Cofiniteness of Local Cohomology Modules over Homomorphic Image of Cohen-Macaulay Rings

Let (R, 𝔪m) be a Noetherian local ring, M a non-zero finitely generated R-module, and let I be an ideal of R. In this paper, we establish some new properties of local cohomology modules HIi(M), i ≥ 0. In particular, we show that if R is catenary, M an equidimensional R-module of dimension d, and x1, x2, … , xt is an I-filter regular sequence on M, then (0:HId−j(M〈x1,x2,…,xi−1〉M)xi) is I-cofinite for all i= 1 , 2 , … , t and all i ≤ j ≤ t if and only if HId−j(M〈x1,x2,…,xi−1〉M) is I-cofinite for all i= 1 , 2 , … , t and all i ≤ j ≤ t. Also we study the cofiniteness of local cohomology modules over homomorphic image of Cohen-Macaulay rings and we show that HIW(I,M)(M)IHIW(I,M)(M) has finite support, where W(I,M):=Max{i:HIi(M)is not weakly Laskerian}. © 2018, Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.