General formal local cohomology modules

Let (R,m) be a local ring, Φ a system of ideals of R and M a finitely generated R-module. In this paper, we define and study general formal local cohomology modules. We denote the ith general formal local cohomology module M with respect to Φ by FiΦ(M) and we investigate some finiteness and Artinianness properties of general formal local cohomology modules.

A finite result for the set of associated prime ideals of formal local cohomology modules

Let [Formula: see text] be a Noetherian local ring, [Formula: see text] two ideals of [Formula: see text], and [Formula: see text] two finitely generated [Formula: see text]-modules. It is first shown that [Formula: see text] is a finite set. We also prove that except the maximal ideal [Formula: see text], the set [Formula: see text] is stable for large [Formula: see text], where we use [Formula: see text] to denote [Formula: see text]-module [Formula: see text] or [Formula: see text] and [Formula: see text] is the eventual value of [Formula: see text].