The aim of this manuscript is to discuss the Hilbert–Kunz functions over an excellent local ring regular in codimension one. We study the shape of the Hilbert–Kunz functions of modules and discuss the properties of the coefficient of the second highest term in the function. Our results extend Huneke, McDermott, and Monsky's result about the shape of the Hilbert–Kunz functions in  and a theorem of the second author in  for rings with weaker conditions. In this article, for a Cohen–Macaulay ring, we also explore an equivalent condition under which the second coefficient vanishes whenever the Hilbert–Kunz function of the ring is considered with respect to an 𝔪-primary ideal of finite projective dimension. We introduce an additive error of the Hilbert–Kunz functions of modules on a short exact sequence and give an estimate of such error.
- Hilbert–Kunz function
- Rational equivalence