Faltings' Local-global Principle and Annihilator Theorem for the finiteness dimensions
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 $H_{\frak a}^i(M)$ such that $\dim{\rm Supp } H_{\frak a}^i(M)/N_i<n$ for all $i<t$ if and only if there is a finitely generated submodule $N_{i,{\frak p}}$ of $H_{{\frak a} R_{\frak p}}^i(M_{\frak p})$ such that $\dim{\rm Supp } H_{{\frak a} R_{\frak p}}^i(M_{\frak p})/N_{i,{\frak p}}<n$ 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 $\inf\{i\in\mathbb N_0\mid\dim{\rm Supp}({\frak b}^tH_{\frak a}^i(M))\geq n\text{ for all } t\in\mathbb N_0\}$ and $\inf\{{\rm depth } M_{\frak p}+{\rm ht}({\frak a}+{\frak p})/{\frak p}\mid{\frak p}\in{\rm Spec } R\setminus V({\frak b}) \text{ and } \dim R/({\frak a}+{\frak p})\geqslant n\}$ are equal, for every finitely generated $R$-module $M$ and for all ideals $\frak a, \frak b$ of $R$ with ${\frak b}\subseteq {\frak a}$. As a consequence, we determine the least integer $i$ where the local cohomology module $H_{\frak a}^i(M)$ is not minimax (resp. weakly laskerian).