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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 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.