About the Theory of Formal Local Cohomology Modules

Let I, a be two ideals of a Noetherian ring R. Let M be an R-module. There exists a systematic study of the formal cohomology modules \(\underleftarrow{lim}_{n\epsilon\mathbb{N}}\) \(H^i_I\)(M/\(\mathfrak{a}^n\)M), 0 \(\le\) i \(\epsilon\) \(\mathbb{Z}\). The purpose of this note is to establish a kind of theorem of nonvanishing on the formal local cohomology module. It is what will be done in this paper. Throughout this paper, R is a commutative ring with non-zero identity. The theory of local cohomology if has developed so much six decades after its introduction by Grothendieck. There exists a relation between local cohomology and formal local cohomology. I study here this latter module.