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

TL;DR

This paper investigates Faltings' local-global principle for minimaxness in local cohomology modules, establishing its validity at certain levels and deriving finiteness results for associated primes in a broad algebraic setting.

Contribution

It extends Faltings' local-global principle for minimaxness to all levels over rings of dimension up to 3 and generalizes finiteness results for associated primes of local cohomology modules.

Findings

Faltings' principle holds at level 2.

The principle holds at all levels for rings of dimension ≤ 3.

Finiteness of associated primes of certain 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}.