The Hilbert–Kunz Function in Graded Dimension Two

Abstract

Let R denote a two-dimensional normal standard-graded K-domain over the algebraic closure K of a finite field of characteristic p, and let I ⊂ R denote a homogeneous R ₊-primary ideal. We prove that the Hilbert–Kunz function of I has the form ϕ (q) = e _HK (I)q ² + γ(q) with rational Hilbert–Kunz multiplicity e _HK (I) and an eventually periodic function γ(q).

Key Words:

• Hilbert–Kunz function
• Hilbert–Kunz multiplicity

Mathematics Subject Classification (2000):

• 13A35
• 13D40
• 14G15
• 14H60

Notes

Communicated by I. Swanson.