On formal local cohomology and connectedness

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 , i∈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 SpecR∖V(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.