Quasi-polynomials associated with Bass and Betti numbers of filtered modules

It is shown that if M is a finite module on a local noetherian ring A which is filtered by an f-good filtration Phi = (M-n) where f is a noetherian filtration on A, then the i-th Betti and the i-th Bass numbers of the modules (M-n) and (M/M-n) define quasi-polynomial functions whose period does not depend on i but only of the Rees ring of f. It is proved that the projective and injective dimension of the modules M/M-n are perodic for large n. In the particular case where fis a good filtration or a strongly AP filtration it is shown that the projective and injective dimension as well as the depth stabilize. As an application, using a result proved by Brodmann, we give an upper bound of the analytic spread of f = (I-n) in terms of the limes inferior of depth(A/I-n).