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Cofiniteness of weakly Laskerian local cohomology modules. (English) Zbl 1340.13010
Summary: Let \(I\) be an ideal of a Noetherian ring \(R\) and \(M\) be a finitely generated \(R\)-module. We introduce the class of extension modules of finitely generated modules by the class of all modules \(T\) with \(\dim T\leq n\) and we denote it by \({\text{FD}_{\leq n}}\) where \(n\geq -1\) is an integer. We prove that for any \({\text{FD}_{\leq 0}}\) (or minimax) submodule \(N\) of \(H^t_I(M)\) the \(R\)-modules \({\text{Hom}}_R(R/I,H^{t}_I(M)/N)\) and \({\text{Ext}}^1_R(R/I,H^{t}_I(M)/N)\) are finitely generated, whenever the modules \(H^0_I(M)\), \(H^1_I(M)\),..., \(H^{t-1}_I(M)\) are \({\text{FD}_{\leq 1}}\) (or weakly Laskerian). As a consequence, it follows that the set of associated primes of \(H^{t}_I(M)/N\) is finite. This generalizes the main results of Kamal Bahmanpour and Reza Naghipour [J. Algebra 321, No. 7, 1997–2011 (2009; Zbl 1168.13016)] and Kamal Bahmanpour and Reza Naghipour [Proc. Am. Math. Soc. 136, No. 7, 2359–2363 (2008; Zbl 1141.13014)], M. P. Brodmann and Faghani A. Lashgari [Proc. Am. Math. Soc. 128, No. 10, 2851–2853 (2000; Zbl 0955.13007)], K. Khashyarmanesh and Sh. Salarian [Commun. Algebra 27, No. 12, 6191–6198 (1999; Zbl 0940.13013)] and Pham Hung Quy [Proc. Am. Math. Soc. 138, No. 6, 1965–1968 (2010; Zbl 1190.13010)]. We also show that the category \(FD^1(R,I)_{cof}\) of \(I\)-cofinite \({\text{FD}_{\leq 1}}\) \(R\)-modules forms an Abelian subcategory of the category of all \(R\)-modules.

13D45 Local cohomology and commutative rings
14B15 Local cohomology and algebraic geometry
13E05 Commutative Noetherian rings and modules
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