Hilbert-Kunz function and Hilbert-Kunz multiplicity of some ideals of the Rees algebra

TL;DR

This paper investigates the Hilbert-Kunz function and multiplicity of certain ideals in Rees algebras, showing they are quasi-polynomials or piecewise polynomials in large characteristic, with explicit formulas in specific cases.

Contribution

It establishes the quasi-polynomial nature of the Hilbert-Kunz function for ideals in Rees algebras and provides explicit descriptions for parameter ideals in Cohen-Macaulay rings.

Findings

Hilbert-Kunz function is a quasi-polynomial in large e.

Explicit formulas for Hilbert-Samuel functions of Frobenius powers.

Generalized Hilbert-Kunz function is piecewise polynomial for parameter ideals.

Abstract

We prove that the Hilbert-Kunz function of the ideal $(I,It)$ of the Rees algebra $\mathcal{R}(I)$, where $I$ is an $\mathfrak{m}$-primary ideal of a $1$-dimensional local ring $(R,\mathfrak{m})$, is a quasi-polynomial in $e$, for large $e.$ For $s \in \mathbb{N}$, we calculate the Hilbert-Samuel function of the $R$-module $I^{[s]}$ and obtain an explicit description of the generalized Hilbert-Kunz function of the ideal $(I,It)\mathcal{R}(I)$ when $I$ is a parameter ideal in a Cohen-Macaulay local ring of dimension $d \geq 2$, proving that the generalized Hilbert-Kunz function is a piecewise polynomial in this case.