Hilbert-Kunz functions over rings regular in codimension one

TL;DR

This paper investigates the shape and properties of Hilbert-Kunz functions over rings regular in codimension one, extending previous results and exploring conditions for the vanishing of the second coefficient in Cohen-Macaulay rings.

Contribution

It extends existing results on Hilbert-Kunz functions to more general rings and introduces an analysis of the second coefficient and additive errors in modules.

Findings

Extended the shape characterization of Hilbert-Kunz functions.

Identified conditions for the vanishing of the second coefficient.

Provided estimates for additive errors in modules.

Abstract

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 (Math. Res. Lett. 11 (2004), no. 4, 539-546) about the shape of the Hilbert-Kunz functions, and a theorem of the second author (J. Algebra 304 (2006), no. 1, 487-499) for rings with weaker conditions. In this paper, for a Cohen-Macaulay ring, we also explores an equivalence condition under which the second coefficient vanishes whenever the Hilbert-Kunz function of the ring is considered with respect to an ideal primary to the maximal ideal and 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.