Graded Betti numbers of good filtrations

The asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field has recently been reviewed. We extend quasi polynomial behavior of graded Betti numbers of powers of homogenous ideals to Z-graded algebra over Notherian local ring. Furthermore our main result treats the Betti table of filtrations which is finite or integral over the Rees algebra.