Faltings' Local-global Principle and Annihilator Theorem for the finiteness dimensions

Abstract

Let $R$ be a commutative Noetherian ring, $M$ a finitely generated $R$-module and $n$ be a non-negative integer. In this article, it is shown that there is a finitely generated submodule $N_i$ of $H_{\mathfrak{a}}^i(M)$ such that $\dim \mathrm{Supp}{H}_{\mathfrak{a}}^i(M)/N_i<n$ for all $i<t$ if and only if there is a finitely generated submodule $N_{i,\mathfrak{p}}$ of $H_{\mathfrak{a}{R}_{\mathfrak{p}}}^i(M_{\mathfrak{p}})$ such that $\dim \mathrm{Supp}{H}_{\mathfrak{a}{R}_{\mathfrak{p}}}^i(M_{\mathfrak{p}})/N_{i,\mathfrak{p}}<n$ for all $i<t$. This generalizes Faltings' Local-global Principle for the finiteness of local cohomology modules (Faltings' in Math. Ann. 255:45-56, 1981). Also, it is shown that whenever $R$ is a homomorphic image of a Gorenstein local ring, then the invariants $\inf \{i\in {\mathbb{N}}_0\mid \dim \mathrm{Supp}({\mathfrak{b}}^t{H}_{\mathfrak{a}}^i(M))\ge n\text{ for all }t\in {\mathbb{N}}_0\}$ and $\inf \{\mathrm{depth}{M}_{\mathfrak{p}}+\mathrm{ht}(\mathfrak{a}+\mathfrak{p})/\mathfrak{p}\mid \mathfrak{p}\in \mathrm{Spec}R\setminus V(\mathfrak{b})\text{ and }\dim R/(\mathfrak{a}+\mathfrak{p})⩾n\}$ are equal, for every finitely generated $R$-module $M$ and for all ideals $\mathfrak{a},\mathfrak{b}$ of $R$ with $\mathfrak{b}\subseteq \mathfrak{a}$. As a consequence, we determine the least integer $i$ where the local cohomology module $H_{\mathfrak{a}}^i(M)$ is not minimax (resp. weakly laskerian).