Asymptotic Hilbert–Kunz multiplicity

Abstract

For a pair (M,I), where M is finitely generated graded module over a standard graded ring R of dimension d≥2, and I is a graded ideal with ℓ(R/I)<∞ and generated by elements of the same degree, we prove that limq→∞⁡e1(M,I[q])/qd exists, where e1(M,I[q]) denotes the first coefficient of the Hilbert–Samuel polynomial of (M,I[q]).We use this to get an expression for limk→∞⁡[eHK(M,Ik)−e0(M,Ik)/d!]/kd−1, where eHK denotes the Hilbert–Kunz multiplicity. In particular, if dim⁡ M=d then we deduce that the difference eHK(M,Ik)−e(M,Ik)/d! grows at least as a fixed positive multiple of kd−1 as k→∞.This is proved using ‘renormalized’ HK density functions.