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Let A be a commutative Noetherian ring (with non-zero identity). The Cousin complex C(A) for A is described in [19, Section 2]: it is a complex of A-modules and A-homomorphismswith the property that, for each n ∈ N0 (we use N0 to denote the set of non-negative integers),Cohen–Macaulay rings can be characterized in terms of the Cousin complex: A is a Cohen–Macaulay ring if and only if C(A) is exact [19, (4.7)]. Also, the Cousin complex provides a natural minimal injective resolution for a Gorenstein ring (see [19,(5.4)]).

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Let R be a commutative Noetherian ring. Let a and b be ideals of R and let N be a finitely generated R-module. We introduce a generalization of the b-finiteness dimension of f b a (N ) relative to a in the context of generalized local cohomology modules aswhere M is an R-module. We also show that f b a (N ) ≤ f b a (M, N ) for any R-module M . This yields a new version of the Local-Global Principle for annihilation of local cohomology modules. Moreover, we obtain a generalization of the Faltings Lemma.

Let [Formula: see text] be an ideal of a commutative Noetherian ring R, and let M and N be finitely generated R-modules. Let [Formula: see text] be the [Formula: see text]-finiteness dimension of N. In this paper, among other things, we show that for each [Formula: see text], (i) the set of associated prime ideals of generalized local cohomology module [Formula: see text] is finite, and (ii) [Formula: see text] is [Formula: see text]-cofinite if and only if [Formula: see text] is so. Moreover, we show that whenever [Formula: see text] is a principal ideal, then [Formula: see text] is [Formula: see text]-cofinite for all n.