Revista de la
Unión Matemática Argentina
Top local cohomology modules over local rings and the weak going-up property
Asghar Farokhi and Alireza Nazari
Volume 60, no. 1 (2019), pp. 209–215

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Let $(R,\mathfrak{m})$ be a Noetherian local ring and let $\widehat{R}$ denote the $\mathfrak{m}$-adic completion of $R$. In this paper, we introduce the concept of the weak going-up property for the extension $R\subseteq \widehat{R}$ 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 $\mathfrak{a}$ and $\mathfrak{b}$ of $R$, we have $\operatorname{Att}_{R}(\operatorname{H}^{d}_{\mathfrak{a}}(M)) = \operatorname{Att}_{R}(\operatorname{H}^{d}_{\mathfrak{b}}(M))$ if and only if $\operatorname{H}^{d}_{\mathfrak{a}}(M)\cong \operatorname{H}^{d}_{\mathfrak{b}}(M)$, and we find a criterion for the cofiniteness of Artinian top local cohomology modules.