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Artinian local cohomology modules. (English) Zbl 1140.13016
Let $$R$$ be a commutative noetherian ring, $$\mathfrak a$$ an ideal of $$R$$ and $$M$$ a finitely generated $$R$$-module. By a result of J. Asadollahi, K. Khashyarmanesh and S. Salarian [J. Aust. Math. Soc. 75, No. 3, 313–324 (2003; Zbl 1096.13522)], it is known that if $$t$$ is the least integer $$i$$ such that $$H^i_{\mathfrak a}(M)$$ is not finitely generated, then the $$R$$-module $$\operatorname{Hom}_R(R/\mathfrak a,H^t_{\mathfrak a}(M))$$ is finitely generated. Now, let $$s$$ be the least integer $$i$$ such that $$H^i_{\mathfrak a}(M)$$ is not Artinian. In the paper under review, the authors have examined the $$R$$-module $$\operatorname{Hom}_R(R/\mathfrak a,H^s_{\mathfrak a}(M))$$. Although, clearly this module need not be Artinian in general, they showed that it possesses a finitely generated submodule $$N$$ such that $$\operatorname{Hom}_R(R/\mathfrak a,H^s_{\mathfrak a}(M))/N$$ is Artinian. In particular, this implies the known fact that $$H^s_{\mathfrak a}(M)$$ has only finitely many associated primes.

MSC:
 13D45 Local cohomology and commutative rings 13E10 Commutative Artinian rings and modules, finite-dimensional algebras 13C05 Structure, classification theorems for modules and ideals in commutative rings
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