Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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SOME FINITENESS RESULTS FOR CO-ASSOCIATED PRIMES OF GENERALIZED LOCAL HOMOLOGY MODULES AND APPLICATIONS

  • Do, Yen Ngoc (Faculty of Mathematics & Computer Science University of Science VNU-HCM) ;
  • Nguyen, Tri Minh (Algebra and Geometry Research Group Ton Duc Thang University) ;
  • Tran, Nam Tuan (Department of Mathematics-Informatics Ho Chi Minh University of Pedagogy)
  • Received : 2018.11.23
  • Accepted : 2020.04.24
  • Published : 2020.09.01
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Abstract

We prove some results about the finiteness of co-associated primes of generalized local homology modules inspired by a conjecture of Grothendieck and a question of Huneke. We also show some equivalent properties of minimax local homology modules. By duality, we get some properties of Herzog's generalized local cohomology modules.

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References

  1. M. Aghapournahr and L. Melkersson, Finiteness properties of minimax and coatomic local cohomology modules, Arch. Math. (Basel) 94 (2010), no. 6, 519-528. https://doi.org/10.1007/s00013-010-0127-z
  2. K. Bahmanpour and R. Naghipour, On the cofiniteness of local cohomology modules, Proc. Amer. Math. Soc. 136 (2008), no. 7, 2359-2363. https://doi.org/10.1090/S0002-9939-08-09260-5
  3. M. P. Brodmann and A. L. Faghani, A finiteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc. 128 (2000), no. 10, 2851-2853. https://doi.org/10.1090/S0002-9939-00-05328-4
  4. 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. https://doi.org/10.1017/CBO9780511629204
  5. N. T. Cuong and T. T. Nam, The I-adic completion and local homology for Artinian modules, Math. Proc. Cambridge Philos. Soc. 131 (2001), no. 1, 61-72. https://doi.org/10.1017/S0305004100004771
  6. N. T. Cuong and T. T. Nam, A local homology theory for linearly compact modules, J. Algebra 319 (2008), no. 11, 4712-4737. https://doi.org/10.1016/j.jalgebra.2007.11.030
  7. K. Divaani-Aazar and A. Mafi, Associated primes of local cohomology modules, Proc. Amer. Math. Soc. 133 (2005), no. 3, 655-660. https://doi.org/10.1090/S0002-9939-04-07728-7
  8. M. R. Doustimehr and R. Naghipour, Faltings' local-global principle for the minimaxness of local cohomology modules, Comm. Algebra 43 (2015), no. 2, 400-411. https://doi.org/10.1080/00927872.2013.843094
  9. G. Faltings, Der Endlichkeitssatz in der lokalen Kohomologie, Math. Ann. 255 (1981), no. 1, 45-56. https://doi.org/10.1007/BF01450555
  10. A. Grothendieck, Cohomologie locale des faisceaux coherents et theoremes de Lefschetz locaux et globaux (SGA 2), North-Holland Publishing Co., Amsterdam, 1968.
  11. R. Hartshorne, Local cohomology, A seminar given by A. Grothendieck, Harvard University, Fall, 1961. Lecture Notes in Mathematics, No. 41, Springer-Verlag, Berlin, 1967.
  12. R. Hartshorne, Affine duality and cofiniteness, Invent. Math. 9 (1969/70), 145-164. https://doi.org/10.1007/BF01404554
  13. J. Herzog, Komplexe, Auflosungen und dualitat in der localen Algebra, Habilitationschrift Univ. Regensburg, 1970.
  14. C. Huneke, Problems on local cohomology, in Free resolutions in commutative algebra and algebraic geometry (Sundance, UT, 1990), 93-108, Res. Notes Math., 2, Jones and Bartlett, Boston, MA, 1992.
  15. C. U. Jensen, Les foncteurs derives de lim et leurs applications en theorie des modules, Lecture Notes in Mathematics, Vol. 254, Springer-Verlag, Berlin, 1972.
  16. 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
  17. I. G. Macdonald, Duality over complete local rings, Topology 1 (1962), 213-235. https://doi.org/10.1016/0040-9383(62)90104-0
  18. T. T. Nam, Co-support and coartinian modules, Algebra Colloq. 15 (2008), no. 1, 83-96. https://doi.org/10.1142/S1005386708000084
  19. T. T. Nam, On the finiteness of co-associated primes of local homology modules, J. Math. Kyoto Univ. 48 (2008), no. 3, 521-527. https://doi.org/10.1215/kjm/1250271382
  20. T. T. Nam, A finiteness result for co-associated and associated primes of generalized local homology and cohomology modules, Comm. Algebra 37 (2009), no. 5, 1748-1757. https://doi.org/10.1080/00927870802216396
  21. T. T. Nam, Left-derived functors of the generalized I-adic completion and generalized local homology, Comm. Algebra 38 (2010), no. 2, 440-453. https://doi.org/10.1080/00927870802578043
  22. T. T. Nam, Generalized local homology for Artinian modules, Algebra Colloq. 19 (2012), Special Issue no. 1, 1205-1212. https://doi.org/10.1142/S1005386712000995
  23. T. T. Nam, Minimax modules, local homology and local cohomology, Internat. J. Math. 26 (2015), no. 12, 1550102, 16 pp. https://doi.org/10.1142/S0129167X15501025
  24. T. T. Nam and D. N. Yen, The finiteness of coassociated primes of generalized local homology modules, Math. Notes 97 (2015), no. 5-6, 738-744. https://doi.org/10.1134/S0001434615050089
  25. A. K. Singh, p-torsion elements in local cohomology modules, Math. Res. Lett. 7 (2000), no. 2-3, 165-176. https://doi.org/10.4310/MRL.2000.v7.n2.a3
  26. N. Suzuki, On the generalized local cohomology and its duality, J. Math. Kyoto Univ. 18 (1978), no. 1, 71-85.
  27. S. Yassemi, Coassociated primes, Comm. Algebra 23 (1995), no. 4, 1473-1498. https://doi.org/10.1080/00927879508825288
  28. D. N. Yen and T. T. Nam, Generalized local homology and duality, Internat. J. Algebra Comput. 29 (2019), no. 3, 581-601. https://doi.org/10.1142/S0218196719500152
  29. H. Zoschinger, Linear-kompakte Moduln uber noetherschen Ringen, Arch. Math. (Basel) 41 (1983), no. 2, 121-130. https://doi.org/10.1007/BF01196867
  30. H. Zoschinger, Minimax-moduln, J. Algebra 102 (1986), no. 1, 1-32. https://doi.org/10.1016/0021-8693(86)90125-0