Let R be a commutative Noetherian ring and let n be a non-negative integer. In this article, by using the theory of Gorenstein dimensions, it is shown that whenever R is a homomorphic image of a Noetherian Gorenstein ring, then the in-M )) ≥ n for all t ∈ N 0 } and inf{λ bRp aRp (M p )| p ∈ Spec R and dim R/p ≥ n} are equal, for every finitely generated R-module M and for every ideals a, b of R with b ⊆ a. This generalizes the Faltings' Annihilator Theorem [G. Faltings,Über die Annulatoren lokaler Kohomologiegruppen, Arch. Math. 30 (1978) 473-476].

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 submoduleThis 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 ∈ N 0 | dim Supp(b t H i a (M )) n for all t ∈ N 0 } and inf{depth M p + ht(a + p)/p | p ∈ Spec(R)\ V(b), dim R/(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 H i a (M ) is not minimax (resp. weakly laskerian).

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Let [Formula: see text] be an ideal of a Noetherian local ring [Formula: see text] with [Formula: see text] and [Formula: see text] be a positive integer. In this paper, it is shown that the top local cohomology module [Formula: see text] (equivalently, its Matlis dual [Formula: see text]) can be written as a direct sum of [Formula: see text] indecomposable summands if and only if the endomorphism ring [Formula: see text] can be written as a direct product of [Formula: see text] local endomorphism rings if and only if the set of minimal primes [Formula: see text] of [Formula: see text] with [Formula: see text] can be written as disjoint union of [Formula: see text] non-empty subsets [Formula: see text] such that for all distinct [Formula: see text] and all [Formula: see text] and all [Formula: see text], we have [Formula: see text]. This generalizes Theorem 3.6 of Hochster and Huneke [Contemp. Math. 159 (1994) 197–208].