It is our intention in this research generalized some concept in local
cohomology such as contravarint functor ext, covariant functor Ext,
covarian functor Tor and ideal transforms with e-exact sequences. The
e-exact sequence was introduced by Akray and Zebari \cite{AZ} in 2020. We
obtain for a torsion-free modules B, eβextRnβ(P,B)=0 while
eβExtRnβ(A,E)=0 for every module A. Also for any torsion-free module B
we have an e-exact sequence 0βΞaβ(B)βBβDaβ(B)βHa1β(B)β0 and an isomorphisms between B and rDaβ(B). Finally we
generalize Mayer-Vietories with e-exact sequences in essential local
cohomology, we get a special e-exact sequences