Skip to main content

Top formal local cohomology module


Let I be an ideal of a local commutative noetherian ring (\(R, {{\mathfrak {m}}}\)) and M a finitely generated R-module. We study some properties of the top formal local cohomology module \({\mathcal {F}}^l_I(M)=\mathop {\varprojlim }H^l_{{{\mathfrak {m}}}}(M{/}I^tM)\) with \(l=\mathrm {dim}M{/}I M\). In particular, we show that, in the case \(M \ne IM\), \({\mathcal {F}}^l_I(M)\) is artinian if and only if \(l>\mathrm {dim}{\overline{M}}/I{\overline{M}}\) where \({\overline{M}}=M/ H^0_I(M)\). As a consequence, we have \( \mathrm {dim}{\overline{M}}/I{\overline{M}}=\sup \{i \in {\mathbb {Z}}\mid {\mathcal {F}}^i_I(M) \text { is not artinian}\},\) provided that \({\overline{M}} \ne I{\overline{M}}\).

This is a preview of subscription content, access via your institution.


  1. 1.

    M. Asgharzadeh, K. Divaani-Aazar, Finiteness properties of formal local cohomology modules and Cohen–Macaulayness. Commun. Algebra 39(3), 1082–1103 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    M. Aghapournahr, L. Melkersson, Finiteness properties of minimax and coatomic local cohomology modules. Arch. Math. 94, 519–528 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    M. Brodmann, R.Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications (Cambridge University Press, Cambridge, 1998)

    Book  MATH  Google Scholar 

  4. 4.

    N.T. Cuong, T.T. Nam, The \(I\)-adic completion and local homology for Artinian modules. Math. Proc. Camb. Philos. Soc. 131, 61–72 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Y. Gu, The Artinianness of formal local cohomology modules. Bull. Malays. Math. Sci. Soc. 2(2), 449–456 (2014)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    R. Hartshorne, Algebraic Geometry (Springer, Berlin, 1977)

    Book  MATH  Google Scholar 

  7. 7.

    I.G. Macdonald, Secondary representation of modules over a commutative ring. Sympos. Math. 11, 23–43 (1973)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    J. Strooker, Homological Questions in Local Algebra (Cambridge University Press, Cambridge, 1990)

    Book  MATH  Google Scholar 

  9. 9.

    P. Schenzel, On formal local cohomology and connectedness. J. Algebra 315, 894–923 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    H. Zöschinger, Minimax Moduln. J. Algebra 102, 1–32 (1986)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Tri Minh Nguyen.

Additional information

The first author and the third author are funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.04-2018.304.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tran, N.T., Nguyen, T.H.H. & Nguyen, T.M. Top formal local cohomology module. Period Math Hung 79, 1–11 (2019).

Download citation


  • Formal local cohomology
  • Minimax module
  • Finitely generated module

Mathematics Subject Classification

  • 13D45