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Cofiniteness of Generalized Local Cohomology Modules for One-Dimensional Ideals

Published online by Cambridge University Press:  20 November 2018

Kamran Divaani-Aazar  and
Alireza Hajikarimi
Kamran Divaani-Aazar
Affiliation:
Department of Mathematics, Az-Zahra University, Vanak, Post Code 19834, Tehran, Iran e-mail: kdivaani@ipm.ir
Alireza Hajikarimi
Affiliation:
Science and Research Branch, Islamic Azad University, Tehran, Iran e-mail: alihajikarimi@yahoo.com
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Abstract

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Let $\mathfrak{a}$ be an ideal of a commutative Noetherian ring $R$ and $M$ and $N$ two finitely generated $R$-modules. Our main result asserts that if $R/\mathfrak{a}\,\le \,1$, then all generalized local cohomology modules $H_{\mathfrak{a}}^{i}(M,\,N)$ are $\mathfrak{a}$-cofinite.

Keywords

13D4513E0513E10cofinite modulesgeneralized local cohomology moduleslocal cohomology modules

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 1 , 01 March 2012 , pp. 81 - 87
Copyright
Copyright © Canadian Mathematical Society 2012

References

[AD] Asgharzadeh, M. and Divaani-Aazar, K., Finiteness properties of formal local cohomology modules and Cohen-Macaulayness. Comm. Algebra, to appear. arXiv:0807.5042Google Scholar
[BN] Bahmanpour, K. and Naghipour, R., Cofiniteness of local cohomology modules for ideals of small dimension. J. Algebra 321(2009), no. 7, 1997–2011. doi:10.1016/j.jalgebra.2008.12.020Google Scholar
[BS] Brodmann, M. 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
[DM] Delfino, D. and Marley, T., Cofinite modules and local cohomology. J. Pure. Appl. Algebra, 121(1997), no. 1, 45–52. doi:10.1016/S0022-4049(96)00044-8Google Scholar
[D] Divaani-Aazar, K., On associated and attached prime ideals of certain modules. Colloq. Math. 89(2001), no. 1, 147–157. doi:10.4064/cm89-1-11Google Scholar
[DH] Divaani-Aazar, K. and Hajikarimi, A., Generalized local cohomology modules and homological Gorenstein dimensions. Comm. Algebra, to appear. arXiv:0803.0107.Google Scholar
[DS] Divaani-Aazar, K. and Sazeedeh, R., Cofiniteness of generalized local cohomology modules. Colloq. Math. 99(2004), no. 2, 283–290. doi:10.4064/cm99-2-12Google Scholar
[Ha] Hartshorne, R., Affine duality and cofiniteness. Invent. Math. 9(1969/1970), 145–164. doi:10.1007/BF01404554Google Scholar
[HV] Hassanzadeh, S. H. and Vahidi, A., On vanishing and cofiniteness of generalized local cohomology modules. Comm. Algebra 37(2009), no. 7, 2290–2299. doi:10.1080/00927870802622718Google Scholar
[He] Herzog, J., Komplex Auflösungen und Dualität in der lokalen Algebra, Habilitationsschrift, Universität Regensburg, 1970.Google Scholar
[KK] Kawakami, S. and Kawasaki, K.-I., On the finiteness of Bass numbers of generalized local cohomology modules. Toyama Math. J. 29(2006), 59–64.Google Scholar
[K] Kawasaki, K.-I., Cofiniteness of local cohomology modules for principle ideals. Bull. London. Math. Soc. 30(1998), no. 3, 241–246. doi:10.1112/S0024609397004347Google Scholar
[MS] Mafi, A. and Saremi, H., Cofinite modules and generalized local cohomology. Houston J. Math. 35(2009), no. 4, 1013–1018.Google Scholar
[MV] Marley, T. and Vassilev, J. C., Cofiniteness and associated primes of local cohomology modules. J. Algebra 256(2002), no. 1, 180–193. doi:10.1016/S0021-8693(02)00151-5Google Scholar
[Ma] Matsumura, H., Commutative Ring Theory. Second edition. Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, Cambridge, 1989.Google Scholar
[Me1] Melkersson, L., On asymptotic stability for sets of prime ideals connected with the powers of an ideal. Math. Proc. Cambridge Philos. Soc. 107(1990), no. 2, 267–271. doi:10.1017/S0305004100068535Google Scholar
[Me2] Melkersson, L., Modules cofinite with respect to an ideal. J. Algebra 285(2005), no. 2, 649–668. doi:10.1016/j.jalgebra.2004.08.037Google Scholar
[Ya] Yassemi, S., Cofinite modules. Comm. Algebra 29(2001), no. 6, 2333–2340. doi:10.1081/AGB-100002392Google Scholar
[Yo] Yoshida, K.-I., Cofiniteness of local cohomology modules for ideals of dimension one. Nagoya Math. J. 147(1997), 179–191.Google Scholar
[Za] Zamani, N., On graded generalized local cohomology. Arch. Math. (Basel) 86(2006), no. 4, 321–330.Google Scholar
[Zo] Zöschinger, H., Minimax-moduln. J. Algebra, 102(1986), no. 1, 1–32. doi:10.1016/0021-8693(86)90125-0Google Scholar