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Mathematische Zeitschrift

, Volume 282, Issue 3–4, pp 819–838 | Cite as

Depth and regularity of powers of sums of ideals

  • Huy Tài Hà
  • Ngo Viet TrungEmail author
  • Trân Nam Trung
Article

Abstract

Given arbitrary homogeneous ideals I and J in polynomial rings A and B over a field k, we investigate the depth and the Castelnuovo–Mumford regularity of powers of the sum \(I+J\) in \(A \otimes _k B\) in terms of those of I and J. Our results can be used to study the behavior of the depth and regularity functions of powers of an ideal. For instance, we show that such a depth function can take as its values any infinite non-increasing sequence of non-negative integers.

Keywords

Power of ideals Sum of ideals Depth Regularity Asymptotic behavior 

Mathematics Subject Classification

13C05 14H20 

References

  1. 1.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bandari, S., Herzog, J., Hibi, T.: Monomial ideals whose depth function has any given number of strict local maxima. Ark. Mat. 52, 11-19 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berlekamp, D.: Regularity defect stabilization of powers of ideals. Math. Res. Lett. 19(1), 109-119 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brodmann, M.: The asymptotic nature of the analytic spread. Math. Proc. Camb. Philos. Soc. 86, 35-39 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bruns, W., Herzog, J.: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)zbMATHGoogle Scholar
  6. 6.
    Chardin, M.: Powers of ideals and the cohomology of stalks and fibers of morphisms. Algebra Number Theory 7(1), 1-18 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Conca, A.: Regularity jumps for powers of ideals. Lect. Notes Pure Appl. Math. 244, 21-32 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cutkosky, S.D., Herzog, J., Trung, N.V.: Asymptotic behaviour of the Castelnuovo-Mumford regularity. Compos. Math. 118(3), 243-261 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Eisenbud, D.: Commutative Algebra: With a View Toward Algebraic Geometry. Springer, New York (1995)CrossRefzbMATHGoogle Scholar
  10. 10.
    Eisenbud, D., Goto, S.: Linear free resolutions and minimal multiplicity. J. Algebra 88, 89-133 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Eisenbud, D., Ulrich, B.: Notes on regularity stabilization. Proc. Am. Math. Soc. 140(4), 1221-1232 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Eisenbud, D., Harris, J.: Powers of ideals and fibers of morphisms. Math. Res. Lett. 17(2), 267-273 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Goto, S., Watanabe, K.: On graded rings $I$. J Math. Soc. Jpn. 30, 179-212 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hà, H.T.: Asymptotic linearity of regularity and a*-invariant of powers of ideals. Math. Res. Lett. 18(1), 1-9 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hà, H.T., Sun, M.: Squarefree monomial ideals that fails the persistence property and non-increasing depth. Acta Math. Vietnam. 40(1), 125-138 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Herzog, J., Hibi, T.: The depth of powers of an ideal. J. Algebra 291(2), 534-550 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Herzog, J., Qureshi, A.A.: Persistence and stability properties of powers of ideals. J. Pure Appl. Algebra 229, 530-542 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Herzog, J., Takayama, Y., Terai, N.: On the radical of a monomial ideal. Arch. Math. 85, 397-408 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Herzog, J., Vladiou, M.: Squarefree monomial ideals with constant depth function. J. Pure Appl. Algebra 217(9), 1764-1772 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hoa, L.T., Tam, N.D.: On some invariants of a mixed product of ideals. Arch. Math. 94(4), 327-337 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kodiyalam, V.: Asymptotic behaviour of Castelnuovo-Mumford regularity. Proceedings of Am. Math. Soc. 128(2), 407-411 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Nam, L.D., Varbaro, M.: When does the depth stabilize soon? J. Algebra 445, 181-192 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Swanson, I.: Powers of ideals. Primary decompositions, Artin-Rees lemma and regularity. Math. Ann. 307, 299-313 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Terai, N., Trung, N.V.: Cohen-Macaulayness of large powers of Stanley-Reisner ideals. Adv. Math. 229, 711-730 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Terai, N., Trung, N.V.: On the associated primes and the depth of the second power of squarefree monomial ideals. J. Pure Appl. Algebra 218(6), 1117-1129 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Trung, N.V., Wang, H.-J.: On the asymptotic linearity of Castelnuovo-Mumford regularity. J. Pure Appl. Algebra 201, 42-48 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Trung, T.N.: Stability of associated primes of integral closures of monomial ideals. J. Combin. Theory Ser. A 116, 44-54 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Trung, T.N.: Stability of depth of power of edge ideals. J. Algebra (preprint 2013 to appear)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Huy Tài Hà
    • 1
  • Ngo Viet Trung
    • 2
    Email author
  • Trân Nam Trung
    • 2
  1. 1.Department of MathematicsTulane UniversityNew OrleansUSA
  2. 2.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam

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