Hilbert-Kunz Density function of tensor product and Fourier transformation

TL;DR

This paper investigates the Hilbert-Kunz density function for tensor products of graded rings, proving it is the convolution of individual functions, and explores its Fourier transform, with applications to multiplicities and projective curves.

Contribution

It proves the Hilbert-Kunz density function of a tensor product is the convolution of the factors' functions and computes its Fourier transform for projective curves.

Findings

HK density function of tensor product is convolution of factors' functions

HK multiplicity of tensor product equals product of individual multiplicities

Fourier transform of HK density function for a projective curve is computed

Abstract

For a standard graded ring $R$ of dimension $\geq 2$ over a perfect field of characteristic $p>0$ and a homogeneous ideal $I$ of finite colength, the HK density function of $R$ with respect to $I$ is a compactly supported continuous function $f_{R, I}:[0, \infty)\longto [0, \infty)$, whose integration yields the \mbox{HK} multiplicity $e_{HK}(R, I)$. Here we answer a question of V. Trivedi about the Hilbert-Kunz density function of the tensor product of standard graded rings and show that it is the convolution of the Hilbert-Kunz density function of the factor rings. Using Fourier transform, as a corollary we get \mbox{HK} multiplicity of the tensor product of rings is product of the HK multiplicity of the factor rings. We compute the Fourier transform of the \mbox{HK} density function of a projective curve.