On the cofiniteness of local cohomology modules
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- by Kamal Bahmanpour and Reza Naghipour
- Proc. Amer. Math. Soc. 136 (2008), 2359-2363
- DOI: https://doi.org/10.1090/S0002-9939-08-09260-5
- Published electronically: March 4, 2008
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Abstract:
In this note we show that if $I$ is an ideal of a Noetherian ring $R$ and $M$ is a finitely generated $R$-module, then for any minimax submodule $N$ of $H^{t}_{I}(M)$ the $R$-module $\textrm {Hom}_{R}(R/I,H_{I}^{t}(M)/N)$ is finitely generated, whenever the modules $H_{I}^{0}(M), H_{I}^{1}(M),..., H_{I}^{t-1}(M)$ are minimax. As a consequence, it follows that the associated primes of $H^{t}_{I}(M)/N$ are finite. This generalizes the main result of Brodmann and Lashgari (2000).References
- M. P. Brodmann and A. Lashgari Faghani, A finiteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc. 128 (2000), no. 10, 2851–2853. MR 1664309, DOI 10.1090/S0002-9939-00-05328-4
- M. Brodmann, Ch. Rotthaus, and R. Y. Sharp, On annihilators and associated primes of local cohomology modules, J. Pure Appl. Algebra 153 (2000), no. 3, 197–227. MR 1783166, DOI 10.1016/S0022-4049(99)00104-8
- M. P. Brodmann and R. Y. Sharp, Local cohomology: an algebraic introduction with geometric applications, Cambridge Studies in Advanced Mathematics, vol. 60, Cambridge University Press, Cambridge, 1998. MR 1613627, DOI 10.1017/CBO9780511629204
- Edgar Enochs, Flat covers and flat cotorsion modules, Proc. Amer. Math. Soc. 92 (1984), no. 2, 179–184. MR 754698, DOI 10.1090/S0002-9939-1984-0754698-X
- Robin Hartshorne, Local cohomology, Lecture Notes in Mathematics, No. 41, Springer-Verlag, Berlin-New York, 1967. A seminar given by A. Grothendieck, Harvard University, Fall, 1961. MR 0224620
- Alexander Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux $(SGA$ $2)$, Advanced Studies in Pure Mathematics, Vol. 2, North-Holland Publishing Co., Amsterdam; Masson & Cie, Editeur, Paris, 1968 (French). Augmenté d’un exposé par Michèle Raynaud; Séminaire de Géométrie Algébrique du Bois-Marie, 1962. MR 0476737
- Robin Hartshorne, Affine duality and cofiniteness, Invent. Math. 9 (1969/70), 145–164. MR 257096, DOI 10.1007/BF01404554
- M. Hellus, On the set of associated primes of a local cohomology module, J. Algebra 237 (2001), no. 1, 406–419. MR 1813886, DOI 10.1006/jabr.2000.8580
- Craig Huneke, Problems on local cohomology, Free resolutions in commutative algebra and algebraic geometry (Sundance, UT, 1990) Res. Notes Math., vol. 2, Jones and Bartlett, Boston, MA, 1992, pp. 93–108. MR 1165320
- Craig L. Huneke and Rodney Y. Sharp, Bass numbers of local cohomology modules, Trans. Amer. Math. Soc. 339 (1993), no. 2, 765–779. MR 1124167, DOI 10.1090/S0002-9947-1993-1124167-6
- Mordechai Katzman, An example of an infinite set of associated primes of a local cohomology module, J. Algebra 252 (2002), no. 1, 161–166. MR 1922391, DOI 10.1016/S0021-8693(02)00032-7
- Gennady Lyubeznik, Finiteness properties of local cohomology modules (an application of $D$-modules to commutative algebra), Invent. Math. 113 (1993), no. 1, 41–55. MR 1223223, DOI 10.1007/BF01244301
- Gennady Lyubeznik, A partial survey of local cohomology, Local cohomology and its applications (Guanajuato, 1999) Lecture Notes in Pure and Appl. Math., vol. 226, Dekker, New York, 2002, pp. 121–154. MR 1888197
- Thomas Marley, The associated primes of local cohomology modules over rings of small dimension, Manuscripta Math. 104 (2001), no. 4, 519–525. MR 1836111, DOI 10.1007/s002290170024
- Leif Melkersson, Properties of cofinite modules and applications to local cohomology, Math. Proc. Cambridge Philos. Soc. 125 (1999), no. 3, 417–423. MR 1656785, DOI 10.1017/S0305004198003041
- Leif Melkersson, Modules cofinite with respect to an ideal, J. Algebra 285 (2005), no. 2, 649–668. MR 2125457, DOI 10.1016/j.jalgebra.2004.08.037
- Le Thanh Nhan, On generalized regular sequences and the finiteness for associated primes of local cohomology modules, Comm. Algebra 33 (2005), no. 3, 793–806. MR 2128412, DOI 10.1081/AGB-200051137
- Anurag K. Singh, $p$-torsion elements in local cohomology modules, Math. Res. Lett. 7 (2000), no. 2-3, 165–176. MR 1764314, DOI 10.4310/MRL.2000.v7.n2.a3
- Thomas Zink, Endlichkeitsbedingungen für Moduln über einem Noetherschen Ring, Math. Nachr. 64 (1974), 239–252 (German). MR 364223, DOI 10.1002/mana.19740640114
- Helmut Zöschinger, Minimax-moduln, J. Algebra 102 (1986), no. 1, 1–32 (German). MR 853228, DOI 10.1016/0021-8693(86)90125-0
- Helmut Zöschinger, Über die Maximalbedingung für radikalvolle Untermoduln, Hokkaido Math. J. 17 (1988), no. 1, 101–116 (German). MR 928469, DOI 10.14492/hokmj/1381517790
Bibliographic Information
- Kamal Bahmanpour
- Affiliation: Department of Mathematics, University of Tabriz, Tabriz, Iran – and – Department of Mathematics, Islamic Azad University-Ardebil Branch, P.O. Box 5614633167, Ardebil, Iran
- Email: bahmanpour@tabrizu.ac.ir
- Reza Naghipour
- Affiliation: Department of Mathematics, University of Tabriz, Tabriz, Iran – and – School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics (IPM), P.O. Box 19395-5746, Tehran, Iran
- Email: naghipour@ipm.ir, naghipour@tabrizu.ac.ir
- Received by editor(s): February 27, 2007
- Received by editor(s) in revised form: May 17, 2007
- Published electronically: March 4, 2008
- Additional Notes: The research of the second author has been in part supported by a grant from IPM (No. 85130042)
- Communicated by: Bernd Ulrich
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 2359-2363
- MSC (2000): Primary 13D45, 14B15, 13E05
- DOI: https://doi.org/10.1090/S0002-9939-08-09260-5
- MathSciNet review: 2390502