A Note on the Artinian Cofinite Modules

In this paper we shall prove the following result, which is a generalization of the Melkersson's main result proved in [16 Melkersson , L. ( 1999 ). Properties of cofinite modules and application to local cohomology . Math. Proc. Cambridge Philos. Soc. 125 : 417 – 423 .[Crossref], [Web of Science ®] , [Google Scholar]]. Let (R, 𝔪) be a Noetherian local ring such that is integral over R. Let I be a proper ideal of R and A be an Artinian R-module. Then A is I-cofinite if and only if Rad(I + Ann R (A)) = 𝔪. Also, we present an example to show that this result does not hold for an arbitrary local Noetherian ring in general. As an application of this result we prove the following generalization of the Lichtenbaum-Hartshorne Vanishing Theorem (see [5 Brodmann , M. P. , Sharp , R. Y. ( 1998 ). Local Cohomology; An Algebraic Introduction with Geometric Applications . Cambridge : Cambridge University Press .[Crossref] , [Google Scholar], Theorem 8.2.1]). Let (R, 𝔪) be a Noetherian local ring such that is integral over R. Let I an ideal of R and M be a nonzero finitely generated R-module of dimension n. Then the following conditions are equivalent: (i) . (ii) There exists a prime ideal 𝔭 in AsshR(M) such that Rad(𝔭 +I) = 𝔪.