A new proof of Faltings' local-global principle for the finiteness of local cohomology modules

Let $R$ denote a commutative Noetherian ring. Brodmann et al. defined and studied the concept of the local-global principle for annihilation of local cohomology modules at level $r\in\mathbb{N}$ for the ideals $\frak a$ and $\frak b$ of $R$. It was shown that this principle holds at levels 1,2, over $R$ and at all levels whenever $\dim R\leq 4$. The goal of this paper is to show that, if the set $\Ass_R(H_{\fa}^{f_{\fa}^{\fb}(M)}(M))$ is finite or $f_{\fa}(M)\neq c_{\fa}^{\fb}(M)$, then the local-global principle holds at all levels $r\in\mathbb{N}_0$, for all ideals $\fa, \fb$ of $R$ and each finitely generated $R$-module $M$, where $c_{\fa}^{\fb}(M)$ denotes the first non $\fb$-cofiniteness of local cohomology module $H^i_{\fa}(M)$. As a consequence of this, we provide a new and short proof of the Faltings' local-global principle for finiteness dimensions. Also, several new results concerning the finiteness dimensions are given.