Abstract. For a given ideal I of a Noetherian ring R and an arbitrary integer k ≥ −1, we apply the concept of k-regular sequences and the notion of k-depth to give some results on modules called k-Cohen Macaulay modules, which in local case, is exactly the k-modules (as a generalization of f-modules). Meanwhile, we give an expression of local cohomology with respect to any k-regular sequence in I, in a particular case. We prove that the dimension of homology modules of the Koszul complex with respect to any k-regular sequence is at most k. Therefore homology modules of the Koszul complex with respect to any filter regular sequence has finite length.