Hilbert-Kunz density function and Hilbert-Kunz multiplicity

Abstract

For a pair $(M, I)$, where $M$ is a finitely generated graded module over a standard graded ring $R$ of dimension $d$, and $I$ is a graded ideal with $\ell (R/I) < \infty$, we introduce a new invariant $HKd(M, I)$ called the Hilbert-Kunz density function. We relate this to the Hilbert-Kunz multiplicity $e_{HK}(M, I)$ by an integral formula. We prove that the Hilbert-Kunz density function satisfies a multiplicative formula for a Segre product of rings. This gives a formula for $e_{HK}$ of the Segre product of rings in terms of the HKd of the rings involved. As a corollary, $e_{HK}$ of the Segre product of any finite number of projective curves is a rational number.