DOI QR코드

DOI QR Code

ARTINIANNESS OF LOCAL COHOMOLOGY MODULES

  • Abbasi, Ahmad (Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Guilan) ;
  • Shekalgourabi, Hajar Roshan (Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Guilan) ;
  • Hassanzadeh-lelekaami, Dawood (Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Guilan)
  • Received : 2015.07.19
  • Accepted : 2016.04.11
  • Published : 2016.06.25

Abstract

In this paper we investigate the Artinianness of certain local cohomology modules $H^i_I(N)$ where N is a minimax module over a commutative Noetherian ring R and I is an ideal of R. Also, we characterize the set of attached prime ideals of $H^n_I(N)$, where n is the dimension of N.

Keywords

Local cohomology modules;generalized Local cohomology modules;Minimax modules;Artinian modules;Cofinite modules

References

  1. A. Abbasi and H. Roshan Shekalgourabi, Serre subcategory properties of generalized local cohomology modules, Korean Ann. Math. 28 (2011), no. 1, 7-19.
  2. I. Bagheriyeh, J. Azami, and K. Bahmanpour, Generalization of the Lichtenbaum-Hartshorne vanishing theorem, Comm. Algebra 40 (2012), no. 1, 134-137. https://doi.org/10.1080/00927872.2010.525225
  3. K. Bahmanpour and R. Naghipour, On the cofiniteness of local cohohomology modules, Proc. Amer. Math. Soc. 136 (2008), no. 7, 2359-2363. https://doi.org/10.1090/S0002-9939-08-09260-5
  4. R. Belshof and C. Wickham, A note on local duality, Bull. London Math. Soc. 29 (1997), 25-31. https://doi.org/10.1112/S0024609396001713
  5. M. H. Bijan-Zadeh, A common generalization of local cohomology theories, Glasgow Math. J. 21 (1980), no. 2, 173-181. https://doi.org/10.1017/S0017089500004158
  6. M. P. Brodmann and R. Y. Sharp, Local cohomology: An algebraic introduction with geometric applications, Cambridge Studies in Advanced Mathematics 60, Cambridge University Press, Cambridge, 1998.
  7. M.T. Dibaei and S. Yassemi, Attached primes of the top local cohomology modules with respect to an ideal, Arch. Math. 84 (2005), 292-297. https://doi.org/10.1007/s00013-004-1156-2
  8. K Divaani-Aazar and A. Hajikarimi, Cofiniteness of generalized local cohomology modules for one-dimensional ideals, Canad. Math. Bull. 55 (2012), no. 1, 81-87. https://doi.org/10.4153/CMB-2011-046-8
  9. K. Divaani-Aazar, R. Sazeedeh, and M. Tousi, On vanishing of generalized local cohomology modules, Algebra Colloq. 12 (2005), 213-218. https://doi.org/10.1142/S1005386705000209
  10. E. Enochs, Flat covers and flat cotorsion modules, Proc. Amer. Math. Soc. 92 (1984), 179-184. https://doi.org/10.1090/S0002-9939-1984-0754698-X
  11. A. Grothendieck, Local cohomology modules, notes by R. Hartshorne, lecture notes in math, vol. 862, 1966.
  12. A. Grothendieck, Cohomologie locale des faisceaux et theoremes de lefshetz locaux et globaux (SGA 2), 1968.
  13. R. Hartshorne, Affine duality and cofinitenesss, Invent. Math 9 (1969/1970), 145-164.
  14. S. H. Hassanzadeh and A. Vahidi, On vanishing and cofiniteness of generalized local cohomology modules, Comm. Algebra 37 (2009), 2290-2299. https://doi.org/10.1080/00927870802622718
  15. J. Herzog, Komplex auflosungen and dualitat in der lockalen algebra, Habilitationss chrift, Universitat Regensburg, 1970.
  16. C. Huneke, Problems on local cohomology modules, free resolution in commutative algebra and algebraic geometry, Res. Notes Math. 2 (1992), 93-108.
  17. K. Khashyarmanesh and F. Khosh-Ahang, A note on the Artinianness and vanishing of local cohomology and generalized local cohomology modules, Algebra Colloq. 16 (2009), no. 3, 517-524. https://doi.org/10.1142/S1005386709000480
  18. K. Khashyarmanesh, M. Yassi, and A. Abbasi, Filter regular sequences and generalized local cohomology modules, Comm. Algebra 32 (2004), no. 1, 253-259. https://doi.org/10.1081/AGB-120027864
  19. A. Mafi, On the associated primes of generalized local cohomology modules, Comm. Algebra 34 (2006), 2489-2494. https://doi.org/10.1080/00927870600650739
  20. A. Mafi, Matlis reflexive and generalized local cohomology modules, Czechoslovak Math. J. 59 (2009), no. 134, 1095-1102. https://doi.org/10.1007/s10587-009-0077-4
  21. A. Mafi and H. Saremi, Cofinite modules and generalized local cohomology, Houston. J. Math. 35 (2009), no. 4, 1013-1018.
  22. L. Melkersson, On asymptotic stability for sets of prime ideals connected with the powers of an ideal, Math. Proc. Cambridge Philos. Soc. 107 (1990), 267-271. https://doi.org/10.1017/S0305004100068535
  23. L. Melkersson, Some applications of a criterion for Artinianness of a module, J. Pure Appl. Algebra 101 (1995), 291-303. https://doi.org/10.1016/0022-4049(94)00059-R
  24. N. Zamani, On graded generalized local cohomology, Arch. Math. (Basel) 86 (2006), no. 4, 321-330. https://doi.org/10.1007/s00013-005-1524-6
  25. T. Zink, Endlichkeitsbedingungen fur Moduln uber einem Notherschen Ring, Math. Nachr 164 (1974), 239-252.
  26. H. Zoschinger, Minimax-moduln, J. Algebra 102 (1986), 1-32. https://doi.org/10.1016/0021-8693(86)90125-0