Rigid Resolution of a Finitely Generated Module Over a Regular Local Ring

Let M be a finitely generated module over a regular local ring (R, n), and let M = {Mi} be an n−stable filtration on M. As a consequence of a recent result by Rossi and Sharifan in [14], we prove that if the i-th Betti numbers of M and (grM({M}) coincide with each other, then for each j geq i the j-th Betti numbers of them are the same and Syzi(M) is a Koszul module provided that Syzi(grM(M) is componentwise linear.