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 Φ = (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 quasipolynomial 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 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 ). Introduction. Let( A, M ) be a local noetherian ring, I an ideal of A and M a finite A-module. We consider the graded homogeneous ring G I ( A) = n∈N I n /I n+1 and the finite graded G I ( A)-module G I (M) = n∈N I n M/I n+1 M associated with the pair (I, M). It is well known that for each integer i, the i-th Betti number and the i-th Bass number of the modules I n M/I n+1 M and M/I n M are polynomial functions in n with degree at most the Krull dimension of G I (M) ⊗ A/M minus 1 or at most the Krull dimension of G I (M) ⊗ A/Mrespectively. Moreover, a classical result asserts, that the projective and injective dimension, as well as the depth of the module M/I n M stabilize for large n. This paper generalizes the above results to the following situation: let f = (I n ) be a noetherian filtration of A that is a decreasing sequence of ideals of A which satisfies I 0 = A and I n I m ֤ I n+m for all n, m such that its Rees ring R( f ) =