Faltings' local-global principle for the minimaxness of local cohomology modules

Abstract

The concept of Faltings' local-global principle for the minimaxness of local cohomology modules over a commutative Noetherian ring $R$ is introduced, and it is shown that this principle holds at level 2. We also establish the same principle at all levels over an arbitrary commutative Noetherian ring of dimension not exceeding 3. These generalize the main results of Brodmann et al. in \cite{BRS}. Moreover, 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 skinny 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}.