Abstract
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.
Highlights
Throughout this paper, let (R, m) be a Noetherian local ring and let R denote the m-adic completion of R
For an R-module M, the m-adic completion of M is denoted by M and the i-th local cohomology module of M with respect to a is defined as
For each Artinian R-module A, we denote by AttR A the set of all attached prime ideals of A
Summary
Throughout this paper, let (R, m) be a Noetherian local ring and let R denote the m-adic completion of R. In the main result of this paper, Theorem 2.4, we will show that the weak going-up property is equivalent to the strong form of the Lichtenbaum–Hartshorne Vanishing Theorem, which is defined as follows: Definition 1.2.
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