Journal of Commutative Algebra

On formal local cohomology modules with respect to a pair of ideals

T.H. Freitas and V.H. Jorge Pérez

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Abstract

We introduce a generalization of the formal local cohomology module, which we call a formal local cohomology module with respect to a pair of ideals, and study its various properties. We analyze their structure, upper and lower vanishing and non-vanishing properties. There are various exact sequences concerning formal cohomology modules, among them we have a Mayer-Vietoris sequence with respect to pair ideals. Also, we give another proof for a generalized version of the local duality theorems for Gorenstein, Cohen-Macaulay rings, and a generalization of the Grothendieck duality theorem for Gorenstein rings. We discuss the concept of formal grade with respect to a pair of ideals and give some results about this.

Article information

Source
J. Commut. Algebra, Volume 8, Number 3 (2016), 337-366.

Dates
First available in Project Euclid: 9 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.jca/1473428564

Digital Object Identifier
doi:10.1216/JCA-2016-8-3-337

Mathematical Reviews number (MathSciNet)
MR3546002

Zentralblatt MATH identifier
1348.13026

Subjects
Primary: 13D45: Local cohomology [See also 14B15]

Keywords
Local cohomology formal local cohomology

Citation

Freitas, T.H.; Pérez, V.H. Jorge. On formal local cohomology modules with respect to a pair of ideals. J. Commut. Algebra 8 (2016), no. 3, 337--366. doi:10.1216/JCA-2016-8-3-337. https://projecteuclid.org/euclid.jca/1473428564


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References

  • M. Aghapournahr, KH. Ahmadi-Amoli and M.Y. Sadegui, The concept of $(I,J)$-Cohen-Macaulay modules, Journal of Algebraic Systems, accepted.
  • M. Asgharzadeh and K. Divaani-Aazar, Finiteness properties of formal local cohomology modules and Cohen- Macaulayness, Comm. Algebra 39 (2011), 1082–1103.
  • N. Bourbaki, Algébre commutative, Hermann, Paris, 1961–1965.
  • M.P. Brodmann and R.Y. Sharp, Local cohomology, An algebraic introduction with geometric applications, Cambridge University Press, Cambridge, 1998.
  • L. Chu and Q. Wang, Some results on local cohomology modules defined by a pair of ideals, J. Math. Kyoto Univ, 49 (2009), 193–200.
  • K. Divaani-Aazar, R. Naghipour and M. Tousi, Cohomological dimension of certain algebraic varieties, Proc. Amer. Math. Soc. 130 (2002), 3537–3544.
  • K. Divaani-Aazar and P. Schenzel, Ideal topology, local cohomology and connectedness, Math. Proc. Cambr. Philos. Soc. 131 (2001), 211–226.
  • M. Eghbali, On formal local cohomology, colocalization and endomorphism ring of top local cohomology modules, Ph.D. thesis, Universitat Halle-Wittenberg, 2011.
  • G. Faltings, Algebraization of some formal vector bundles, Ann. Math. 110 (1979), 501–514.
  • A. Grothendieck, Local cohomology, Notes by R. Hartshorne, Lect. Notes Math. 20, Springer, Berlin, 1966.
  • A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique III, Publ. Math. IHES 11 (1961).
  • J. Herzog, Komplexe, Auflsungen und Dualitt in der lokalen Algebra, Habilitationsschrift, Universität Regensburg, 1970.
  • C. Huneke, Problems on local cohomology, in Free resolutions in commutative algebra and algebraic geometry, Res. Notes Math. 2 (1992), 93–108.
  • S.B. Iyengar, G.J. Leuschke, A. Leykin, C. Miller, E. Miller, A.K. Singh and U. Walther, Twenty-four hours of local cohomology, Grad. Stud. Math. 87, American Mathematical Society, 2007.
  • A. Kianezhad, A.J. Taherizadeh and A. Tehranian, Formal local cohomology modules and serre subcategories, J. Sci. Kharazmi University 13 (2013), 337–346.
  • A. Mafi, Some results on the local cohomology modules, Arch. Math (Basel) 87 (2006), 211–216.
  • ––––, Results on formal local cohomology modules, Bull. Malays. Math. Sci. Soc. 36 (2013), 173–177.
  • C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale, Publ. Math. I.H.E.S. 42 (1972), 47–119.
  • P. Schenzel,On formal local cohomology and connectedness, J. Alg. 315 (2007), 897–923.
  • ––––, On the use of local cohomology in algebra and geometry, in Six lectures in commutative algebra, J. Elias, J.M. Giral, R.M. Miró-Roig and S. Zarzuela, eds., Progr. Math. 166, Birkhäuser, Berlin, 1998.
  • ––––, Proregular sequences, local cohomology, and completion, Math. Scand. 92 (2003), 161–180.
  • A. Tehranian and A.P.E Talemi, Non-Artinian local cohomology with respect to a pair of ideals, A Colloquium 20 (2013), 637–642.
  • T. Takahashi, Y. Yoshino and T. Yoshizawa, Local cohomology based on a nonclosed support defined by a pair of ideals, J. Pure Appl. Alg. 213 (2009), 582–600.
  • C.A. Weibel, An introduction to homological algebra, Cambridge University Press, Cambridge, 1994.