DOI QR코드

DOI QR Code

AN INDEPENDENT RESULT FOR ATTACHED PRIMES OF CERTAIN TOR-MODULES

  • Received : 2014.02.21
  • Published : 2015.03.31

Abstract

Let (R, m) be a Noetherian local ring, I an ideal of R, and A an Artinian R-module. Let $k{\geq}0$ be an integer and $r=Width_{>k}(I,A)$ the supremum of length of A-cosequence in dimension > k in I defined by Nhan-Hoang [8]. It is shown that for all $t{\leq}r$ the sets $$(\bigcup_{i=0}^{t}Att_R(Tor_i^R(R/I^n,A)))_{{\geq}k}\;and\\(\bigcup_{i=0}^{t}Att_R(Tor_i^R(R/(a_1^{n_1},{\cdots},a_l^{n_l}),A)))_{{\geq}k}$$ are independent of the choice of $n,n_1,{\cdots},n_l$ for any system of generators ($a_1,{\cdots},a_l$) of I.

Acknowledgement

Supported by : Vietnam National Foundation for Science and Technology Development (Nafosted)

References

  1. M. Brodmann, Asymptotic stability of AssR(M/InM), Proc. Amer. Math. Soc. 74 (1979), no. 1, 16-18. https://doi.org/10.1090/S0002-9939-1979-0521865-8
  2. M. Brodmann and R. Y. Sharp, Local Cohomology: an algebraic introduction with geo-metric applications, Cambridge University Press, 1998.
  3. N. T. Cuong, P. Schenzel, and N. V. Trung, Verallgemeinerte Cohen-Macaulay Moduln, Math. Nachr. 85 (1978), 57-73. https://doi.org/10.1002/mana.19780850106
  4. M. Katzman, An example of an infinite set of associated primes of a local cohomology module, J. Algebra 252 (2002), no. 1, 161-166. https://doi.org/10.1016/S0021-8693(02)00032-7
  5. I. G. Macdonald, Secondary representation of modules over a commutative ring, Sym-posia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Rome, 1971), pp. 23-43. Academic Press, London, 1973.
  6. L. Melkersson and P. Schenzel, Asymptotic prime ideals related to derived functions, Proc. Amer. Math. Soc. 117 (1993), no. 4, 935-938. https://doi.org/10.1090/S0002-9939-1993-1124148-8
  7. L. T. Nhan and N. T. Dung, A Finiteness Result for Attached Primes of Certain Tor- Modules, Algebra Colloq. 19 (2012), 787-796. https://doi.org/10.1142/S1005386712000673
  8. L. T. Nhan and N. V. Hoang, A finiteness result for attached primes of Artinian local cohomology, J. Algebra Appl. 13 (2014), no. 1, 1350063, 14 pp.
  9. A. Ooishi, Matlis duality and the width of a module, Hiroshima Math. J. 6 (1976), no. 3, 573-587.
  10. L. J. Ratliff, On prime divisors of $I^n$, n large, Michigan Math. J. 23 (1976), no. 4, 337-352. https://doi.org/10.1307/mmj/1029001769
  11. R. Y. Sharp, Some results on the vanishing of local cohomology modules, Proc. Lond. Math. Soc. 30 (1975), 177-195.
  12. R. Y. Sharp, Asymptotic behaviour of certain sets of attached prime ideals, J. Lond. Math. Soc. 34 (1986), no. 2, 212-218.