\(d\)-transform functor and some finiteness and isomorphism results (Q2510593)

Let \(R\) denote a Noetherian ring with \(\mathcal{I}(R)\) the set of its ideals and \(d\geq 0\) an integer. Let \(\Sigma_d = \{\mathfrak{a} \in \mathcal{I}(R)| \dim R/\mathfrak{a} \leq d\}\). Then, \(\Sigma_d\) with the inclusion forms an inductive system of ideals. The authors investigate the section functor \(L_d (-) \) with respect to \(\Sigma_d\) its derived local cohomology functors \(H^i_d(-)\) and the global transform \(T_d(-) = \varinjlim{}_{\mathfrak{a}\in \Sigma_d} \Hom_R(\mathfrak{a}, -)\). Note that this is a particular case of Hartshorne's variation 1 on local cohomology see \textit{R. Hartshorne} [Residues and duality. Appendix: Cohomologie à support propre et construction du foncteur \(f^!\). par P. Deligne. York: Springer-Verlag, 423 p (1966; Zbl 0212.26101)]. In the situation of the authors', they give a module theoretic description of \(T_d(M)\) for an \(R\)-module \(M\), characterize when the natural homomorphism \(M \to T_d(M)\) is an isomorphism, prove a finiteness criterion for \(T_d(M)\), describe the associated primes and study the behaviour of localization of \(T_d(M)\).