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On the finiteness of associated primes of local cohomology modules. (English) Zbl 1190.13010
Summary: Let \( R\) be a Noetherian ring, \( \mathfrak{a}\) be an ideal of \( R\) and \( M\) be a finitely generated \( R\)-module. The aim of this paper is to show that if \( t\) is the least integer such that neither \( H^t_{\mathfrak{a}}(M)\) nor supp\((H^t_{\mathfrak{a}}(M))\) is non-finite, then \( H^t_{\mathfrak{a}}(M)\) has finitely many associated primes. This combines the main results of M. P. Brodmann and A. Lashgari Faghani [Proc. Am. Math. Soc. 128, No. 10, 2851–2853 (2000; Zbl 0955.13007)] and independently of K. Khashyarmanesh and Sh. Salarian [Commun. Algebra 27, No. 12, 6191–6198 (1999; Zbl 0940.13013)].

13D45 Local cohomology and commutative rings
Full Text: DOI
[1] 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. · Zbl 0955.13007
[2] 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. · Zbl 0903.13006
[3] Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. · Zbl 0788.13005
[4] Craig L. Huneke and Rodney Y. Sharp, Bass numbers of local cohomology modules, Trans. Amer. Math. Soc. 339 (1993), no. 2, 765 – 779. · Zbl 0785.13005
[5] Mordechai Katzman, An example of an infinite set of associated primes of a local cohomology module, J. Algebra 252 (2002), no. 1, 161 – 166. · Zbl 1083.13505
[6] K. Khashyarmanesh and Sh. Salarian, On the associated primes of local cohomology modules, Comm. Algebra 27 (1999), no. 12, 6191 – 6198. · Zbl 0940.13013
[7] Gennady Lyubeznik, Finiteness properties of local cohomology modules (an application of \?-modules to commutative algebra), Invent. Math. 113 (1993), no. 1, 41 – 55. · Zbl 0795.13004
[8] Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. · Zbl 0603.13001
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