Asymptotic properties of filtrations on a module

Abstract

Let M be a module over a commutative ring A. For any filtrat ions F = (M_n) and G = (N_n) on M, we define the numbers [abar]_F(G). Their existence in [bbar]_F(G). and their symetrie properties are proved. The relationship between [abar]_F(G), [bbar]_F(G). the multiplicities of F and G with respect to an integer s defined by Bishop, is established. The numbers [abar]_F(G).and [bbar]_F(G). are studied in particular case when F or G is a weakly good filtration. It is proved that, if F is weakly f-good and if G is weakly g-good with f=(I_n)and g=(J_n)then we have

Several examples to show that each relation in (*) may be a strict inequality when f is not AP, even in a noetherian module, are given.