A note on Hilbert-Kunz multiplicity

TL;DR

This paper provides new bounds on Hilbert-Kunz multiplicity for invariant rings, extends reciprocity formulas, and explores their implications in complete intersection rings with isolated singularities.

Contribution

It introduces a new bound on Hilbert-Kunz multiplicity using Noether's bound and extends reciprocity formulas to broader classes of rings and ideals.

Findings

New bound on Hilbert-Kunz multiplicity for invariant rings

Extension of reciprocity formulas to complete intersection rings

Equivalence of reciprocity formula validity with finite projective dimension

Abstract

In this note we first give a new bound on $e_{HK}(\sim)$ the Hilbert-Kunz multiplicity of invariant rings, by the help of the Noether's bound. Then, we simplify, extend and present applications of the reciprocity formulae due to L. Smith. He proved the formula over polynomial rings and his result is tight in the following sense: Over complete intersection rings with isolated singularity we show that the reciprocity formulae "$e_{HK}(R/I)+ e_{HK}(R/J) = e_{HK}(R/\underline{f})$" is equivalent with $\pd(I)<\infty$ when $I$ is an $\fm$-primary unmixed ideal linked to $J$ along with a regular sequence $\underline{f}$.