A Hilbert–Kunz function with a periodic term that has a given period

Abstract

A result of Monsky states that the Hilbert–Kunz function of a one-dimensional local ring of prime characteristic has a term ϕ \phi that is eventually periodic. For example, in the case of a power series ring in one variable over a prime-characteristic field, ϕ \phi is the zero function and is therefore immediately periodic with period 1. In additional examples produced by Kunz [Amer. J. Math. 91 (1969), pp. 772–784] and Monsky [Math. Ann. 263 (1983), pp. 43–49], ϕ \phi is immediately periodic with period 2. We show that, for every positive integer π \pi , there exists a ring for which ϕ \phi is immediately periodic with period π \pi .