General local cohomology modules in view of low points and high points

Abstract

UDC 512.5

Let $R$ be a commutative Noetherian ring, let $\Phi $ 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 $H_{\Phi}^{i}(M)$ is a finitely generated $R$-module for all $i<t$ if and only if $\mathrm{Ass}_{R} (H_{\Phi}^{i}(M))$ is a finite set and $H_{\Phi_{\mathfrak{p}}}^{i}(M_{\mathfrak{p}})$ is a finitely generated $R_{\mathfrak{p}}$-module for all $i<t$ and all $\mathfrak{p}\in \mathrm{Spec}(R)$. Then, as a consequence, we prove that if $(R,\mathfrak{m})$ is a complete local ring, $\Phi$ is countable, and $n\in \mathbb{N}_{0}$ is such that $\big(\mathrm{Ass}_{R} \big(H_{\Phi}^{h_{\Phi}^{n}(M)}(M)\big)\big)_{\geq n}$ is a finite set, then $f_{\Phi}^{n}(M)=h_{\Phi}^{n}(M)$. In addition, we show that the properties of vanishing and finiteness of general local cohomology modules are equivalent on high points over an arbitrary Noetherian (not necessary local) ring. For each covariant $R$-linear functor $T$ from $\mathrm{Mod}(R)$ into itself, which has  the global vanishing property on $\mathrm{Mod}(R)$ and for an arbitrary Serre subcategory $\mathcal{S}$ and $t\in \mathbb{N},$ we prove that  $\mathcal{R}^{i}T(R)\in \mathcal{S}$ for all $i\geq t$ if and only if $\mathcal{R}^{i}T(M)\in \mathcal{S}$ for any finitely generated  $R$-module $M$ and all $i\geq t$.  Then we obtain some results on general local cohomology modules.