Filtrations, rees rings, and ideal transforms

Abstract

Let I be an ideal, and let f = { K n | n ≥ 0 } be a filtration of the Noetherian ring R, such that I n ⊆ K n for all n ≥ 0. We study when the Rees ring R (f) is either finite or integral over the Rees ring R ( I), for two types of filtrations f which have recently drawn interest. If I and J are ideals in R, and if m( n) is the least power of J such that ( I n : J m( n) + 1 ), we show that the function m( n) is eventually non-decreasing. For J regular, we characterize when it is eventually constant.