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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.

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