Quasi-pure resolutions and some lower bounds of Hilbert coefficients of Cohen-Macaulay modules

Abstract

Let (A,m) be a Noetherian local ring and let M be a finitely generated Cohen Macaulay A module. Let G(A)=⨁n≥0mn/mn+1 be the associated graded ring of A and G(M)=⨁n≥0mnM/mn+1M be the associated graded module of M. If A is regular and if G(M) has a quasi-pure resolution then we show that G(M) is Cohen-Macaulay. If A is Gorenstein and G(A) is Cohen-Macaulay and if M has finite projective dimension then we give lower bounds on e0(M) and e1(M). Finally let A=Q/(f1,…,fc) be a strict complete intersection with ord(fi)=s for all i, where Q is a regular local ring. Let M be an Cohen-Macaulay module with complexity cxA(M)=r<c. We give lower bounds on e0(M) and e1(M).