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

A result of Monsky states that the Hilbert–Kunz function of a one-dimensional local ring of prime characteristic has a term \phi that is eventually periodic. For example, in the case of a power series ring in one variable over a prime-characteristic field, \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], \phi is immediately periodic with period 2. We show that, for every positive integer \pi, there exists a ring for which \phi is immediately periodic with period \pi. [ABSTRACT FROM AUTHOR]