Generalized Hilbert-Kunz function in graded dimension two

TL;DR

This paper establishes a formula for the generalized Hilbert-Kunz function in two-dimensional graded normal domains, showing it has a quadratic form with rational multiplicity and analyzing its behavior over different characteristics.

Contribution

It proves the quadratic form of the generalized Hilbert-Kunz function and the rationality of the multiplicity, including its limit as characteristic varies.

Findings

The generalized Hilbert-Kunz function has the form $gHK(M,q)=e_{gHK}(M)q^{2}+ ext{bounded function}$.

The generalized Hilbert-Kunz multiplicity $e_{gHK}(M)$ is rational.

The limit of $e_{gHK}^{R_p}(M_p)$ exists and is rational as $p o o + abla$.

Abstract

We prove that the generalized Hilbert-Kunz function of a graded module $M$ over a two-dimensional standard graded normal $K$-domain over an algebraically closed field $K$ of prime characteristic $p$ has the form $gHK(M,q)=e_{gHK}(M)q^{2}+\gamma(q)$, with rational generalized Hilbert-Kunz multiplicity $e_{gHK}(M)$ and a bounded function $\gamma(q)$. Moreover we prove that if $R$ is a $\mathbb{Z}$-algebra, the limit for $p\rightarrow+\infty$ of the generalized Hilbert-Kunz multiplicity $e_{gHK}^{R_p}(M_p)$ over the fibers $R_p$ exists and it is a rational number.