Eakin-Sathaye-Type Theorems for Joint Reductions and Good Filtrations of Ideals

Abstract

Analogues of Eakin-Sathaye theorem for reductions of ideals are proved for \(\mathbb N^{s}\)-graded good filtrations. These analogues yield bounds on joint reduction vectors for a family of ideals and reduction numbers for \(\mathbb N\)-graded filtrations. Several examples related to lex-segment ideals, contracted ideals in 2-dimensional regular local rings and the filtration of integral and tight closures of powers of ideals in hypersurface rings are constructed to show effectiveness of these bounds.

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References

  1. 1.

    Atiyah, M.F., Macdonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley Publishing Co., Reading (1969). Mass.-London-Don Mills, Ont

    Google Scholar 

  2. 2.

    Caviglia, G.: A theorem of Eakin and Sathaye and Green’s hyperplane restriction theorem. In: Commutative Algebra, Lecture Notes in Pure and Applied Mathematics, vol. 244, pp 1–5. Chapman & Hall/CRC, Boca Raton (2006)

  3. 3.

    Cortadellas, T., Zarzuela, S.: On the depth of the fiber cone of filtrations. J. Algebra 198(2), 428–445 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Eakin, P., Sathaye, A.: Prestable ideals. J. Algebra 41(2), 439–454 (1976)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Goel, K., Mukundan, V., Verma, J.K.: Tight closure of powers of ideals and tight Hilbert polynomials. to appear in Math. Proc. Cambridge Philos. Soc.

  6. 6.

    Goto, S., Shimoda, Y.: On the Rees algebras of Cohen-Macaulay local rings. In: Commutative Algebra (Fairfax, Va., 1979), Lecture Notes in Pure and Applied Mathematics, vol. 68, pp 201–231. Dekker, New York (1982)

  7. 7.

    Herzog, J., Saem, M.M., Zamani, N.: On the number of generators of powers of an ideal. arXiv preprint arXiv:1707.07302 (2017)

  8. 8.

    Hoa, L.T., Zarzuela, S.: Reduction number and a-invariant of good filtrations. Comm. Algebra 22(14), 5635–5656 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Hochster, M.: Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes. Ann. of Math. (2) 96, 318–337 (1972)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Huckaba, S., Marley, T.: Hilbert coefficients and the depths of associated graded rings. J. London Math. Soc. (2) 56(1), 64–76 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Huneke, C., Sally, J.D.: Birational extensions in dimension two and integrally closed ideals. J. Algebra 115(2), 481–500 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Huneke, C., Swanson, I.: Integral Closure of Ideals, Rings, and Modules. London Mathematical Society Lecture Note Series, vol. 336. Cambridge University Press, Cambridge (2006)

    Google Scholar 

  13. 13.

    Lipman, J.: On complete ideals in regular local rings. In: Algebraic geometry and commutative algebra, vol. I, pp 203–231. Kinokuniya, Tokyo (1987)

  14. 14.

    Lyubeznik, G.: A property of ideals in polynomial rings. Proc. Am. Math. Soc. 98(3), 399–400 (1986)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Matsumura, H.: Commutative Ring Theory. Cambridge Studies in Advanced Mathematics. 2nd. Translated from the Japanese by M. Reid, vol. 8. Cambridge University Press, Cambridge (1989)

  16. 16.

    Northcott, D.G., Rees, D.: Reductions of ideals in local rings. Proc. Cambridge Philos. Soc. 50, 145–158 (1954)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    O’Carroll, L.: Around the Eakin-Sathaye theorem. J. Algebra 291(1), 259–268 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Okuma, T., Watanabe, K.I., Yoshida, K.I.: Normal reduction numbers for normal surface singularities with application to elliptic singularities of Brieskorn type. arXiv preprint arXiv:1804.03795 (2017)

  19. 19.

    Rees, D.: \(\mathfrak {a}\)-transforms of local rings and a theorem on multiplicities of ideals. Proc. Cambridge Philos. Soc. 57, 8–17 (1961)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Rees, D.: A note on analytically unramified local rings. J. London Math. Soc. 36, 24–28 (1961)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Rees, D.: Hilbert functions and pseudorational local rings of dimension two. J. London Math. Soc. (2) 24(3), 467–479 (1981)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Rees, D.: Generalizations of reductions and mixed multiplicities. J. London Math. Soc. (2) 29(3), 397–414 (1984)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Sally, J.D.: On the number of generators of powers of an ideal. Proc. Am. Math. Soc. 53(1), 24–26 (1975)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Teissier, B.: Cycles évanescents, sections planes et conditions de Whitney pp. 285–362. Astérisque, Nos. 7 et 8 (1973)

  25. 25.

    Trung, N.V.: Constructive characterization of the reduction numbers. Compositio Math. 137(1), 99–113 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Viêt, D.Q.: A note on the Cohen-Macaulayness of Rees algebras of filtrations. Comm. Algebra 21(1), 221–229 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Zariski, O.: Polynomial Ideals Defined by Infinitely Near Base Points. Am. J. Math. 60(1), 151–204 (1938)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Zariski, O., Samuel, P.: Commutative Algebra. Vol. II. The University Series in Higher Mathematics. D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-London-New York (1960)

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Acknowledgments

We thank the referee for a very careful reading of the manuscript and suggesting several improvements.

Funding

Both of the first author and the second author are supported by UGC fellowship, Govt. of India.

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Correspondence to J. K. Verma.

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Dedicated to Le Tuan Hoa on the occasion of his sixtieth birthday

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Goel, K., Roy, S. & Verma, J.K. Eakin-Sathaye-Type Theorems for Joint Reductions and Good Filtrations of Ideals. Acta Math Vietnam 44, 671–689 (2019). https://doi.org/10.1007/s40306-019-00341-6

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Keywords

  • Eakin-Sathaye theorem
  • Good filtrations
  • Equimultiple good filtrations
  • Joint reduction

Mathematics Subject Classification (2010)

  • 13A30