Algebra Colloquium c  2014 AMSS CAS & SUZHOU UNIV Algebra Colloquium 21 : 3 (2014) 517–520 DOI: 10.1142/S1005386714000455 On the Finiteness Dimension of Local Cohomology Modules Hero Saremi Department of Mathematics, Sanandaj Branch Islamic Azad University, Sanandaj, Iran E-mail: herosaremi@yahoo.com Amir Mafi † Department of Mathematics, University of Kurdistan Pasdaran St., P.O. Box 416, Sanandaj, Iran E-mail: a mafi@ipm.ir Received 9 March 2011 Revised 25 April 2012 Communicated by Zhongming Tang Abstract. Let R be a commutative Noetherian ring, a an ideal of R, and M a non-zero finitely generated R-module. Let t be a non-negative integer. In this paper, it is shown that dim Supp H i a (M) ≤ 1 for all i<t if and only if there exists an ideal b of R such that dim R/b ≤ 1 and H i a (M) ∼ = H i b (M) for all i<t. Moreover, we prove that dimSupp H i a (M) ≤ dim M - i for all i. 2010 Mathematics Subject Classification: 13D45, 13E99 Keywords: local cohomology modules, cofinite modules, finiteness dimension 1 Introduction Throughout this paper, we assume that R is a commutative Noetherian ring with non-zero identity, a an ideal of R, and M a non-zero finitely generated R-module. For a non-negative integer i, the i-th local cohomology module of M with respect to a is denoted by H i a (M ). The reader should consult [4] for the definition of local cohomology and its basic properties. An R-module N is called a-cofinite if Supp(N ) ⊆ V (a) and Ext i R (R/a,N ) is finitely generated for all i. This notion was introduced by Hartshorne in [9]. The reader is referred to [3, 5–7, 10–12, 14, 16, 18–20] for more information about cofiniteness with respect to an ideal. We denote by dim Supp H i a (M ) the maximum † Corresponding author.