GENERALIZED HILBERT–KUNZ FUNCTION IN GRADED DIMENSION 2

HOLGER BRENNER, ALESSIO CAMINATA
2016 Nagoya mathematical journal  
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 $gHK(M,q)=e_{gHK}(M)q^{2}+\unicode[STIX]{x1D6FE}(q)$ , with rational generalized Hilbert–Kunz multiplicity $e_{gHK}(M)$ and a bounded function $\unicode[STIX]{x1D6FE}(q)$ . Moreover, we prove that if $R$ is a $\mathbb{Z}$ -algebra, the limit for $p\rightarrow +\infty$ of the
more » ... $ of the generalized Hilbert–Kunz multiplicity $e_{gHK}^{R_{p}}(M_{p})$ over the fibers $R_{p}$ exists, and it is a rational number.
doi:10.1017/nmj.2016.66 fatcat:7zevpccvgvapfbs4ivvi2wizdu