On formal local cohomology and connectedness

Let it 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 (lim) under left arrow H-m(i) (M/a" M), i is an element of Z. 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 Spec R \ V (in). 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. (c) 2007 Elsevier Inc. All rights reserved.