Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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A Generalization of Formal Local Cohomology Modules

  • Received : 2016.01.22
  • Accepted : 2016.10.12
  • Published : 2016.09.23
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Abstract

Let a and b be two ideals of a commutative Noetherian ring R, M a finitely generated R-module and i an integer. In this paper we study formal local cohomology modules with respect to a pair of ideals. We denote the i-th a-formal local cohomology module M with respect to b by ${\mathfrak{F}}^i_{a,b}(M)$. We show that if ${\mathfrak{F}}^i_{a,b}(M)$ is artinian, then $a{\subseteq}{\sqrt{(0:{\mathfrak{F}}^i_{a,b}(M))$. Also, we show that ${\mathfrak{F}}^{\text{dim }M}_{a,b}(M)$ is artinian and we determine the set $Att_R\;{\mathfrak{F}}^{\text{dim }M}_{a,b}(M)$.

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References

  1. M. Asgharzadeh, K. Divaani-Aazar, Finiteness properties of formal local cohomology modules and Cohen-Macaulayness, Comm. Algebra, 39(2011), 1082-1103. https://doi.org/10.1080/00927871003610312
  2. M. H. Bijan-Zadeh, Sh. Rezaei, Artinianness and attached primes of formal local cohomology modules, Algebra Colloquium, 21(2)(2014), 307-316. https://doi.org/10.1142/S1005386714000261
  3. M. Brodmann, R. Y. Sharp, Local cohomology: an algebraic introduction with geometric applications , Cambridge Univ. Press, 60(1988).
  4. M. T. Dibaei, S. Yassemi, Attached primes of the top local cohomology modules with respect to an ideal, Archiv der Mathematik, 84(2005), 292-297. https://doi.org/10.1007/s00013-004-1156-2
  5. K. Divaani-Aazar, R. Naghipour and M. Tousi, Cohomological dimension of certain algebraic varieties, Proc. Amer. Math. Soc., 130(2002), 3537-3544. https://doi.org/10.1090/S0002-9939-02-06500-0
  6. M. Eghbali, On Artinianness of formal local cohomology, colocalization and coassoci- ated primes, Math. Scand., 113(1)(2013), 5-19. https://doi.org/10.7146/math.scand.a-15478
  7. R. Hartshorne, On the De Rham cohomology of algebraic varieties, Publ. Math. IHES, 45(1976), 5-99.
  8. I. G. MacDonald, Secondary representations of modules over a commutative ring , in Symposia Mat. 11, Istituto Nazionale di alta Matematica, Roma, (1973), 23-43.
  9. H. Matsumura, Commutative ring theory , Cambridge University Press, (1986).
  10. C. Peskine, L. Szpiro, Dimension projective nie et cohomologie locale, Publ. Math. Inst. Hautes tud. Sci., 42, (1972), 47-119.
  11. Sh. Rezaei, Minimaxness and niteness properties of formal local cohomology modules, Kodai Math. J., 38(2)(2015), 430-436. https://doi.org/10.2996/kmj/1436403898
  12. P. Schenzel, On formal local cohomology and connectedness, J. Algebra, 315(2)(2007), 894-923. https://doi.org/10.1016/j.jalgebra.2007.06.015