General formal local cohomology modules

Abstract

Let \((R,\mathfrak{m})\) be a local ring, \(\Phi\) 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 \(i\)th general formal local cohomology module \(M\) with respect to \(\Phi\) by \(\mathfrak{F}_{\Phi}^{i}(M)\) and we investigate some finiteness and Artinianness properties of general formal local cohomology modules.