A splitting theorem for local cohomology and its applications

Let R be a commutative Noetherian ring and M a finitely generated R-module. We show in this paper that, for an integer t, if the local cohomology module H(a)(i)(M) with respect to an ideal a is finitely generated for all i < t, then H(a)(i)(M/xM) congruent to H(a)(i)(M) circle plus H(a)(i+1) (M) for all a-filter regular elements x contained in a enough large power of a and all i < t - 1. As consequences we obtain generalizations, by very short proofs, of the main results of M. Brodmann and A.L. Faghani [M. Brodmann, A.L Faghani, A finiteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc. 128 (2000) 2851-2853] and of HI. Truong and the first author [N.T. Cuong, H.L. Truong, Asymptotic behavior of parameter ideals in generalized Cohen-Macaulay module, J. Algebra 320 (2008) 158-168]. (c) 2010 Elsevier Inc. All rights reserved.