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Archiv der Mathematik

, Volume 112, Issue 1, pp 33–39 | Cite as

Faltings’ local–global principle for finiteness dimension of cofinite modules

  • Leila Abdi
  • Reza NaghipourEmail author
  • Monireh Sedghi
Article
  • 18 Downloads

Abstract

Let R denote a commutative Noetherian ring, \(\mathfrak {a}\) an ideal of R, and M an \(\mathfrak {a}\)-cofinite R-module. The purpose of this article is to show that for a positive integer t, the R-module \(H_\mathfrak {a}^i(M)\) is finitely generated for all \(i<t\) if and only if the \(R_{\mathfrak {p}}\)-module \(H_{\mathfrak {a}R_{\mathfrak {p}}}^i(M_{{\mathfrak {p}}})\) is finitely generated for all \(i<t\) and all \(\mathfrak {p}\in \mathrm {Spec}(R)\). As a consequence, we provide a generalization and short proof of Faltings’ local–global principle for finiteness dimensions; i.e., \(f_\mathfrak {a}(M)=\mathrm{{inf}}\{f_{\mathfrak {a}R_{\mathfrak {p}}}(M_{{\mathfrak {p}}}) | \ \mathfrak {p}\in \mathrm {Spec}(R)\}.\)

Keywords

Cofinite module Faltings’ local–global principle Finiteness dimension Local cohomology 

Mathematics Subject Classification

13D45 14B15 13E05 

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Notes

Acknowledgements

The authors are deeply grateful to the referee for a very careful reading of the manuscript and many valuable suggestions in improving the quality of the paper and for drawing the authors’ attention to Theorem 2.6. We also would like to thank Professor Kamal Bahmanpour for his reading of the first draft and valuable discussions. Finally, we would like to thank the Institute for Research in Fundamental Sciences (IPM) for the financial support.

References

  1. 1.
    Asadollahi, D., Naghipour, R.: A new proof of Faltings’ local–global principle for the finiteness of local cohomology modules. Arch. Math. 103, 451–459 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Asadollahi, D., Naghipour, R.: Faltings’ local–global principle for the finiteness of local cohomology modules. Commun. Algebra 43, 953–958 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bahmanpour, K., Naghipour, R.: On the cofiniteness of local cohomology modules. Proc. Am. Math. Soc. 136, 2359–2363 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bahmanpour, K., Naghipour, R.: Cofiniteness of local cohomology modules for ideals of small dimension. J. Algebra 321, 1997–2011 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bahmanpour, K., Naghipour, R., Sedghi, M.: Minimaxness and cofiniteness properties of local cohomology modules. Commun. Algebra 41, 2799–2814 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brodmann, M.P., Rothaus, Ch., Sharp, R.Y.: On annihilators and associated primes of local cohomology modules. J. Pure Appl. Algebra 153, 197–227 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brodmann, M.P., Sharp, R.Y.: Local Cohomology; An Algebraic Introduction with Geometric Applications. Cambridge University Press, Cambridge (2013)zbMATHGoogle Scholar
  8. 8.
    Dibaei, M.T., Yassemi, S.: Associated primes and cofiniteness of local cohomology modules. Manuscripta Math. 117, 199–205 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Faltings, G.: Der Endlichkeitssatz in der lokalen Kohomologie. Math. Ann. 255, 45–56 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Grothendieck, A.: Local Cohomology, Notes by R. Hartshorne, Lecture Notes in Mathematics, vol. 862. Springer, New York (1966)Google Scholar
  11. 11.
    Hartshorne, R.: Affine duality and cofiniteness. Invent. Math. 9, 145–164 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Huneke, C., Koh, J.: Cofiniteness and vanishing of local cohomology modules. Math. Proc. Camb. Phil. Soc. 110, 421–429 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lyubezink, G.: Finiteness properties of local cohomology modules (an application of D-modules to commutative algebra). Invent. Math. 113, 41–55 (1993)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Mehrvarz, A.A., Naghipour, R., Sedghi, M.: Faltings’ local–global principle for the finiteness of local cohomology modules over Noetherian rings. Commun. Algebra 43, 4860–4872 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Melkersson, L.: Modules cofinite with respect to an ideal. J. Algebra 285, 649–668 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Melkersson, L.: Cofiniteness with respect to ideals of dimension one. J. Algebra 372, 459–462 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TabrizTabrizIran
  2. 2.School of MathematicsInstitute for Research in Fundamental Sciences (IPM)TehranIran
  3. 3.Department of MathematicsAzarbaijan Shahid Madani UniversityTabrizIran

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