Faltings' local-global principle for the in dimension $\bf< n$ of local cohomology modules

Abstract: The concept of Faltings' local-global principle for the in dimension $< n$ of local cohomology modules over a Noetherian ring $R$ is introduced, and it is shown that this principle holds at levels 1, 2. We also establish the same principle at all levels over an arbitrary Noetherian ring of dimension not exceeding 3. These generalize the main results of Brodmann et al. in \cite{BRS}. Moreover, as a generalization of Raghavan's result, we show that the Faltings' local-global principle for the in dimension $<n$ of local cohomology modules holds at all levels $r\in \mathbb{N}$ whenever the ring $R$ is a homomorphic image of a Noetherian Gorenstein ring. Finally, it is shown that if $M$ is a finitely generated $R$-module, $\frak a$ an ideal of $R$ and $r$ a non-negative integer such that $\frak a^tH^i_{\frak a}(M)$ is in dimension $< 2$ for all $i<r$ and for some positive integer $t$, then for any minimax submodule $N$ of $H^r_{\frak a}(M)$, the $R$-module $\Hom_R(R/\frak a, H^r_{\frak a}(M)/N)$ is finitely generated. As a consequence, it follows that the associated primes of $H^r_{\frak a}(M)/N$ are finite. This generalizes the main results of Brodmann-Lashgari \cite{BL} and Quy \cite{Qu}.