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Finiteness properties of local cohomology modules for $$\mathfrak a$$-minimax modules. (English) Zbl 1157.13014
Let $$\mathfrak a$$ be an ideal of a commutative noetherian ring $$R$$ and $$N$$ a finitely generated $$R$$-module. In 1992 Huneke has conjectured that for any nonnegative integer $$t$$, the local cohomology module $$H^t_{\mathfrak{a}}(N):={\varinjlim}_n\text{Ext}^t_R(R/{\mathfrak{a}}^n,N)$$ has only finitely many associated prime ideals. A. K. Singh [Math. Res. Lett. 7, No. 2–3, 165–176 (2000; Zbl 0965.13013)] gave a counterexample to this conjecture. However in many situations, it is shown that local cohomology modules of finitely generated modules have finitely many associated prime ideals. For instance, Brodmann and Lashgari and independently Salarian and Khashyarmanesh have shown that if for a nonnegative integer $$t$$, $$H^i_{\mathfrak{a}}(N)$$ is finitely generated for all $$i<t$$, then the set of associated prime ideals of $$H^t_{\mathfrak{a}}(N)$$ is finite [see M. P. Brodmann and A. Lashgari Faghani, Proc. Am. Math. Soc. 128, No. 10, 2851–2853 (2000; Zbl 0955.13007) and K. Khashyarmanesh and Sh. Salarian, Commun. Algebra 27, No. 12, 6191–6198 (1999; Zbl 0940.13013)].
The main achievement of the paper under review is a generalization of the above mentioned result. The authors introduce the notion of $$\mathfrak{a}$$-minimax modules and they examine some of the properties of these modules. According to their definition, an $$R$$-module $$M$$ is said to be $$\mathfrak{a}$$-minimax if the Goldie dimension of $$H^0_{\mathfrak{a}}(M/L)$$ is finite for all submodules $$L$$ of $$M$$. The authors prove that if $$M$$ is $$\mathfrak{a}$$-minimax $$R$$-module such that for a nonnegative integer $$t$$, $$H^i_{\mathfrak{a}}(M)$$ is $$\mathfrak{a}$$-minimax for all $$i<t$$, then $$\text{Hom}_R(R/\mathfrak{a},H^t_{\mathfrak{a}}(M))$$ is $$\mathfrak{a}$$-minimax. This immediately implies that the Goldie dimension of $$H^t_{\mathfrak{a}}(M)$$ is finite, and so the set of associated prime ideals of $$H^t_{\mathfrak{a}}(M)$$ is finite.

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