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On the cofiniteness of local cohomology modules. (English) Zbl 1141.13014
Let \(I\) be an ideal of a noetherian ring \(R\) and let \(M\) be a finitely generated \(R\)-module. The local cohomology modules \(H^l_I(M)\) are not finitely generated in general. Even worse, in general, they have infinitely many associated prime ideals. However, there are certain positive finiteness results on local cohomology: For example, a result of M. P. Brodmann and A. Lashgari Faghani [Proc. Am. Math. Soc. 128, No. 10, 2851–2853 (2000; Zbl 0955.13007)] says that if \(H^0_I(M),\ldots H^{t-1}_I(M)\) are finitely generated and \(N\) is a finitely generated submodule of \(H^t_I(M)\), then \(\text{Ass}_R(H^t_I(M)/N)\) is finite. This paper contains a generalization of the Faghani-Brodmann theorem: The assumptions are weaker in the sense that \(H^0_I(M),\ldots ,H^{t-1}_I(M)\) and \(N\) are only required to be minimax (a module is called minimax if it has a finitely generated submodule such that the quotient by this submodule is Artinian; over a complete local ring the minimax modules are precisely the Matlis reflexive ones). The statement in the general version is that \(\operatorname{Hom}_R(R/I,H^t_I(M)/N)\) is finitely generated (as an immediate consequence, \(H^t_I(M)/N\) has only finitely many associated prime ideals).

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