Faltings' local-global principle for the finiteness of local cohomology modules over Noetherian rings

Let $R$ denote a commutative Noetherian (not necessarily local) ring, $\frak a$ an ideal of $R$ and $M$ a finitely generated $R$-module. The purpose of this paper is to show that $f^n_{\frak a}(M)=\inf \{0\leq i\in\mathbb{Z}|\, \dim H^{i}_{\frak a}(M)/N \geq n\, \, \text{for any finitely generated submodule}\,\, N \subseteq H^{i}_{\frak a}(M)\}$, where $n$ is a non-negative integer and the invariant $f^n_{\frak a}(M):=\inf\{f_{\frak a R_{\frak p}}(M_{\frak p})\,\,|\,\,{\frak p}\in \Supp M/\frak a M\,\,{\rm and}\,\,\dim R/{\frak p}\geq n\}$ is the $n$-th finiteness dimension of $M$ relative to $\frak a$. As a consequence, it follows that the set $$ \Ass_R(\oplus _{i=0}^{f^n_{\frak a}(M)}H^{i}_{\frak a}(M))\cap \{\frak p\in \Spec R|\, \dim R/\frak p\geq n\}$$ is finite. This generalizes the main result of Quy \cite{Qu}, Brodmann-Lashgari \cite{BL} and Asadollahi-Naghipour \cite{AN}.