Generalized local cohomology modules and homological Gorenstein dimensions

The finiteness dimension of local cohomology modules and its dual notion

“…This implies that q a (M, N) ≤ t. iii) is clear by Lemma 3.6. iv) For any finitely generate R-module C, one has H i a (M, C) = 0 for all i > pd M + ara(a), see e.g. [DH,Theorem 2.5]. Hence by using decreasing induction on q a (M, L) ≤ i ≤ pd M + ara(a) + 1, one can prove the claim by slight modification of the proof of Theorem 2.1. v) this can be deduced by an argument similar to that used in the proof of Corollary 2.2 iii).…”

“…iv) [DH,Lemma 2.11] implies that H i a (C, L) ∼ = Ext i R (C, L) for all a-torsion R-modules C and all i. So, the exact sequence 0 N) , L).…”

“…Proof. We use induction on n := dim N. By [DH,Theorem 2.5], it follows that H i a (M, L) = 0 for all finitely generated R-modules L and all i > pd M + dim L. So, since the functor H i a (M, •) commutes with direct limits, it turns out that H i a (M, N) = 0 for all i > pd M + dim N. Thus the claim clearly holds for n = 0. Now, assume that n > 0 and that the claim holds for n − 1.…”

“…By induction on n, it is enough to show the claim for the case n = 1. So, let b = a + (x) for some x ∈ R. By [DH,Lemma 3.1], there is the following long exact sequence of generalized local cohomology modules…”