GENERALIZED HILBERT-KUNZ FUNCTION IN GRADED DIMENSION 2

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(g) (H K)(M)q(2) gamma(q), with rational generalized Hilbert-Kunz multiplicity e(g H K) (M) and a bounded function gamma(q). Moreover, we prove that if R is a Z-algebra, the limit for p -> infinity of the generalized Hilbert-Kunz multiplicity eg H K-Rp(M-P) over the fibers R-p exists, and it is a rational number.