Let R be a commutative Noetherian ring, a be an ideal of R and M be a finitely generated R-module. Melkersson and Schenzel asked whether the set AssRExti R(R/a 6 j , M) 7 becomes stable for a fixed integer i and sufficiently large j . This paper is concerned with this question. In fact, we prove that if s ≥ 0 and n ≥ 0 such that dim(SuppRHi a 8 (M)) ≤ s 9 for all i with i < n, then (i) the set
j>0 SuppRExti R
R/a j , M≥s 10 is finite for all i with i < n, and (ii) the set
j>0 AssRExti R
R/a j , M≥s 11 is finite for all i with i ≤ n, 12 where for a subset T of Spec(R), we set (T )≥s := {p ∈ T | dim(R/p) ≥ s}. Also, among other things, we show that if n ≥ 0, R is semi-local and SuppRHi a 13 (M) is finite for all i with i < n, then j>0 AssRExti R(R/a 14 j , M) is finite for all i with i ≤ n.

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