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 ∈ Z. This note shows that the least integer i such that dim 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_$\frak{p}$)|$\frak{p}$ ∈ 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 = 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.