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Czechoslovak Mathematical Journal

, Volume 69, Issue 1, pp 75–86 | Cite as

Cominimaxness of Local Cohomology Modules

  • Moharram AghapournahrEmail author
Article
  • 12 Downloads

Abstract

Let R be a commutative Noetherian ring, I an ideal of R. Let t ∈ ℕ0 be an integer and M an R-module such that Ext R i (R/I,M) is minimax for all it+1. We prove that if H I i (M) is FD⩽1 (or weakly Laskerian) for all i < t, then the R-modules H I i (M) are I-cominimax for all i < t and Ext R i (R/I,H I t (M) is minimax for i = 0, 1. Let N be a finitely generated R-module. We prove that Ext R j (N,H I i (M)) and Tor j R (N,H I i (M)) are I-cominimax for all i and j whenever M is minimax and H I i (M) is FD⩽1 (or weakly Laskerian) for all i.

Keywords

local cohomology FDn modules cofinite modules cominimax modules 

MSC 2010

13D45 13E10 13C05 

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References

  1. [1]
    A. Abbasi, H. Roshan-Shekalgourabi, D. Hassanzadeh-Lelekaami: Some results on the local cohomology of minimax modules. Czech. Math. J. 64 (2014), 327–333.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    M. Aghapournahr, K. Bahmanpour: Cofiniteness of weakly Laskerian local cohomology modules. Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 57(105) (2014), 347–356.Google Scholar
  3. [3]
    J. Asadollahi, K. Khashyarmanesh, S. Salarian: A generalization of the cofiniteness problem in local cohomology modules. J. Aust. Math. Soc. 75 (2003), 313–324.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    J. Azami, R. Naghipour, B. Vakili: Finiteness properties of local cohomology modules for a-minimax modules. Proc. Amer. Math. Soc. 137 (2009), 439–448.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    K. Bahmanpour: On the category of weakly Laskerian cofinite modules. Math. Scand. 115 (2014), 62–68.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    K. Bahmanpour: Cohomological dimension, cofiniteness and Abelian categories of cofinite modules. J. Algebra 484 (2017), 168–197.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    K. Bahmanpour, R. Naghipour: On the cofiniteness of local cohomology modules. Proc. Am. Math. Soc. 136 (2008), 2359–2363.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    K. Bahmanpour, R. Naghipour: Cofiniteness of local cohomology modules for ideals of small dimension. J. Algebra 321 (2009), 1997–2011.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    K. Bahmanpour, R. Naghipour, M. Sedghi: Minimaxness and cofiniteness properties of local cohomology modules. Commun. Algebra 41 (2013), 2799–2814.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    K. Bahmanpour, R. Naghipour, M. Sedghi: On the category of cofinite modules which is Abelian. Proc. Am. Math. Soc. 142 (2014), 1101–1107.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    M. P. Brodmann, R. Y. Sharp: Local Cohomology. An Algebraic Introduction with Geometric Applications. Cambridge Studies in Advanced Mathematics 60, Cambridge University Press, Cambridge, 1998.Google Scholar
  12. [12]
    W. Bruns, J. Herzog: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge, 1993.zbMATHGoogle Scholar
  13. [13]
    D. Delfino, T. Marley: Cofinite modules and local cohomology. J. Pure Appl. Algebra 121 (1997), 45–52.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    M. T. Dibaei, S. Yassemi: Associated primes and cofiniteness of local cohomology modules. Manuscr. Math. 117 (2005), 199–205.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    M. T. Dibaei, S. Yassemi: Associated primes of the local cohomology modules. Abelian Groups, Rings, Modules, and Homological Algebra (P. Goeters, O. M. G. Jenda, eds.). Lecture Notes in Pure and Applied Mathematics 249, Chapman & Hall/CRC, Boca Raton, 2006, pp. 51–58.Google Scholar
  16. [16]
    K. Divaani-Aazar, A. Mafi: Associated primes of local cohomology modules. Proc. Am. Math. Soc. 133 (2005), 655–660.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    E. Enochs: Flat covers and flat cotorsion modules. Proc. Am. Math. Soc. 92 (1984), 179–184.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    A. Grothendieck: Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz loceaux et globeaux (SGA 2). Advanced Studies in Pure Mathematics 2, North-Holland Publishing, Amsterdam, 1968. (In French.)zbMATHGoogle Scholar
  19. [19]
    R. Hartshorne: Affine duality and cofiniteness. Invent. Math. 9 (1970), 145–164.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    D. Hassanzadeh-Lelekaami, H. Roshan-Shekalgourabi: Extension functors of cominimax modules. Commun. Algebra 45 (2017), 621–629.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    C. Huneke, J. Koh: Cofiniteness and vanishing of local cohomology modules. Math. Proc. Camb. Philos. Soc. 110 (1991), 421–429.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Y. Irani: Cominimaxness with respect to ideals of dimension one. Bull. Korean Math. Soc. 54 (2017), 289–298.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    K.-I. Kawasaki: On a category of cofinite modules which is Abelian. Math. Z. 269 (2011), 587–608.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    K. B. Lorestani, P. Sahandi, S. Yassemi: Artinian local cohomology modules. Can. Math. Bull. 50 (2007), 598–602.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    I. G. MacDonald: Secondary representation of modules over a commutative ring. Symposia Mathematica 11. Academic Press, London, 1973, pp. 23–43.Google Scholar
  26. [26]
    A. Mafi: On the local cohomology of minimax modules. Bull. Korean Math. Soc. 48 (2011), 1125–1128.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    T. Marley, J. C. Vassilev: Cofiniteness and associated primes of local cohomology modules. J. Algebra 256 (2002), 180–193.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    H. Matsumura: Commutative Ring Theory. Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, Cambridge, 1968.Google Scholar
  29. [29]
    L. Melkersson: Properties of cofinite modules and applications to local cohomology. Math. Proc. Camb. Philos. Soc. 125 (1999), 417–423.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    L. Melkersson: Modules cofinite with respect to an ideal. J. Algebra 285 (2005), 649–668.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    L. Melkersson: Cofiniteness with respect to ideals of dimension one. J. Algebra 372 (2012), 459–462.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    P. H. Quy: On the finiteness of associated primes of local cohomology modules. Proc. Am. Math. Soc. 138 (2010), 1965–1968.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    P. Schenzel: Proregular sequences, local cohomology, and completion. Math. Scand. 92 (2003), 161–180.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    K.-I. Yoshida: Cofiniteness of local cohomology modules for ideals of dimension one. Nagoya Math. J. 147 (1997), 179–191.MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    T. Yoshizawa: Subcategories of extension modules by Serre subcategories. Proc. Am. Math. Soc. 140 (2012), 2293–2305.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    T. Zink: Endlichkeitsbedingungen für Moduln über einem Noetherschen Ring. Math. Nachr. 64 (1974), 239–252. (In German.)MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    H. Zöschinger: Minimax-Moduln. J. Algebra 102 (1986), 1–32. (In German.)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceArak UniversityArakIran

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