Elsevier

Journal of Algebra

Volume 315, Issue 2, 15 September 2007, Pages 894-923
Journal of Algebra

On formal local cohomology and connectedness

https://doi.org/10.1016/j.jalgebra.2007.06.015Get rights and content
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Abstract

Let a denote an ideal of a local ring (R,m). Let M be a finitely generated R-module. There is a systematic study of the formal cohomology modules limHmi(M/anM), iZ. We analyze their R-module structure, the upper and lower vanishing and non-vanishing in terms of intrinsic data of M, and its functorial behavior. These cohomology modules occur in relation to the formal completion of the punctured spectrum SpecRV(m).

As a new cohomological data there is a description on the formal grade fgrade(a,M) defined as the minimal non-vanishing of the formal cohomology modules. There are various exact sequences concerning the formal cohomology modules. Among them a Mayer–Vietoris sequence for two ideals. It applies to new connectedness results. There are also relations to local cohomological dimensions.

Keywords

Local cohomology
Formal cohomology
Connectedness properties

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