Kyoto Journal of Mathematics

On the cofiniteness of generalized local cohomology modules

Nguyen Tu Cuong, Shiro Goto, and Nguyen Van Hoang

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Abstract

Let R be a commutative Noetherian ring, let I be an ideal of R, and let M, N be two finitely generated R-modules. The aim of this paper is to investigate the I-cofiniteness of generalized local cohomology modules HIj(M,N)=limnExtRj(M/InM,N) of M and N with respect to I. We first prove that if I is a principal ideal, then HIj(M,N) is I-cofinite for all M, N and all j. Secondly, let t be a nonnegative integer such that dimSupp(HIj(M,N))1 for all j<t. Then HIj(M,N) is I-cofinite for all j<t and Hom(R/I,HIt(M,N)) is finitely generated. Finally, we show that if dim(M)2 or dim(N)2, then HIj(M,N) is I-cofinite for all j.

Article information

Source
Kyoto J. Math. Volume 55, Number 1 (2015), 169-185.

Dates
First available in Project Euclid: 13 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1426252134

Digital Object Identifier
doi:10.1215/21562261-2848151

Mathematical Reviews number (MathSciNet)
MR3323531

Zentralblatt MATH identifier
1316.13024

Subjects
Primary: 13D45: Local cohomology [See also 14B15] 13E99: None of the above, but in this section 18G60: Other (co)homology theories [See also 19D55, 46L80, 58J20, 58J22]

Keywords
Generalized local cohomology $I$-cofiniteness Noetherian

Citation

Cuong, Nguyen Tu; Goto, Shiro; Van Hoang, Nguyen. On the cofiniteness of generalized local cohomology modules. Kyoto J. Math. 55 (2015), no. 1, 169--185. doi:10.1215/21562261-2848151. https://projecteuclid.org/euclid.kjm/1426252134


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References

  • [1] K. Bahmanpour and R. Naghipour, On the cofiniteness of local cohomology modules, Proc. Amer. Math. Soc. 136 (2008), 2359–2363.
  • [2] K. Bahmanpour and R. Naghipour, Cofiniteness of local cohomology modules for ideals of small dimension, J. Algebra 321 (2009), 1997–2011.
  • [3] M. H. Bijan-Zadeh, A common generalization of local cohomology theories, Glasgow Math. J. 21 (1980), 173–181.
  • [4] K. Borna, P. Sahandi, and S. Yassemi, Cofiniteness of generalized local cohomology modules, Bull. Aust. Math. Soc. 83 (2011), 382–388.
  • [5] M. Brodmann and A. L. Faghani, A finiteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc. 128 (2000), 2851–2853.
  • [6] M. Brodmann and R. Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge Stud. Adv. Math 60, Cambridge Univ. Press, Cambridge, 1998.
  • [7] M. Chardin and K. Divaani-Aazar, A duality theorem for generalized local cohomology, Proc. Amer. Math. Soc. 136 (2008), 2749–2754.
  • [8] M. Chardin and K. Divaani-Aazar, Generalized local cohomology and regularity of Ext modules, J. Algebra 319 (2008), 4780–4797.
  • [9] N. T. Cuong and N. V. Hoang, Some finite properties of generalized local cohomology modules, East-West J. Math. 7 (2005), 107–115.
  • [10] N. T. Cuong and N. V. Hoang, On the vanishing and the finiteness of supports of generalized local cohomology modules, Manuscripta Math. 126 (2008), 59–72.
  • [11] N. T. Cuong and L. T. Nhan, On the Noetherian dimension of Artinian modules, Vietnam J. Math. 30 (2002), 121–130.
  • [12] D. Delfino and T. Marley, Cofinite modules and local cohomology, J. Pure Appl. Algebra 121 (1997), 45–52.
  • [13] K. Divaani-Aazar and A. Mafi, Associated primes of local cohomology modules, Proc. Amer. Math. Soc. 133 (2005), 655–660.
  • [14] K. Divaani-Aazar and R. Sazeedeh, Cofiniteness of generalized local cohomology modules, Colloq. Math. 99 (2004), 283–290.
  • [15] A. Grothendieck, Cohomologie local des faisceaux coherents et théorèmes de Lefschetz locaux et globaux, Séminaire de Géométrie Algébrique du Bois Marie 1962 (SGA 2), North-Holland, Amsterdam, 1968.
  • [16] R. Hartshorne, Affine duality and cofiniteness, Invent. Math. 9 (1969/1970), 145–164.
  • [17] J. Herzog, Komplexe, Auflösungen und Dualität in der Lokalen Algebra, Habilitationsschrift, Universität Regensburg, 1970.
  • [18] J. Herzog and N. Zamani, Duality and vanishing of generalized local cohomology, Arch. Math. (Basel) 81 (2003), 512–519.
  • [19] N. V. Hoang, On the associated primes and the supports of generalized local cohomology modules, Acta Math. Vietnam 33 (2008), 163–171.
  • [20] S. Kawakami and K. I. Kawasaki, On the finiteness of Bass numbers of generalized local cohomology modules, Toyama Math. J. 29 (2006), 59–64.
  • [21] K. I. Kawasaki, On the finiteness of Bass numbers of local cohomology modules, Proc. Amer. Math. Soc. 124 (1996), 3275–3279.
  • [22] K. I. Kawasaki, Cofiniteness of local cohomology modules for principal ideals, Bull. London Math. Soc. 30 (1998), 241–246.
  • [23] K. I. Kawasaki, On a category of cofinite modules which is Abelian, Math. Z. 269 (2011), 587–608.
  • [24] I. G. Macdonald, “Secondary representation of modules over a commutative ring” in Symposia Mathematica, Vol. XI (Rome, 1971), Academic Pres, London, 1973, 23–43.
  • [25] T. Marley, Associated primes of local cohomology module over rings of small dimension, Manuscripta Math. 104 (2001), 519–525.
  • [26] T. Marley and J. C. Vassilev, Cofiniteness and associated primes of local cohomology modules, J. Algebra 256 (2002), 180–193.
  • [27] 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.
  • [28] L. Melkersson, Some applications of a criterion for Artinianness of a module, J. Pure Appl. Algebra 101 (1995), 291–303.
  • [29] L. Melkersson, Properties of cofinite modules and applications to local cohomology, Math. Proc. Cambridge Phil. Soc. 125 (1999), 417–423.
  • [30] L. Melkersson, Modules cofinite with respect to an ideal, J. Algebra 285 (2005), 649–668.
  • [31] N. Suzuki, On the generalized local cohomology and its duality, J. Math. Kyoto Univ. 18 (1978), 71–85.
  • [32] S. Yassemi, Generalized section functors, J. Pure Appl. Algebra 95 (1994), 103–119.
  • [33] K. I. Yoshida, Cofiniteness of local cohomology modules for ideals of dimension one, Nagoya Math. J. 147 (1997), 179–191.
  • [34] H. Zöschinger, Minimax moduln, J. Algebra 102 (1986), 1–32.