General Local Cohomology Modules in View of Low Points and High Points

Let R be a commutative Noetherian ring, let Φ be a system of ideals of R, let M be a finitely generated R-module, and let t be a nonnegative integer. We first show that a general local cohomology module ${H}_{{\Phi }_{\mathfrak{p}}}^{i}\left({M}_{\mathfrak{p}}\right)$ is a finitely generated R-module for all i < t if and only if ${\mathrm{Ass}}_{R}\left({H}_{{\Phi }_{\mathfrak{p}}}^{i}\left({M}_{\mathfrak{p}}\right)\right)$ is a finite set and ${H}_{{\Phi }_{\mathfrak{p}}}^{i}\left({M}_{\mathfrak{p}}\right)$ is a finitely generated ${R}_{\mathfrak{p}}$-module for all i < t and all 𝔭 ∈ Spec(R). Then, as a consequence, we prove that if (R, 𝔪) is a complete local ring, Φ is countable, and n ∈ ℕ₀ is such that ${\left({\mathrm{Ass}}_{R}\left({H}_{{\Phi }_{\mathfrak{p}}}^{{h}_{\Phi }^{n}\left(M\right)}\left(M\right)\right)\right)}_{\ge n}$ is a finite set, then ${f}_{\Phi }^{n}\left(M\right)={h}_{\Phi }^{n}\left(M\right).$ In addition, we show that the properties of vanishing and finiteness of general local cohomology modules are equivalent at high points over an arbitrary Noetherian (not necessarily local) ring. For each covariant R-linear functor T from Mod(R) into itself, which has the property of global vanishing on Mod(R) , for any Serre subcategory 𝒮, and t ∈ ℕ, we prove that ℛⁱT(R) ∈ 𝒮 for all i ≥ t if and only if ℛⁱT(M) ∈ 𝒮 for any finitely generated R-module M and all i ≥ t. We also obtain some results on general local cohomology modules.