Essential ideal transforms

TL;DR

This paper generalizes concepts in local cohomology, such as Ext, Tor, and ideal transforms, using e-exact sequences, and extends Mayer-Vietoris sequences within this framework.

Contribution

It introduces and develops the theory of e-exact sequences in local cohomology, generalizing classical functors and sequences with new isomorphisms and exact sequences.

Findings

For torsion-free modules B, $_eext^n_R(P,B)=0$.

For torsion-free modules B, $_eExt^n_R(A,E)=0$ for all modules A.

Generalization of Mayer-Vietoris sequences using e-exact sequences.

Abstract

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 $B$, $_eext^n_R(P,B)=0$ while $_eExt^n_R(A,E)=0$ for every module $A$. Also for any torsion-free module $B$ we have an $e$-exact sequence $0\to \Gamma_{a}(B) \to B\to D_{a}(B)\to H^1_{a}(B)\to 0$ and an isomorphisms between $B$ and $r D_{a}(B)$. Finally we generalize Mayer-Vietories with $e$-exact sequences in essential local cohomology, we get a special $e$-exact sequences.