On Faltings' local-global principle of generalized local cohomology modules

Abstract

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M, N$ finitely generated $R$-modules. Let $0 ≤ n \in \mathbf Z$. This note shows that the least integer $i$ such that $\dim \mathrm{Supp}(H^i_I(M, N)/K) ≥ n$ for any finitely generated submodule $K$ of $H^i_I(M, N)$ equal to the number $\inf\{f_{I_{\frak p}}(M_{\frak p},N_{\frak p})|{\frak p}\in \mathrm{Supp}(N/I_MN), \dim R/{\frak p} ≥ n\}$, where $f_{I_{\frak p}}(M_{\frak p},N_{\frak p})$ is the least integer $i$ such that $H^i_{I_{\frak p}}(M_{{\frak p}},N_{{\frak p}})$ is not finitely generated, and $I_M = \mathrm{ann}(M/IM)$. This extends the main result of Asadollahi-Naghipour [1] and Mehrvarz-Naghipour-Sedghi [8] for generalized local cohomology modules by a short proof.