Resumen de Top local cohomology modules over local rings and the weak going-up property

Asghar Farokhi, Alireza Nazari

• Let (R,m) be a Noetherian local ring and let ˆR denote the m-adic completion of R. In this paper, we introduce the concept of the weak going-up property for the extension R⊆ˆR and we give some characterizations of this property. In particular, we show that this property is equivalent to the strong form of the Lichtenbaum–Hartshorne Vanishing Theorem. Also, when R satisfies the weak going-up property, we show that for a finitely generated R-module M of dimension d, and ideals a and b of R, we have AttR(Hda(M))=AttR(Hdb(M)) if and only if da(M)≅Hd(M), and we find a criterion for the cofiniteness of Artinian top local cohomology modules.