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Artinian Local Cohomology Modules

Published online by Cambridge University Press:  20 November 2018

Keivan Borna Lorestani ,
Parviz Sahandi  and
Siamak Yassemi
Keivan Borna Lorestani
Affiliation:
Department of Mathematics, University of Tehran, Tehran, Iran e-mail: borna@ipm.ir
Parviz Sahandi
Affiliation:
Center of Excellence in Biomathematics, School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, Iran and School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics, Tehran, Iran e-mail: sahandi@ipm.ir yassemi@ipm.ir
Siamak Yassemi
Affiliation:
Center of Excellence in Biomathematics, School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, Iran and School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics, Tehran, Iran e-mail: sahandi@ipm.ir yassemi@ipm.ir
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Abstract

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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.

Keywords

13D4513E1013C05local cohomology moduleArtinian modulereflexive module

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 50 , Issue 4 , 01 December 2007 , pp. 598 - 602
Copyright
Copyright © Canadian Mathematical Society 2007

References

[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, 313–324.Google Scholar
[B] Belshoff, R., Some change of ring theorems for Matlis reflexive modules. Comm. Algebra 22(1994), no. 9, 3545–3552.Google Scholar
[BER] Belshoff, R., Enochs, E. E., and Rozas, J. R. García, Generalized Matlis duality. Proc. Amer. Math. Soc. 128(2000), no. 5, 1307–1312.Google Scholar
[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, 2851–2853.Google Scholar
[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.Google Scholar
[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.Google Scholar
[H] Hartshorne, R., Affine duality and cofiniteness. Invent. Math. 9(1969/1970) 145–164.Google Scholar
[KS] Khashyarmanesh, K. and Salarian, Sh., On the associated primes of local cohomology modules. Comm. Algebra 27(1999), no. 12, 6191–6198.Google Scholar
[M] Melkersson, L., Some application of a criterion for Artinianness of a module. J. Pure and Appl. Algebra 101(1995), no. 3, 291–303.Google Scholar
[R] Rudlof, P., On minimax and related modules. Canad. J. Math. 44 1992), no. 1, 154–166.Google Scholar
[X] Xue, W., Generalized Matlis duality and linear compactness. Comm. Algebra 30(2002), no. 4, 2075–2084.Google Scholar
[Z] Zöschinger, H., Minimax-moduln. J. Algebra 102(1986), no. 1, 1–32.Google Scholar