Filtrations on a ring and asymptotic deviations

Abstract

The Asymptotic Theory of ideals originated with the investigation in noetherian ring A of the Samuel numbers ⊢_I(J) and [wbar]_I(J) associ-ated to each pair (I, J) of nonnilpotent ideals having the same radical where , the limit being reached from below and . The number [wbar]_I(J) is defined in a symetric situation. As an answer of a question raised by Samuel, Nagata has shown that the set of deviations is bounded. Here we extend these numbers to pairs (f,g), where f = (I _n) and g= (J _n) are filtrations on A, as follows: , where f(^r) is filtration (I_nr ) . We prove in particular that if f and g are separated, nonnilpotent, strongly AP filtrations and such then the deviation sequence is bounded in R₊. A similar study is done concerning the sequence and b_f (g) = ∞ if the last set is empty.