The Frobenius-Poincare Function and Hilbert-Kunz Multiplicity

Abstract

For a triple (M,R, I), where I is a finite co-length homogeneous ideal in an N-graded noetherian ring R and M is a finitely generated Z-graded R-module, we prove that the following limit exists for any complex number y: lim n→∞ ( 1 p^n )dim(M) sum λ([ M I^[p^n]M ]j)e−iyj/pn . We call the limiting function in the complex variable ‘y′, the Frobenius-Poincar´e function of the triple (M,R, I) and develop a theory of Frobenius-Poincar´e functions. We prove that the Frobenius-Poincar´e function is entire and its value at the origin is the corresponding Hilbert- Kunz multiplicity. We show that the Frobenius-Poincar´e function is the Fourier transform of the Hilbert-Kunz density function when R is standard graded, and provide applications to the theory of Hilbert-Kunz density functions. Our examples of Frobenius-Poincar´e functions motivate questions on the structure of Frobenius-Poincar´e functions in general.