, 276–290 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Yanagawa, K.: Alexander duality for Stanley–Reisner rings and squarefree \(\mathbb {N}^{n}\)-graded modules. J. Algebra 225, 630–645 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam
  2. 2.Hanoi University of Science and TechnologyHanoiVietnam

Personalised recommendations