The Index of Reducibility of Parameter Ideals and Mostly Zero Finite Local Cohomology

We prove that if M is a finitely-generated module of dimension d with finite local cohomologies over a Noetherian local ring, and if the ith local cohomology module of M is zero unless i = d, i = 0, and i = r for some r strictly between 0 and d, then there is an integer l such that every parameter ideal for M contained in the lth power of the maximal ideal has the same index of reducibility on M. This theorem generalizes work of the second author (math.AC/0408143) and is closely related to work of S. Goto, H. Sakurai (math.AC/0306148), and N. Suzuki.