Skip to main content Accessibility help
×
Home

Artinian Local Cohomology Modules

  • Keivan Borna Lorestani (a1), Parviz Sahandi (a2) and Siamak Yassemi (a2)

Abstract

Let $R$ be a commutative Noetherian ring, $\mathfrak{a}$ an ideal of $R$ and $M$ a finitely generated $R$ -module. Let $t$ be a non-negative integer. It is known that if the local cohomology module $\text{H}_{\mathfrak{a}}^{i}\,\left( M \right)$ is finitely generated for all $i\,<\,t$ , then $\text{Ho}{{\text{m}}_{R}}\,\left( R/\mathfrak{a},\,\text{H}_{\mathfrak{a}}^{t}\,\left( M \right) \right)$ is finitely generated. In this paper it is shown that if $\text{H}_{\mathfrak{a}}^{i}\,\left( M \right)$ is Artinian for all $i\,<\,t$ , then $\text{Ho}{{\text{m}}_{R}}\,\left( R/\mathfrak{a},\,\text{H}_{\mathfrak{a}}^{t}\,\left( M \right) \right)$ need not be Artinian, but it has a finitely generated submodule $N$ such that $\text{Ho}{{\text{m}}_{R}}\left( R/\mathfrak{a},\text{H}_{\mathfrak{a}}^{t}\left( M \right) \right)/N$ is Artinian.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Artinian Local Cohomology Modules
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Artinian Local Cohomology Modules
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Artinian Local Cohomology Modules
      Available formats
      ×

Copyright

References

Hide All
[AKS] Asadollahi, J., Khashyarmanesh, K., Salarian, Sh., A generalization of the cofiniteness problem in local cohomology modules. J. Aust. Math. Soc. 75(2003), no. 3, 313324.
[B] Belshoff, R., Some change of ring theorems for Matlis reflexive modules. Comm. Algebra 22(1994), no. 9, 35453552.
[BER] Belshoff, R., Enochs, E. E., and Rozas, J. R. García, Generalized Matlis duality. Proc. Amer. Math. Soc. 128(2000), no. 5, 13071312.
[BL] Brodmann, M. P. and Faghani, A. L., A finiteness result for associated primes of local cohomology modules. Proc. Amer. Math. Soc. 128(2000), no. 10, 28512853.
[BS] Brodmann, M. P. and Sharp, R. Y., Local Cohomology: An Algebraic Introduction with Geometric Applications. Cambridge Studies in Advanced Mathematics 60, Cambridge University Press, Cambridge, 1998.
[G] Grothendieck, A., Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2). Advanced Studies in Pure Mathematics, vol. 2, North-Holland, Amsterdam 1968.
[H] Hartshorne, R., Affine duality and cofiniteness. Invent. Math. 9(1969/1970) 145164.
[KS] Khashyarmanesh, K. and Salarian, Sh., On the associated primes of local cohomology modules. Comm. Algebra 27(1999), no. 12, 61916198.
[M] Melkersson, L., Some application of a criterion for Artinianness of a module. J. Pure and Appl. Algebra 101(1995), no. 3, 291303.
[R] Rudlof, P., On minimax and related modules. Canad. J. Math. 44 1992), no. 1, 154166.
[X] Xue, W., Generalized Matlis duality and linear compactness. Comm. Algebra 30(2002), no. 4, 20752084.
[Z] Zöschinger, H., Minimax-moduln. J. Algebra 102(1986), no. 1, 132.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed