Hilbert-Kunz Density function of tensor product and Fourier transformation

Abstract

For a standard graded ring $R$ of dimension $\ge 2$ 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.