Journal of the
Korean Mathematical Society JKMS

For a given ideal $I$ of a Noetherian ring $R$ and an arbitrary integer $k\geq{-1}$, 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 $\rm{f}$-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.

Keywords: $k$-regular $M$-sequences, $k$-depth, $k$-ht, local cohomology modules, $k$-Cohen Macaulay modules, ${\rm{f}}$-modules, $k$-modules, Koszul complexes

MSC numbers: 13D45, 13C15