Let \(R\) be a commutative Noetherian ring with identity and \(I\) an ideal of \(R\). The author of this paper establishes a generalization of Faltings’ local-global principle for the generalized local cohomology modules \[ \text{H}_{I}^i(M,N): =\underset{n}{\varinjlim} \text{Ext}^i_R(M/{I}^nM,N); i\in \mathbb{N}_0. \]
Let \(M,N\) be two finitely generated \(R\)-modules and \(n\in \mathbb{N}_0.\) Recall that the finiteness dimension of \(M\) and \(N\) with respect to \(I\) is defined as \[ f_I(M,N):=\inf \{i\in \mathbb{N}_0|\text{H}_I^i(M,N) \text{ is not finitely generated} \}. \] Set \(I_M:=\text{Ann}_R(M/IM)\) and for each non-negative integer \(i\), let \(\Omega_i(M,N)\) denote the set of all finitely generated submodules of \(\text{H}_{I}^i(M,N)\). The main result of this paper asserts that the two integers \[ \inf\{i\in \mathbb{N}_0|\dim_R(\mathrm{Supp}_R(\text{H}_{I}^i(M,N)/K))\geq n \;\text{for\;all} \;K\in \Omega_i(M,N) \} \] and
\[ \inf \{f_{I_{\mathfrak p}}(M_{\mathfrak p},M_{\mathfrak p})|{\mathfrak p}\in \mathrm{Supp}_R(N/I_MN) \text{ and } \dim R/{\mathfrak p}\geq n\} \] are the same.