On formal local cohomology and connectedness

Let denote an ideal of a local ring . Let M be a finitely generated R-module. There is a systematic study of the formal cohomology modules , . 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 . As a new cohomological data there is a description on the formal grade 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.