Abstract. The notion of weakly Laskerian modules was introduced recently by the authors. Let R be a commutative Noetherian ring with identity, a an ideal of R, and M a weakly Laskerian module. It is shown that if a is principal, then the set of associated primes of the local cohomology module H i a (M ) is finite for all i ≥ 0. We also prove that when R is local, then Ass R (H
Abstract. Let a be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. Let t be a natural integer. It is shown that there is a finite subset X of Spec R, such that Ass R (H t a (M )) is contained in X union with the union of the sets Ass R (Ext j R (R/a, H i a (M ))), where 0 ≤ i < t and 0 ≤ j ≤ t 2 + 1. As an immediate consequence, we deduce that the first non-a-cofinite local cohomology module of M with respect to a has only finitely many associated prime ideals.
Abstract. Let a be an ideal of a commutative Noetherian ring R. For finitely generated R-modules M and N with Supp N ⊆ Supp M , it is shown that cd(a, N) ≤ cd(a, M). Let N be a finitely generated module over a local ring (R, m) such that MinRN = AsshRN . Using the above result and the notion of connectedness dimension, it is proved that cd(a, N) ≥ dim N − c(N/aN ) − 1. Here c(N ) denotes the connectedness dimension of the topological space Supp N . Finally, as a consequence of this inequality, two previously known generalizations of Faltings' connectedness theorem are improved.
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