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

Abstract

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 f is a good filtration or a strongly A P 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 ).