The index of reducibility of parameter ideals in low dimension

In this paper we present results concerning the following question: if M is a finitely-generated module with finite local cohomologies over a Noetherian local ring , does there exist an integer ℓ such that every parameter ideal for M contained in has the same index of reducibility? We show that the answer is yes if dimM=1 or if dimM=2 and depthM>0. This research is closely related to work of Goto–Suzuki and Goto–Sakurai; Goto–Sakurai have supplied an answer of yes in case M is Buchsbaum.