Growth of Hilbert coefficients of Syzygy modules

Let (A,m) be a local complete intersection ring of dimension d and let I be an m-primary ideal. Let M be a maximal Cohen-Macaulay A-module. For i=0,1,⋯,d, let eiI(M) denote the ith Hilbert-coefficient of M with respect to I. We prove that for i=0,1,2, the function j↦eiI(SyzjA(M)) is of quasi-polynomial type with period 2. Let GI(M) be the associated graded module of M with respect to I. If GI(A) is Cohen-Macaulay and dim⁡A≤2 we also prove that the functions j↦depthGI(Syz2j+iA(M)) are eventually constant for i=0,1. Let ξI(M)=liml→∞⁡depthGIl(M). Finally we prove that if dim⁡A=2 and GI(A) is Cohen-Macaulay then the functions j↦ξI(Syz2j+iA(M)) are eventually constant for i=0,1.