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

Abstract

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 such that for all i < t if and only if there is a finitely generated submodule of such that 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 and are equal, for every finitely generated R-module M and for all ideals of R with . As a consequence, we determine the least integer i where the local cohomology module is not minimax (resp. weakly Laskerian).