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Acta Mathematica Vietnamica

, Volume 44, Issue 1, pp 285–306 | Cite as

Bockstein Cohomology of Associated Graded Rings

  • Tony J. PuthenpurakalEmail author
Article
  • 38 Downloads

Abstract

Let \((A,\mathfrak {m})\) be a Cohen-Macaulay local ring of dimension d and let I be an \(\mathfrak {m}\)-primary ideal. Let G be the associated graded ring of A with respect to I and let \(\mathcal R = A[It,t^{-1}]\) be the extended Rees ring of A with respect to I. Notice t− 1 is a nonzero divisor on \(\mathcal R\) and \(\mathcal R/t^{-1}\mathcal R = G\). So, we have Bockstein operators\(\beta ^{i} {\colon } {H}^{i}_{{G}_{+}}(G)(-1) \rightarrow {H}^{i + 1}_{{G}_{+}}(G)\) for i ≥  0. Since βi+ 1(+ 1) ∘ βi = 0, we have Bockstein cohomology modules BHi(G) for i = 0,…,d. In this paper, we show that certain natural conditions on I implies vanishing of some Bockstein cohomology modules.

Keywords

Associated graded rings Rees Algebras Local cohomology 

Mathematics Subject Classification (2010)

Primary 13A30 Secondary 13D40 13D07 

Notes

References

  1. 1.
    Blancafort, C.: On Hilbert functions and cohomology. J. Algebra 1, 439–459 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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)CrossRefGoogle Scholar
  3. 3.
    Bruns, W., Herzog, J.: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)Google Scholar
  4. 4.
    Heinzer, W., Lantz, D., Shah, K.: The Ratliff-Rush ideals in a Noetherian ring. Comm. Algebra 20, 591–622 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Huckaba, S., Huneke, C.: Normal ideals in regular rings. J. Reine Angew. Math. 510, 63–83 (1999)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Huckaba, S., Marley, T.: On associated graded rings of normal ideals. J. Algebra 222, 146–163 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Huckaba, S.: On associated graded rings having almost maximal depth. Comm. Algebra 26(3), 967–976 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Itoh, S.: Coefficients of normal Hilbert polynomials. J. Algebra 150, 101–117 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Itoh, S.: Hilbert coefficients of integrally closed ideals. J. Algebra 176, 638–652 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Johnston, B., Verma, J.K.: Local cohomology of Rees algebras and Hilbert functions. Proc. Am. Math. Soc. 123, 1–10 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Matsumura, H.: Commutative Ring Theory. Translated from the Japanese by M. Reid. Cambridge Studies in Advanced Mathematics, 2nd edn., vol. 8. Cambridge University Press, Cambridge (1989)Google Scholar
  12. 12.
    Narita, M.: A note on the coefficients of Hilbert characteristic functions in semi-regular local rings. Proc. Cambridge Philos. Soc. 59, 269–275 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Puthenpurakal, T.J.: Hilbert coeffecients of a Cohen-Macaulay module. J. Algebra 264, 82–97 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Puthenpurakal, T.J.: Ratliff-Rush filtration, regularity and depth of higher associated graded modules. I. J. Pure Appl. Algebra 208(1), 159–176 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Puthenpurakal, T.J.: Ratliff-Rush filtration, regularity and depth of higher associated graded modules. II. J. Pure Appl. Algebra 221(3), 611–631 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Rossi, M.E., Valla, G.: A conjecture of J. Sally. Comm. Algebra 24(3), 4249–4261 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Rossi, M.E., Valla, G.: Cohen-Macaulay local rings of dimension two and an extended version of a conjecture of J. Sally. J. Pure Appl. Algebra 122(3), 293–311 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Rossi, M.E., Valla, G.: Hilbert Functions of Filtered Modules Lecture Notes of the Unione Matematica Italiana, vol. 9. Springer-Verlag, Berlin (2010)Google Scholar
  19. 19.
    Sally, J.D.: Tangent cones at Gorenstein singularities. Compositio Math. 40, 167–175 (1980)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Sally, J.D.: Reductions, local cohomology and Hilbert functions of local ring. Commutative Algebra: Durham 1981 (R. Y. Sharp, ed.), London Math. Soc. Lecture Note Ser, vol. 72, pp 231–241. Cambridge Univ. Press, Cambridge (1982)Google Scholar
  21. 21.
    Singh, A., Walther, U.: Bockstein homomorphisms in local cohomology. J. Reine Angew Math. 655, 147–164 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Trung, N.V.: The Castelnuovo regularity of the Rees algebra and the associated graded ring. Trans. Am. Math. Soc. 350(7), 2813–2832 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Valla, G.: Problems and results on Hilbert functions of graded algebras. In: Six Lectures in Commutative Algebra, Progress in Mathematics, vol. 166, pp 293–344. BIR, Basel (1998)Google Scholar
  24. 24.
    Vasconcelos, W.V.: Cohomological degrees of graded modules. In: Six Lectures in Commutative Algebra, Progress in Mathematics, vol. 166, pp 345–398. BIR, Basel (1998)Google Scholar
  25. 25.
    Wang, H.: On Cohen-Macaulay local rings with embedding dimension e + d − 2. J. Algebra 190(1), 226–240 (1997)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.Department of MathematicsIIT BombayPowaiIndia

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