Cofiniteness of local cohomology modules for ideals of small dimension

Let R be a commutative Noetherian ring and let M be a non-zero finitely generated R-module. Let 1 be an ideal of R and t a non-negative integer such that dim Supp H(1)(t) (M) <= 1 for all i < t. It is shown that the R-modules H(1)(0) (M), H(1)(t)(M),...,H(1)(t-1)(M) are 1-cofinite and the R-module Hom(R)(R/1, H(1)(t)(M)) is finitely generated. This immediately implies that if I has dimension one (i.e., dim R/1 = 1), then H(1)(t)(M) is 1-cofinite for all i >= 0. This is a generalization of the main results of Delfino and Marley [D. Delfino, T Marley, Cofinite modules and local cohomology, J. Pure Appl. Algebra 121 (1997) 45-52] and Yoshida [K.I. Yoshida, Cofiniteness of local cohomology modules for ideals of dimension one, Nagoya Math. J. 147 (1997) 179-191] for an arbitrary Noetherian ring R. Also, we prove that if R is local and dim Supp H(1)(t)(M) <= 2 for all i < t, then the R-modules Ext(R)(J)(R/1, H(1)(t)(M)) and Hom(R) (R/1, H(1)(t)(M)) are weakly Laskerian for all i < t and all j >= 0. As a consequence, it follows that the set of associated primes of H(1)(t)(M) is finite for all i >= 0, whenever dim R/1 <= 2. (c) 2009 Elsevier Inc. All rights reserved.