Results on Hilbert coefficients of a Cohen-Macaulay module

Let (R,m) be a commutative Noetherian local ring, M a finitely generated R-module of dimension d, and let I be an ideal of definition for M. In this paper, we extend \cite[Corollary ۱۰(۴)]{P} and also we show that if M is a Cohen-Macaulay R-module and d=۲, then \lambda(\frac{\widetilde{I^nM}}{J\widetilde{I^{n-۱}M}}) does not depend on J for all n\geq ۱, where J is a minimal reduction of I.

Keywords:

Cohen-Macaulay rings , Hilbert series , Hilbert function