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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.

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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