Faltings’ local–global principle and annihilator theorem for the finiteness dimensions

Let R be a commutative Noetherian ring, M be a finitely generated R-module and n be a non-negative integer. In this article, it is shown that for a positive integer t, there is a finitely generated submodule Ni of Hai(M) such that dimSupp Hai(M)/Ni<n for all i < t if and only if there is a finitely generated submodule Ni,p of HaRpi(Mp) such that dimSupp HaRpi(Mp)/Ni,p<n for all i < t and all p ∈ Spec(R). This generalizes Faltings’ Local–global Principle for the finiteness of local cohomology modules (Faltings’ in Math. Ann. 255:45–56, 1981). Also, it is shown that whenever R is a homomorphic image of a Gorenstein local ring, then the invariants inf{i∈N0|dimSupp (btHai(M))⩾n for all t∈N0} and inf{depth Mp+ht(a+p)/p|p∈Spec(R)∖V(b),dimR/(a+p)⩾n} are equal, for every finitely generated R-module M and for all ideals a,b of R with b⊆a. As a consequence, we determine the least integer i where the local cohomology module Hai(M) is not minimax (resp. weakly Laskerian).